Just Dollars and Cents: Making Change via a Virtual Money Manipulative

Abstract

Providing high-quality services to students with disabilities in rural settings can be challenging, as rural schools often lack resources both in and out of the classroom. One potential option to provide high-quality mathematics instruction to students with disabilities in rural areas is using virtual manipulatives and online instruction. In this study, a single-case research design was used to examine the progress of three high school students with intellectual and developmental disabilities (IDDs) as they used a virtual money manipulative within an online learning environment to solve making change problems. The researchers found a functional relation between the intervention package (i.e., a virtual money manipulative, modeling, system of least prompts) and the dependent variable of accuracy in solving making change problems across three students. In addition, all three students demonstrated an increase in independence during each session and maintained their accuracy and independence after the intervention concluded.

Keywords

rural special education mathematics virtual instruction special education

All students should have access to high-quality instruction; however, researchers have noted the challenges of providing high-quality services to students with disabilities in rural schools (Brownell et al., 2005; Collins & Ludlow, 2018; McKissick et al., 2017). Rural schools often face resource challenges—in and out of the classroom, which include access to classroom supplies and curriculum resources as well as recruiting qualified teachers, maintaining high-quality instructional programs, and offering continued professional development (Hott et al., 2021; Kormos & Wisdom, 2021; Wieczorek & Manard, 2018). For students with intellectual and developmental disabilities (IDDs), who may participate in community-based instruction (CBI), limited access to resources and locations to support CBI in rural communities may necessitate alternative means of providing life skills engagement, such as simulation or other creative options (Collins & Ludlow, 2018; Hopkins & Dymond, 2020).

McKissick et al. (2017) presented computer-assisted instruction (CAI) as a means of supplementing the challenges rural schools face with providing high-quality services and opportunities to students with disabilities. Computer-assisted instruction is considered an evidence-based practice, or, at a minimum, a research-supported practice for students with IDD (Knight et al., 2013; Root et al., 2017). Computer-assisted instruction can be defined two ways: instruction provided just by technology (McKissick et al., 2017) or instruction where computers or similar technology act as a medium through which learning or instruction is provided (McKissick et al., 2018). Effective examples of the narrow definition of CAI include video-based self-operating prompting system (Kellems et al., 2021), whereas the more broadly conceptualized CAI involves web- and mobile-based applications (Snyder & Huber, 2019).

One example of an evidence-based CAI for supporting students with IDD in mathematics is virtual manipulatives (Long et al., 2022). Virtual manipulatives are often digitalized versions of concrete manipulatives available online through websites or via apps (Satsangi & Miller, 2017). Despite the amassing research base regarding the efficacy of virtual manipulatives to support students with IDD in mathematics over the past decade, most of the research focuses on middle school students with IDD and mathematics delivered in face-to-face learning environments (Long et al., 2022).

Researchers found students with IDD acquired, and in many cases maintained, mathematical skills following interventions involving virtual manipulatives, virtual manipulative-based instructional sequences, or intervention packages involving virtual manipulatives (Bouck et al., 2020, 2021; Park, Bouck, & Fisher, 2020; Park, Bouck, & Smith, 2020; Root et al., 2021). For example, Bouck et al. (2020) taught three middle school students with autism spectrum disorder (ASD) and learning disabilities to solve division with remainder problems using virtual Cuisenaire rods. All three students acquired and maintained the skill of division with remainders. In another study, Bouck et al. (2021) used a virtual number line and corrective feedback to teach four middle school students with ASD and IDD to solve addition of integer problems. All four students were successful in acquiring and maintaining the skill at higher levels than baseline.

While one benefit of virtual manipulatives is the digital nature of the tool, currently, there is limited research exploring virtual manipulatives as a part of online instruction. With the COVID-19 pandemic, school-based online instruction became prevalent (Mcelrath, 2020), and Bouck and Long (in press) explored the use of virtual manipulatives (i.e., money manipulative) to support mathematics within online instruction to support high school students with IDD in solving problems involving making change with just coins (e.g., U.S.$1.00–U.S.$0.96). Posed as word problems, Bouck and Long taught three high school students with IDD to solve such problems using a virtual money manipulative, modeling, and the system of least prompts (SLPs). The researchers found a functional relation between the intervention package and student accuracy. Students’ independence increased throughout the intervention phase and students maintained their skills after the intervention ended.

