Adding It Up

Abstract

Manipulatives are a common tool in mathematics teaching and learning, including for students with disabilities. The most common manipulatives are concrete manipulatives, yet app-based manipulatives are a viable age-appropriate option for secondary students with disabilities. Through an adapted alternating treatment design with three middle school students—two with mild intellectual disability and one with a learning disability, researchers explored the impact of virtual and concrete manipulatives on students’ accuracy, independence, and task completion time for solving addition of fractions with unlike denominators. Students were equally successful in terms of accuracy and differences with independence were minimal. When comparing the two manipulative types, the results were idiosyncratic; two students were more independent with the concrete manipulative and one with the app-based manipulative. Implications for research regarding mathematics instruction and use of concrete and app-based manipulatives are discussed.

Keywords

math content/curriculum area apps technology perspectives manipulatives middle school age/grade level

Concrete and App-Based Manipulatives to Support Students With Disabilities to Add Fractions With Unlike Denominators

Manipulatives are considered an evidence-based or research-informed practice in the area of mathematics instruction for students with disabilities (Lai & Berkeley, 2012; Maccini & Gagnon, 2000). The use of manipulatives is supported in mathematics teaching and learning for students with and without disabilities for multiple reasons. For one, scholars and educators often report that manipulatives help make abstract concepts more concrete (Goodman, Seymour, & Anderson, 2016). In other words, scholars assert manipulatives—particularly simple manipulatives—help students to develop conceptual understanding of mathematical concepts (McNeil & Jarvin, 2007). Manipulatives can also engage students and make mathematical concepts appealing (Moyer, 2001). Finally, researchers found use of manipulatives can result in greater or improved mathematical performance, particularly when manipulatives are part of retention-focused instruction (Carbonneau, Marley, & Selig, 2013).

Typically, the term manipulative refers to concrete manipulatives. Concrete manipulatives are physical objects one can manipulate to help build conceptual understanding and solve mathematical problems (Bouck & Flanagan, 2010); common examples of concrete manipulatives include base 10 blocks, fraction pieces, and tiles. Yet another equally useful type of manipulative exists—virtual manipulatives (Bouck & Flanagan, 2010). Virtual manipulatives are digital manipulatives, often digital recreations of concrete manipulatives, and exist both as online manipulatives for use on a computer with Internet and app-based manipulatives for use on a tablet (i.e., iPad; Bouck & Flanagan, 2010; Bouck, Working, & Bone, 2018). Virtual manipulatives can allow access to mathematical supports beyond concrete manipulatives and offer additional benefits (Satsangi & Miller, 2017).

An emerging research base exists on comparing the use of concrete and virtual manipulatives as tools to support students with disabilities in mathematics. Bouck, Satsangi, Taber-Doughty, and Courtney (2014) compared concrete base 10 blocks to virtual online base 10 blocks to support elementary students with autism in solving subtraction with regrouping problems. The three students were supported with both types of manipulatives, but were slightly more independent (i.e., needed few prompts) with the virtual base 10 blocks. The students also indicated a preference for the virtual manipulatives. More recently, Root, Browder, Saunders, and Lo (2017) found elementary students with autism and moderate intellectual disability correctly performed more or an equal number of steps in solving comparison problems (i.e., problems comparing quantities) with virtual manipulative as compared to concrete manipulatives. The three students in the study by Root et al. (2017) also indicated a preference for virtual manipulatives.

At the secondary level, Satsangi, Bouck, Taber-Doughty, Bofferding, and Roberts (2016) compared concrete and virtual (online) algebraic balance scales for supporting three high school students with learning disabilities in solving linear algebra problems. The students were effective with both types of manipulatives but expressed a preference for using the virtual manipulatives. More recently, Bouck, Chamberlain, and Park (2017) compared concrete base 10 blocks to an app-based manipulative of base 10 blocks as tools to support secondary students with mild intellectual disability and learning disabilities to solve multidigit subtraction with regrouping problems. Similar to results from other research, Bouck, Chamberlain, and Park (2017) found both types of manipulatives effective but the three students were more independent (i.e., needed fewer prompts) with the app-based manipulative. Two of the three students also reported a preference for the app-based manipulative.