Current Study

Although the aforementioned study (Bouck & Long, in press) demonstrated the efficacy of virtual manipulatives for addressing problem-solving related to making change, the researchers focused only on problems with the change amount less than one dollar (i.e., U.S.$0.65–U.S.$0.23). Realistically, most items cost more, creating a need to explore money manipulatives to target problems involving coins and bills (i.e., problems involving prices over U.S.$1). This study sought to answer the following questions to extend the previous study:

Research Question 1 (RQ1): Are students able to increase their accuracy in solving making change problems with dollars and coins following receipt of an intervention package consisting of a virtual money manipulative, modeling, and the SLP?

Research Question 2 (RQ2): Are students able to increase the percentage of steps of a task analysis completed independently in solving making change problems with dollars and coins following throughout receipt of an intervention package consisting of a virtual money manipulative, modeling, and the SLP?

Research Question 3 (RQ3): Do students maintain their accuracy and percentage of steps of a task analysis completed independently in making change when instruction does not proceed?

Research Question 4 (RQ4): What are student and teacher perceptions of the intervention package?

Method

Participants

Three students with IDD participated in the study. All received special education services in the same self-contained classroom taught by the same special education teacher in a rural high school. These students attended both special education and general education classes at the high school, and also engaged in community-based experiences, such as visiting job sites (e.g., farm, food bank) and athletics (e.g., skiing, bowling). For inclusion in the study, the researchers sought students who had an identification of an IDD, were able to use Zoom, navigate a virtual manipulative website with limited support, identify money values (e.g., recognize a dime and know it is U.S.$0.10), and were reported by their teacher to struggle with making change. All students had parental consent to participate, and they assented to their own participation. One participant, Gina, turned 18 during the time of the study and provided her consent to continue participating in the study upon her 18th birthday.

Ryan

Ryan, a White, 17-year-old, male student in Grade 11 had special education eligibility as intellectual disability (ID) according to his Individualized Education Program (IEP); his secondary eligibility included autism and attention-deficit/hyperactivity disorder (ADHD) under other health impairment. His school records indicate his IQ was 50 on the Wechsler Intelligence Scale for Children-Fifth Edition (WISC-V; Wechsler, 2014). On the researcher-administered KeyMath-3 assessment (Connolly, 2007), Ryan scored a 3 on the Mental Computation subtest (K.5 grade equivalency) and scored a 4 on the Addition and Subtraction subtest (1.9 grade equivalency). These scores were consistent with Ryan’s scores on the Wechsler Individual Achievement Test (4th ed., WIAT-4; Breaux, 2020) as reported in his school records (range K.6-1.5). Based on teacher-collected data reported on his IEP, Ryan demonstrated 80% accuracy with verbal prompting counting coin combinations involving nickels and dimes or nickels and quarters. He had an IEP goal of counting money and writing the value when given dollars and coins. Ryan demonstrated strong technology skills and could independently navigate Zoom and the manipulative app. Ryan previously participated with the researchers on a study examining the money virtual manipulative website and successfully determined change using coins with values up to U.S.$1.00, which gave him familiarity with the manipulative used during the study.

Sam

Sam, a White, 15-year-old, male student, in Grade 10 had special education eligibility as ID (identified with having Down syndrome) according to his IEP. His school records indicated an IQ of 46 on the WISC-V. On the researcher-administered KeyMath-3 assessment (Connolly, 2007), Sam scored a 0 on the mental computation and estimation subtest (< K.0 grade equivalency) and a score of 3 on the Addition and Subtraction subtest (K.8 grade equivalency). Sam’s IEP goals included following a teacher’s model to select three different combinations of coins to represent the cost of an item, and he was actively working on increasingly his fluency in calculations with the use of money. Sam demonstrated relative independence on Zoom and the money manipulative website, with which he had prior exposure but no training as it was used as a generalization probe in a previous study that he participated in.

Gina

Gina, a white, female student, in Grade 12 started the study at the age of 17 but turned 18 near the end. According to her IEP, she was identified with ID, emotional disturbance, speech-language impairment, and orthopedic impairment (i.e., limited use of one hand and arm). Gina’s school records indicated an IQ of 42 on the WISC-V. On the researcher-administered KeyMath-3 assessment, Gina achieved a score of 1 on the Mental Computation and Estimation subtest (less than K.0 grade equivalency) and a score of 3 on the Addition and Subtraction subtest (K.8 grade equivalency). These scores were slightly lower than her reported current math grade equivalency of 1.9 on the STAR assessment, as mentioned in her IEP. Gina had IEP goals pertaining to adding mixed combinations of coins and providing the correct number of mixed coins when instructed. Similar to Sam, Gina had prior exposure to the money manipulative website but not instruction on how to use, as it was used as a generalization probe on a previous study in which she had participated.