Given the increased in advocacy for the use of virtual manipulatives in mathematics teaching and learning for students with disabilities (cf., Satsangi & Miller, 2017; Shin et al., 2017), additional research regarding their effectiveness, support of independence, and efficiency is needed. First, few published studies examine app-based virtual manipulatives as opposed to online virtual manipulatives. Yet app-based manipulatives may pose an advantage over online virtual manipulatives with regard to social desirability, portability, and feasibility (Bouck et al., 2018). Second, the current existing literature on virtual manipulatives is largely focused on the mathematical domains of whole number operations (e.g., Bouck, Chamberlain, & Park, 2017; Bouck, Satsangi, Taber-Doughty, & Courtney, 2014; Root, Browder, Saunders, & Lo, 2017). In an effort to build the research base of virtual manipulatives across all domains of mathematics, examination of use in other areas, such as fractions, is needed.

Although fractions are an important mathematical domain, limited literature has explored research regarding manipulatives and fraction instruction for students with disabilities (Jordan et al., 2013; National Mathematics Advisory Panel [NMAP], 2008). Two published studies examining fractions and concrete manipulatives for only students with disabilities involved the use of manipulatives within the concrete–representational–abstract (CRA) instructional sequence (Butler, Miller, Crehan, Babbitt, & Pierce, 2003; Jordan, Miller, & Mercer, 1999). Both studies used group designs to compare the CRA instructional sequence to another type of instructional sequence (representational–abstract; Butler et al., 2003) or traditional instruction (Jordan et al., 1999). Butler, Miller, Crehan, Babbitt, and Pierce (2003) and Jordan, Miller, and Mercer (1999) reported improved student performance following the CRA instructional sequence. Jordan et al. (1999) determined that students who received the intervention (e.g., CRA) improved to a greater extent from the pretest to the posttest than student who received traditional instruction. Butler et al. (2003) found higher posttest scores for students who received the CRA sequence than for students who just received the RA sequence.

More recently, Bouck, Bassette, et al. (2017) and Bouck, Park, et al. (2017) explored virtual manipulatives within an instructional sequence to support secondary students with disabilities with fractions. In a single case design, Bouck, Bassette, et al. (2017) found the virtual–representational–abstract (VRA) instructional sequence (with virtual fraction tiles) to be an effective intervention in support three middle students in solving equivalent fraction problems. Bouck, Park, et al. (2017) similarly found the virtual–abstract instructional sequence—an adaption of the VRA but without a representational phase—to be effective with regard to three middle school students with disabilities in adding fractions with unlike denominators; the virtual manipulative in this study was also virtual fraction tiles.

Given the importance of understanding fractions for both advanced mathematical concepts, such as algebra, and daily living activities (e.g., baking, home repairs; Jordan et al., 2013; NMAP, 2008), the limited literature on both concrete and virtual manipulatives for instruction involving fractions is disconcerting. This study sought to address the limited published research on fraction manipulatives for students with disabilities, particularly targeting the lack of comparison between concrete and virtual fraction manipulatives. The research questions, all focused on addition of fractions with unlike denominators, included the following: (a) What number of problems do students with disabilities solve accurately when using app-based manipulatives and concrete manipulatives? (b) what percentage of problems do students with disabilities solve independently (i.e., without prompts) when using app-based manipulatives and concrete manipulatives? and (c) what are student preferences when considering app-based or concrete fraction manipulatives to solve the problems?

Method

Participants

Three middle school students with disabilities participated in this study; informed consent and assent were obtained for all three participants. All three were initially educated in the same self-contained mathematics class taught by a special education teacher prior to the study; however, one student’s program changed as the study started. The students in the same class received the same mathematics instruction. The inclusion criteria for participation involved the following: (a) educated in the self-contained math class, (b) teacher recommendation for students struggling with math content previously taught (i.e., fractions), (c) confirmation of struggles with fraction equivalence and addition through independent KeyMath™ subtests, (d) fine motor ability to move concrete manipulative blocks and navigate a touch-based iPad application, and (e) parental consent and student assent to participate.

Kelly

Kelly was a 13-year-old, female, Caucasian, seventh-grade student. She was quiet but eager to work with researchers. Kelly was eligible for special education in the area of intellectual disability and received 10–15 hr per week in a self-contained special program. Over the years of receiving special education services, Kelly was evaluated with a variety of assessments. From the most recent data, Kelly’s Full Scale IQ on the Wechsler Intelligence Scale for Children, Fourth Edition (WISC, 4th ed.; Wechsler, 2004) was 53, with a score of 57 for verbal comprehension, 77 for perceptual reasoning, 54 for working memory, and 56 for processing speed. On the Vineland-II adaptive behavior assessment (Sparrow, Cicchetti, & Balla, 2005), Kelly scored a 64 for communication, 74 for daily living, and 69 for socialization. On the Kaufman Test of Educational Achievement, Second Edition (KTEA™-II; Kaufman & Kaufman, 2004), Kelly’s score for math was a 73–68 for math concepts and 85 for math computation. On the KeyMath-3, Kelly’s numeration score was a 22 (4.5 grade equivalent and 9:11 age equivalent), and she was unable to answer questions related to adding or subtracting fractions, but she could identify fractions. Kelly did not have any individualized education program (IEP) mathematics goals related to fractions.

Miles

Miles was a 13-year-old, male, Caucasian, seventh-grade student. He was eligible for special education services under the category of intellectual disability. Just prior to the start of the study, Miles was receiving 10–15 hr per week of services in the self-contained program. However, as the study was starting, Miles’s program was changed via an IEP meeting initiated by his parents. Miles went from having his mathematics class in the self-contained class to a general education seventh-grade math class with support from a peer mentor. Miles always enjoyed working one-on-one with the researcher and frequently expressed a desire to have longer or more sessions. On the WISC-IV (Wechsler, 2004), Mile’s Full Scale IQ was 68, with a processing speed score of 80, working memory of 65, perceptual reasoning 77, and verbal comprehension 69. The Vineland-II adaptive behavior assessment (Sparrow et al., 2005) testing indicated scores of 69 for communication, 74 for daily living, and 83 for socialization. Miles’s numeration score on the KeyMath-3 was 14 (2.2 grade equivalency and 7:11 age equivalency). On the assessment, he was unable to correctly answer the questions related to adding and subtracting fractions, but he could identify fractions. Miles did not have any IEP mathematics goals related to fractions

Doug

Doug was a 12-year-old, male, Caucasian, eighth-grade student. Supported by both teacher report and independent researcher observation, Doug was a hard worker and wanted to learn and do well in math. Doug’s special education eligibility was in the area of learning disability. Doug’s Full Scale IQ from the WISC-IV (Wechsler, 2004) was 78: Verbal comprehension was 69, perceptual reasoning 75, working memory 99, and processing speed 94. On the KTEA-II (Kaufman & Kaufman, 2004), Doug’s score for math composition was 79–78 for math concepts and applications and 85 for math computation. On the KeyMath-3, Doug’s numeration score was 18 (3.5 grade equivalency, 8:11 age equivalence) and he struggled with adding and subtraction fractions. Doug attended the self-contained class 5 hr per week for mathematics. Doug had an IEP goal relative to demonstrating an understanding of fractions.

Setting

The setting for the study was a rural public middle school in a small Midwestern town; the school was selected due to the teacher’s expressed interest in exploring new ways to support her students in learning mathematics. Four hundred and thirty-nine students were enrolled in the sixth-, seventh-, and eighth-grade school at the time of study. The study population was predominantly Caucasian (95%), followed by 3% Hispanic, and less than 1% each identifying as Multiracial, African American, or Alaska Native/American Indian. Twenty-six percent of the student body received free or reduced lunch, and 8% received special education services.