Setting

The students’ high school is located in a rural district in a U.S. Western Mountain state. According to the National Center for Education Statistics-Education Demographic and Geographic Estimates (NCES-EDGE, 2023), the area where the school was located was considered rural-fringe, defined as “Census-defined rural territory that is less than or equal to 5 miles from an urbanized area, as well as rural territory that is less than or equal to 2.5 miles from an urban cluster” (p. 2) according to the National Center for Education Statistics-Education Demographic and Geographic Estimates (NCES-EDGE, 2023). The district enrolled under 2,500 students across the one high school, one middle school, and two elementary schools. The high school had between 600 and 650 students enrolled in Grades 9 through 12. For the study, the researchers and students connected virtually via Zoom. One Zoom link existed for the classroom and researchers worked individually with students using breakout rooms. The teacher and paraprofessionals were within the space but not next to the students as they worked. Each session lasted no more than 30 minutes per student per day, and researchers met with the class 2 days per week. When students engaged with the researchers, they were in their life skills/study support class. Some students used this class period to work on other class work, others engaged with related service providers (e.g., speech), and several others participated in CBI (i.e., volunteering at the food bank).

Materials

During the study, researchers used Zoom, probes, a data collection sheet, and a virtual money manipulative. The students accessed Zoom and the virtual money manipulative through their classroom Chromebooks. The virtual money manipulative was the free web-based manipulative Money Pieces from the Math Learning Center. The manipulative was available to students during baseline, intervention, and maintenance sessions. The manipulative provided U.S.$1, U.S.$0.50, U.S.$0.25, U.S.$0.10, U.S.$0.05, and U.S.$0.01 options, represented as both the symbolic or picture form and base 10 form (e.g., a penny was one cube, a dime was a rod of 10 cubes, and a dollar was a square of 100 cubes). Each coin and dollar were represented in base 10 and was labeled with the symbolic or picture form (see Figure 1). For this study, researchers and students always used the base 10 representation of the money value. The web-based app itself was a blank whiteboard, with the money to the left and other features (pen and eraser) on the bottom. Students accessed the manipulative via a link provided by the researchers using the chat feature of Zoom.

Figure 1.

Screenshot From the Math Learning Center Money Pieces Application.

Data Collection

Each probe sheet consisted of two-word problems, presented orally by the researcher, that involved making change with dollars and coins (e.g., You purchased items at a store. Your total bill came to U.S.$1.20. You gave the cashier U.S.$3. How much change did you get back?). Problems in all phases involved costs ending in 0 (e.g., U.S.$1.20 or U.S.$2.70); thus, researchers and students used the dollar bills and dimes to solve problems. All problems involved providing the cashier with U.S.$4 or less, given space constraints on the virtual money manipulative. Given the online environment and students only having one screen, problems were presented orally to students; researchers repeated the problems multiple times as needed for the student to complete the problem. The data collection sheet, which was used by researchers but not provided to students, listed the task analysis steps for solving the problems—six steps for intervention and maintenance sessions. Researchers did not collect independence data during baseline. For each session, researchers recorded if students were independent or the level of prompting (e.g., independent, indirect verbal, direct verbal, or model) needed across the two problems on the data collection sheet.

Experimental Design

Researchers employed a single-case multiple probe across participants design study to explore the relationship between the intervention package and student accuracy and independence. Consistent with multiple probe across participant designs, all students entered baseline at the same time but transitioned into the intervention phase in a staggered fashion (Ledford & Gast, 2018). The researchers followed the Council for Exceptional Children (CEC; Cook et al., 2014) quality indicators for single-case design research. As such, when the first student completed at least three baseline sessions and their baseline accuracy data were either decelerating or zero-celerating and stable, they entered intervention. The other students continued in baseline, with the second student entering intervention when they had at least four baseline sessions with stable and decelerating or zero-celerating accuracy data and the first student had two intervention sessions with an increase in accuracy and independence. The other student systematically entered intervention in a similar fashion. All students continued in intervention until they achieved 100% accuracy and over 90% independence on the intervention probe for two consecutive sessions. When students achieved both, they entered the maintenance phase.