Data collection for all sessions occurred at a table outside of the self-contained class and all occurred during the students’ mathematics class (note, Miles’s mathematics class was the same hour as the self-contained mathematics class). The table was large and rectangular with four chairs. The first author worked with each student one-on-one at the table for all sessions, except when interobserver agreement (IOA) was collected and a second researcher was present. The space was reasonably quiet as surrounding classrooms closed their doors during instruction.

Materials

The main materials for this study included researcher-constructed probe sheets, pencils, task analysis data collection sheet with codes for the system of least prompts data, concrete fraction blocks, and an iPad with a fraction pieces app (Fraction Tiles by Brainingcamp, 2017). During each session, researchers gave the student a probe containing five problems of adding fractions with unlike denominators, presented horizontally, and a pencil. To develop the assessments, the researchers developed all possible addition of fractions problems with the available fractions. In other words, the fractions in the problems (i.e., the addends) were limited due to the manipulatives, which both included fractions only with denominators of halves, thirds, fourths, fifths, sixths, eighths, tenths, and twelfths. Hence, problems were limited due to the availability of fractions represented by the manipulatives, which were consistent across both concrete and virtual modes. The problem answers (i.e., sum) included fractions with only these denominators as well as one whole. Once all possible addition of fractions problems were developed, the researchers distributed them across assessments (baseline, intervention, best treatment, and generalization), so that individual problems were repeated the fewest number of times and each probe was unique.

The concrete manipulatives were plastic blocks all the same size that were divided evenly depending on the fraction they represented (e.g., eighths divided into eight equal pieces; see Figure 1). The fraction pieces connected to each other to represent a whole block. The concrete manipulatives—fraction tower equivalency cubes each contained a different colored tower labeled 1, $\frac{1}{2}$ (two equal pieces), $\frac{1}{3}$ (three equal pieces), $\frac{1}{4}$ (four equal pieces), $\frac{1}{5}$ (five equal pieces), $\frac{1}{6}$ (two equal pieces), $\frac{1}{8}$ (eight equal pieces), $\frac{1}{10}$ (10 equal pieces), and $\frac{1}{12}$ (12 equal pieces). The app-based manipulatives were essentially two-dimensional representations of the fraction blocks; on the left side of the screen was a column of all possible fraction tiles (Brainingcamp, 2017; see Figure 1). The app-based manipulatives included a one block as well as the aforementioned fractions. Students could move the fraction tiles around on the screen and they would snap together when placed next to each other. The app included a whiteboard capacity, and students would write on the screen with digital markers as well as clear the screen.

Figure 1.

Concrete and app-based fraction manipulatives used in the study.

Independent and Dependent Variables

The independent variable for this study was use of the concrete and app-based manipulatives. Using the concrete fraction blocks, manipulative was defined as setting up the fraction addition problem using the blocks and then using the blocks to find the common denominator as well as solve the problem. Using the app-based fraction tiles manipulative was defined as setting up the fraction addition problem using the digital tiles (i.e., dragging the tiles) and then using the tiles to find the common denominator as well as solve the problem. To find the common denominator—for either condition—students used with the concrete or app-based fraction manipulative to determine equivalent fractions (i.e., aligning blocks) or used mathematical means. Once the students determined the common denominator, the student then set up the concrete or app-based fraction manipulatives to represent the addition problem with both addends expressed in their equivalent fractions of the same common denominator.

The dependent variables for this study included (a) accuracy, defined as the number of addition of fractions with unlike denominators problems answered correctly out of five; (b) task completion time, defined as the amount of time it took for a student to complete each probe; and (c) task independence, defined as the percentage of steps in the task analysis of solving the addition of fractions with unlike denominator problems students completed without any prompting. Duration was used to measure task completion time and event recording data were recorded for accuracy and independence. To calculate accuracy, researchers summed the problems students solved correctly out of five. To measure task completion time, researchers used a stopwatch app on an iPhone, starting the stopwatch when the researcher gave the student the probe and provided the prompt to start and ending the stopwatch when the student completed the last problem. To determine independence, the researchers calculated the number of steps the student did not need prompting on for each problem across the five-problem probe and divided that number by the number of steps (the task analysis recording sheet is available upon request). Depending on the fraction addition problem, there were 9 or 11 steps to the task analysis. The variation depended on the problem. For example, a problem such as $\frac{1}{2} + \frac{1}{6}$ required nine steps as only one fraction $(\frac{1}{2})$ needed to be rewritten into an equivalent fraction with a common denominator (.). Whereas with a problem such as $\frac{1}{3} + \frac{1}{4}$ required 11 steps as both fractions needed to be rewritten into equivalent fractions with a common denominator $(\frac{4}{12} + \frac{3}{12})$ . Hence, if a student completed eight of the nine task analysis steps without any prompting (e.g., gesture, indirect verbal), their task independence would be 88.9%.