Independent and Dependent Variables

The independent variable in the study was the intervention package consisting of the virtual money manipulative, the SLP, and modeling. Researchers modeled one problem using the virtual money manipulative and then provided students with the link to the virtual money manipulative in the chat. Researchers utilized the SLP to support students in solving the problems throughout the intervention. For this study, the SLP included three prompts: indirect verbal, direct verbal, and modeling. Researchers initiated the SLP when the student indicated they did not know what to do next, engaged in an incorrect action, or failed to engage in any action within 10 seconds. For indirect verbal prompts, the researcher asked the student if they knew what to do next (e.g., “Do you remember what to do next?”). For direct verbal prompts, the researcher explicitly told the participant what the error was or the next step (i.e., “You need to represent the total amount you need to pay for the items” or “The amount was $2.60; do you have enough dimes to represent sixty cents?”). For modeling prompts, the researcher stopped student screen sharing, shared their screen, and demonstrated the step of the problem on which the student needed prompting. Given limitations in technology (i.e., the feature being blocked on the school-issued Chromebooks students used), the researchers were unable to use the guided access feature in Zoom where they could control a student’s screen to model how to complete the step.

The dependent variables in the study involved accuracy, independence, and number of prompts. Researchers operationally defined accuracy as the percentage of the two making-change-with-dollars-and-coins problems students solved correctly—regardless of the use of prompting during intervention—on each probe (solved correctly was defined as the student identifying the correct answer according to an answer key). The students were not required to write the dollar sign but were required to state or write the dollar and coin amount (i.e., 1.20 or zero dollars and 30 cents). Researchers defined independence as the percentage of task analysis steps completed without prompting during each probe when solving the two problems involving making change with dollars and coins. On the intervention probes, each problem consisted of six steps for a total of 12 per intervention probe. The steps were (a) represent the amount given to the cashier, (b) cross out the total purchase amount in dollars, (c) represent the total purchase in dimes, (d) cross-out the total purchase in dimes, (e) count up from the purchase amount (represent with coins), and (f) say/write the answer. Researchers also calculated the total number of prompts given per probe. Researchers defined the number of prompts as the total number of prompts provided across the 12 total task analysis steps of each intervention probe. Since the SLPs was used, students could receive more than one prompt on each step (i.e., indirect verbal and direct verbal), for a maximum of 36 prompts per intervention session (i.e., 3 levels of prompts × 2 steps × 2 problems).

Procedures

Students and researchers connected via Zoom calls. Researchers enabled breakout rooms to allow researchers to work individually with students. Generally, researchers worked one-on-one, except when inter-observer agreement (IOA) data were collected and two researchers were present. When a second researcher was present for IOA, they kept their video off but the researcher leading the intervention and the students kept their videos on. Students participated in one-to-two sessions per week, depending on their availability and attendance, with each session lasting about 30 minutes. The primary data collectors and co-developers of the intervention were a fourth-year special education doctoral candidate and a faculty member, both of whose research focused on mathematical interventions for students with IDD. Both the doctoral candidate and faculty member (authors H.M.L. and E.C.B.) were the interventionists in the study. The two interventionists collaborated to develop the study and the faculty member trained the doctoral candidate on implementing the intervention with fidelity. The two interventionists worked individually with two students, but not always the same students during each session. Procedural fidelity data were taken consistently throughout the study to ensure both interventionists were implementing the study the same way. Two education doctoral students assisted with IOA data collection. The fourth-year doctoral student trained the IOA data collectors. Training sessions for IOA data collectors were repeated until both IOA data collectors were 100% reliable to control for threated to internal validity.

Baseline

During baseline, students solved two making-change-with-dollars-and-coins problems with access to the money manipulative but no training. Researchers also provided no prompts, although they did orally present the problem two times, and more if requested. The first student completed at least three baseline sessions, the second at least four, and the third five. Researchers collected accuracy but not independence data during baseline.

Intervention

During intervention, students used the virtual money manipulative Money Pieces from the Math Learning Center (https://www.mathlearningcenter.org/apps/money-pieces; see Figure 1) to solve the making-change-with-dollars-and-coins problems. In each session, researchers modeled how to solve one problem and then implemented the SLP as students solved two problems. To model, the researcher provided a verbal narration (i.e., think aloud) while demonstrating with the money manipulative. The researcher began by providing a problem (e.g., “You went to a store and the total of the items you purchased was $3.40. You gave the cashier $4. How much change will you get back?”). The researcher both orally presented and then wrote the mathematical notation on the whiteboard of the app (U.S.$4.00–U.S.$3.40); the step of writing the problem was optional for students when they solved as students tended to write big on the screen and space was limited. The researcher then modeled first representing the amount of money given to the cashier—in this case, U.S.$4—by bringing out the correct number of dollar bills, each shown as a square of 100 cubes. Next, the researcher explained representing how much was spent by first subtracting—placing an X through—the dollars (i.e., in this case 3). After addressing the dollars spent, the researcher focused on representing the amount spent with coins. For the remaining dollars, the researcher placed the number of dimes needed to represent the coins spent—in this case, four dimes (i.e., U.S.$0.40). As with crossing out the dollars spent, the researcher put lines or Xs through these dimes to indicate these coins were spent and would not be part of the change given back. Finally, the researcher filled in dollar square of 100 cubes with dimes (10-cube rods) until it was complete, representing the change back. The researcher than added the change to any dollars left and both wrote and verbally stated the answer—U.S.$0.60. After researchers modeled how to solve one problem, the students solved two problems independently. During the independent problems, the researchers used the SLP is the student made a mistake, forgot what to do next, or did not initiate the next step within 10 seconds of completing the previous step. Researchers collected data on student accuracy, student independence, and number of prompts. After the student completed two sessions at 100% accuracy and over 90% independence, the student entered the maintenance phase.

Maintenance

Starting 1 week after the last intervention session, students completed one maintenance session per week for 2 weeks. During maintenance sessions, researchers provided students with the virtual money manipulative, orally presented two problems, and used the SLP to support students as they solved two problems; however, they provided no modeling. As with intervention, researchers collected data on student accuracy, independence, and number of prompts.

IOA and Procedural Fidelity

Researchers collected IOA data on at least one third of the baseline and intervention sessions, and one-half of maintenance sessions. To collect IOA, a second data collector attended the session, keeping their camera off. They used the same data collection sheet as the primary data collected and recorded accuracy and independence (number of prompts per task analysis step and level of prompting). The two data collectors (primary data collector and IOA data collector) compared and used the point-by-point IOA process by dividing the number of agreements by the total number of opportunities to compare and multiplying by 100. Inter-observer agreement for baseline for all three students for accuracy was 100%. Intervention IOA for Ryan was 100% for accuracy, and 97% (92%–100%) for independence. For Sam, IOA was 100% for accuracy, and 98% (86%–100%) for independence. Finally, IOA was 100% for accuracy, and 98% (86%–100%) for independence for Gina. In maintenance, IOA for accuracy and independence was 100% for all three students.

Researchers used a checklist to evaluate procedural fidelity. The second data collector collected procedural fidelity data during the baseline, intervention, and maintenance sessions they attended. Procedural fidelity data were collected on at least 30% of baseline, intervention, and maintenance sessions. The checklist for intervention involved the following: (a) the student has access to the money manipulative in all sessions, including baseline and maintenance, (b) the researcher models one problem (procedural fidelity checklist captured each step of modeling, including representing the amount given to the cashier, representing how much money was spent in dollars and coins, crossing out dollars and coins spent, representing the dollars and coins they would get back, and stating the answer), (c) the student completes two problems independently or with the SLP if needed, (d) the SLP is initiated if the student does not respond in 10 seconds, if they make a mistake, or indicate they do not know the next step, and (e) the researcher first provides an indirect verbal prompt; if the student does not respond the researcher administers a direct verbal prompt; and finally a model prompt if needed. For baseline, the checklist only involved Step a, and, for maintenance, Steps (a), (c), (d), and (e). Procedural fidelity was 100% for all three students during baseline, intervention, and maintenance sessions.

Social Validity

After the completion of the maintenance phases, researchers conducted a social validity interview with the students’ teacher regarding the intervention. The researchers asked the following questions:

Do you think the students benefited from learning how to use the virtual money manipulative to solve the making change problems?

Do you see any changes or growth in these areas?

Do you see the students applying and of the skills they have learned in the classroom?

Due to the end of the school year, researchers were unable to interview the students about their perceptions of the intervention.

Data Analysis

Consistent with single-case design studies, researchers analyzed the data via visual analysis and conducted calculations to analyze data within and across phases (Ledford & Gast, 2018). Researchers entered and graphed data in Excel and visually compared baseline and intervention phase data to determine overlap as well as the immediacy of the effect. To determine the trend of baseline and intervention phase data, researchers employed the split middle technique (White & Haring, 1980). Using this technique, researchers found the mid-point, mid-rate, and mid-date for both baseline and intervention phases and then drew a line between the mid-rate and mid-date. Lines that sloped up were accelerating, lines that sloped down decelerating, and lines with a zero slope were zero-celerating (White & Haring, 1980). For stability, researchers calculated if 80% of both baseline and intervention data for each student fell within 25% of the phase’s respective median. If this occurred, data were deemed stable; if not, variable (Ledford & Gast, 2018).

Results

Researchers determined a functional relation existed between the intervention package of the virtual manipulative, modeling, and the SLP and the dependent variable of accuracy (see Figure 2). Because researchers did not collect independence data during baseline, they were unable to determine a functional relation between the intervention package and both the dependent variables of the independence and total number of prompts. However, students successfully increased their independence and decreased their number of prompts needed (i.e., 100% accurate and over 90% independence for two consecutive sessions) in 12 to 17 sessions. Students were also able to maintain their accuracy, independence, and fewer number of prompts for up to 2 weeks following intervention.

Figure 2.

Graphed Accuracy, Independence, and Number of Prompts Data for Making Change Problems.

Ryan

Ryan’s accuracy was 0% for his three baseline sessions. Ryan’s baseline data were zero-celerating and stable. The effect of the intervention was gradual. For the first intervention sessions, Ryan was 0% accurate, 16.7% independent when completing the steps of the task analysis and required 17 prompts. For his second intervention session, Ryan improved to 50% accurate, 41.7% independent, and required 11 prompts. Starting in Session 3, and for all that followed, Ryan was 100% accurate and always over 75% independent; the number of prompts he needed ranged from 0 to 5. Ryan achieved mastery—100% independent and over 90% independent for two consecutive sessions in 12 sessions. His intervention data for accuracy were stable and accelerating and his independence data were variable and accelerating. Ryan was 100% accurate, 91.7% independent, and needed one prompt on both maintenance sessions.

Sam

Sam’s accuracy was 0% for his four baseline sessions; his baseline data were stable and zero-celerating. The effect of the intervention was gradual. For his first intervention session, Sam was unable to complete a full session as he ran out of time. For his second and third intervention sessions, Sam was 0% accurate and 0% independent. He needed at least one prompt on every step, with 22 prompts and 24 prompts, respectively. Starting in Session 4 and for all that followed, Sam was 100% accurate and 41.7% to 100% independent. In Session 11, a paraprofessional in the classroom provided Sam with the answer unrequested by Sam or researchers. The session was terminated, and no data were recorded for this session. Sam achieved mastery—100% accurate and over 90% independent for two consecutive sessions in 17 sessions. Sam’s intervention data for accuracy were stable and accelerating and his independence data were variable and accelerating. In terms of maintenance, Sam was 100% accurate and independent on his first maintenance session and 100% accurate and 91.7% independent on his second maintenance session. He needed one prompt on his second maintenance session.

Gina

Gina’s accuracy was 0% for her five baseline sessions. Gina’s baseline data were stable and zero-celerating. The effect of the intervention on accuracy was immediate. Gina was 100% accurate for all 13 of her intervention sessions. On her first and second intervention sessions she was 25% independent (17 prompts), 33.3% independent (15 prompts), respectively. In Sessions 3 through 8, Gina was 75% to 83.3% independent and needed a range of two-to-six prompts, with the exception of Session 5. In Session 5, Gina and her teacher reported she was not having a good day, and she struggled to stay focused, which resulted in 33.3% independence and needing 15 prompts. For Sessions 9 through 11, Gina was 75% to 83% independent, and she needed two-to-four prompts per session. Gina achieved mastery criteria after completing session 12 with 100% accuracy and 91.7% independence with only one prompt and Session 13 at 100% accuracy and independence. Her intervention data for accuracy were stable and accelerating; for independence, her data were variable and accelerating. In terms of maintenance, Gina was 100% accurate and 83.3% independent on both maintenance sessions, needing three prompts each session.

Social Validity

The teacher indicated the students benefited from learning how to use the virtual money to solve making change problems. She noted growth in all three participants when working in the classroom with plastic or real money, including counting coins up to U.S.$1.00. Specifically, she commented on student improvement relative to counting combinations of dimes, nickels, and pennies. Furthermore, the teacher stated that she observed the students using these skills when paying for items in the community. She observed the students working together to determine the different menu items they would be able to purchase with the amount of money they had and how much money they would have left over.

Discussion

The purpose of the current study was to extend the use of virtual manipulatives to high school students with IDD in an online environment, with a focus on life skills. This study resulted in two main findings (a) students with IDD were able to successfully solve making change problems as a result of the intervention package of virtual manipulatives, the SLP, and modeling and (b) students with IDD were successful in learning to solve life skills mathematics problems in an online environment. These findings support prior research exploring virtual manipulatives in online environments and have implications for rural schools and future online instruction for students with IDD.

The three students with IDD successfully solving the mathematics problems using an intervention package involving virtual manipulatives is unsurprising. Previous research determined virtual manipulatives in and of themselves as well as part of manipulative-based instructional sequences are an evidence-based practice for students with IDD (Long et al., 2022). However, this study extends the current literature in multiple ways, including (a) involving high school students with IDD as opposed to middle schools students with IDD (e.g., Bouck et al., 2020; Long et al., 2022); (b) focusing on life skills mathematics (e.g., making change) as opposed to mathematics aligned to grade-level content standards (Bouck et al., 2020; Root et al., 2021); and (c) teaching students within an online environment rather than face-to-face (Bouck & Long, 2021a, 2021b).

This study supports the hypothesis that secondary students with IDD can be successfully taught in an online environment. Online instruction, and attention to such, has increased over the past few years due to the COVID-19 pandemic, which resulted in prolonged school closures, reduced face-to-face services, and increased online education (Mcelrath, 2020). Researchers previously found positive results for the use of virtual manipulatives for students with disabilities provided during online instruction, specifically with mathematics skills (e.g., Bouck & Long, 2021a, 2021b). For students in rural settings, online instruction could be a beneficial way to expand the resources and learning opportunities not readily available to them in their district (McKissick et al., 2017). High school students with IDD, especially, should receive a portion of their instruction in non-school settings, such as in the community, to practice the generalization of the skills they have learned in the classroom, which is often referred to as CBI (Collins & Ludlow, 2018). However, for programs in which it is not always feasible to take students into the community, such as those in more rural location with limited transportation, online instruction can provide access to additional support unavailable in the same physical location (Rowe & Test, 2012).

While the researchers were unable to perform generalization probes to the community as would be realistic in this case, we recognize this is a fundamental part of instruction for students with IDD (Shurr et al., 2019). Previous research has confirmed that mathematics manipulatives that are perceptually rich, or manipulatives that are more realistic (e.g., manipulatives that look like money) may hinder a students’ ability to acquire and generalize the concept that is being targeted (Carbonneau et al., 2012). The same researchers suggests that plain or bland manipulatives like base 10 blocks or fraction tiles, may be more beneficial for students in not only the acquisition but also generalization of skills. In the current study, researchers opted to use virtual manipulatives that represented money within base 10 (see Figure 1) which represent them in a more bland, but more mathematically appropriate representation. Furthermore, previous research on virtual manipulative based interventions consistently show students with IDD are able to generalize to solving the target problem without the use of manipulatives, or to similar problem types (Bouck et al., 2020, 2021; Root et al., 2021). However, there are currently no published studies exploring the effects of virtual money manipulative-based instructional sequences that involve generalization to a community-based setting. As we continue to explore the effectiveness of virtual manipulative to support students with IDD, researchers should begin to turn their attention to generalization to real life settings and scenarios.

In addition to CBI, rural schools have also struggled with both providing innovative instruction and retaining high-quality teachers (Collins & Ludlow, 2018; Daniel et al., 2019; McKissick et al., 2017). Although virtual manipulatives do not necessarily provide an alternative to CBI, they do provide an innovative approach to supporting mathematics and life skills instruction. As such, they offer rural special educators another option to expose their students to content and activities that may otherwise be inaccessible. Furthermore, virtual manipulatives expose students to technology-infused instruction, which occurs less in rural settings (Kabel et al., 2021; Sundeen & Kalos, 2022). In the case of secondary students with IDD, using virtual manipulatives exposes students to an evidence-based practice in mathematics (Long et al., 2022).

This study allows for the consideration of the intersection of online instruction, life skills mathematics instruction, and students with IDD, and to do so within a rural context. The rural context is important as all students deserve access to high-quality experiences, evidence-based practices, and innovative instruction, such as CBI and technology-infused teaching, as well as online opportunities that create flexibility in instruction, regardless of their geographical location (McKissick et al., 2018; Rowe et al., 2020). Due to the limited resources those in rural settings may have access to, free products—such as virtual manipulatives—that target both life skills and academic skills provide materials to rural educators (Stenhoff et al., 2020) and allow for educators to infuse life skills and academics. The practice of incorporating both life skills and grade-aligned mathematics content when teaching mathematics to students with IDD is receiving increased attention and advocacy over the past decade (Collins & Ludlow, 2018; Hyer & Cooper-Duffy, 2019). Data from this study suggest that students were able to become relatively independent is further support for rural teachers using the virtual manipulatives during online instruction, as needing caregiver support for materials during online or remote learning was noted as a concern for students with IDD (Stenhoff et al., 2020).

Implications for Practice

This study offers important implications for practice. First, the results demonstrate high school students with IDD can be successful, and relatively independent, using virtual manipulatives to solve mathematics problems involving making change. The students in this study were able to acquire and maintain the skill of calculating change using virtual manipulatives while significantly decreasing the number of prompts required to complete the problems. All three students also maintained their accuracy and independence at higher than baseline levels for 2 weeks following the end of intervention with minimal prompting (one to three prompts across the two sessions). In the social validity interview, the teacher noted growth when using plastic or real money in the classroom. Using virtual money manipulatives that presented the money within base 10, was beneficial for students’ generalization to using plastic or real money in the classroom and community. Teachers should feel confident implementing virtual manipulatives in their mathematics classrooms for secondary students with IDD for a range of mathematics skills including grade-aligned content as well as life skills-related mathematics.

Finally, teachers of students with IDD in rural school districts may be faced with unique challenges due to their limited resources and lack of proximity to places suitable for CBI (Collins & Ludlow, 2018; Hopkins & Dymond, 2020). In the current study, virtual manipulatives were also shown to be a successful way to engage students in practice with purchasing skills. While this is a viable option to support students with IDD in learning personal finance skills, teachers in rural schools may consider additional ways to supplement the instruction. Participating in fundraisers or other events or projects within the school in which the students can practice handling money is a great alternative to CBI when it is not accessible (Collins & Ludlow, 2018). However, not all students with IDD who would benefit from CBI have access to these learning opportunities. While school location, such as being in a rural school district, may be a barrier to accessing CBI, students with IDD can practice these skills effectively in a classroom setting with the use of technology.

Limitations and Future Directions

This study is not without limitations. First, the researchers were unable to collect baseline data on independence (percentage of steps or number of prompts). Because independence was calculated based on the percentage of steps completed independently on the task analysis, and researchers did not deliver prompts in baseline, they were unable to determine a functional relation for independence. Another limitation was that researchers did not explore generalization as part of this study. Generalization is an important component of all learning opportunities and should be planned for throughout instruction and intervention (Shurr et al., 2019). For life skills-related interventions, the fundamental focus should be generalization to the community within real life contexts. However, the researchers were unable to do so in this study. Furthermore, the researchers did not collect data on a simulation of CBI. In this study, generalization could have been explored with different problem types (e.g., problems involving pennies, and amounts using larger bills), as well as generalization to completing these skills in realistic settings (e.g., the grocery store) or even with physical (i.e., concrete) bills and coins. Future research should include generalization to real life settings or an opportunity to participate in purchasing skills in a simulated setting in their studies. In addition, one could suggest that a limitation was the lack of a true maintenance phase in that the SLP was delivered during maintenance sessions. However, the maintenance phase did involve a lack of modeling prior to students engaging in the task.

Another limitation involves the problems students were asked to solve in the study. The problems presented to students involved dollar amounts U.S.$4.00 or less, and the cost of the items were always multiples of 10 (i.e., U.S.$4.00–U.S.$2.40; U.S.$2.00–U.S.$1.80). When solving these problems, students were only required to use the dollar and dime representations. This is not realistic as prices are generally more than U.S.$4.00 and involve all coins. Future research should seek to explore problems with a greater variety of costs that reflect more realistic situations in which students need to calculate cost. The number of baseline data points could be viewed as a limitation of this study. When designing this study, the authors attended to the quality indicators from the CEC (Cook et al., 2014), which supports the use of at least three data points in the baseline condition. Other quality indicators, such as What Works Clearinghouse (WWC, 2020), require at least six data points in baseline for a study to qualify as meets without reservations. While this study meets the CEC standards for quality, it will not for the WWC. An additional limitation was the authors’ limited focus of social validity on the teacher’s perception of the outcomes, rather than other aspects, such as the acceptability of the procedures and implementation feasibility. Future research should plan for a more expansive social validity measure.

An additional limitation was Sam’s Session 11, when a paraprofessional in the classroom provided Sam with the answer and therefore led to termination of the session. While this is concerning, it is a reality of online instruction as researchers do not have control over the physical classroom environment. At the beginning of the study, the researchers set expectations with the teacher and students that the students were to engage in the study on their own, and after this session the researcher again emphasized the importance of independence. Finally, the classroom distractions and limited control researchers had on classroom behavior due to being online was a limitation. Because this research was conducted online, the researchers had no way of controlling such classroom factors as other students in the room talking to the participants. For example, during some sessions, other students in the classroom were having conversations with the participants that took their focus away from the mathematics problems, sometimes resulting in errors. While this is the reality of being in a classroom, future research should seek to determine if there is a way to control more of the classroom environment when conducting online research, such as students moving to another location.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Holly M. Long

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