Experimental Design

Researchers used an adapted alternating treatment design across four phases: baseline, intervention, best treatment (i.e., the most effective condition; also referred to as best alone), and generalization (Sindelar, Rosenberg, & Wilson, 1985; Wolery, Gast, & Ledford, 2014). In this study, students solved adding fractions with unlike denominators across three alternating conditions during intervention: concrete fraction blocks, app-based fraction tiles, and an extended baseline with no manipulatives. Each condition consisted of five sessions, which alternated via a random order with no more than two consecutive sessions of the same condition in a row. The condition order was randomized by the first author writing the three conditions on a piece of paper and putting those papers into a cup. The first author then drew out a piece of paper to represent each session. The slip of paper was returned to the cup before each draw unless the same condition had been drawn twice in a row. In that case, the slip of paper was removed from the cup for that draw and returned afterward.

The adapted alternating treatment design allowed researchers to evaluate the two manipulative interventions for efficiency (i.e., effectiveness with superiority; Wolery et al., 2014) on a nonreversible behavior chain (adding fractions with unlike denominators). The first author, a faculty member in special education with over a decade of experience developing and implementing mathematical interventions for students with disabilities as well as specific experience with virtual and concrete manipulatives, delivered all sessions in a one-on-one format. Two other authors, both education doctoral students, served as collectors for IOA; both doctoral students were trained in the study prior to IOA collection. Training consisted of practicing with the first author, watching the first author deliver an actual intervention, and using the data collection protocol during a simulated session with the first author.

Procedures

Pre-assessment

Prior to baseline, students were assessed using KeyMath-3 to confirm current mathematics abilities. Students were also given a brief pre-assessment consisting of five problems to determine students could identify fractions (i.e., a picture of a circle separated into X equal pieces and inquiring what was the fraction when Y of the X equal pieces were shaded, such as a circle evenly divided into six pieces and four of those pieces were shaded; hence, the fraction would be $\frac{4}{6}$ ) as well as five problems inquiring about equivalent fractions (i.e., $\frac{1}{3} + \frac{}{6}$ ).

Baseline

In the baseline phase, each student completed problems involving the addition of fractions with unlike denominators. Each baseline session involved students completing five problems via paper and pencil. Students moved out of baseline when they had completed at least three baseline sessions and data were stable with a zero-celerating or decelerating trend.

Pretraining

After baseline but prior to intervention, each student completed a pretraining phase. During pretraining, each student was individually trained on both types of manipulatives. Using explicit instruction, the researcher modeled two problems using the manipulative—concrete or app-based; the researcher used think alouds as she solved the problems to make her use of the manipulative and her approach to solving the problems explicit. Next, the researcher guided or prompted students as they solved two problems using the same manipulative. Finally, the students attempted to solve five problems independently with the given manipulative. If the student solved 80% of the problems correctly, training was considered successful and the student moved onto the next training or into the intervention phase. If the student did not answer 80% of problems correctly, the training protocol was repeated the next session. For both concrete and app-based manipulatives, each student was trained in one session.

During the concrete manipulative condition, each student was trained to read the problem on the probe sheet and set up the fraction blocks. Specifically, students were trained to set out the one block (i.e., the block representing one whole) for each problem and then to connect the correct fraction block pieces to represent each fraction in the addition problem (i.e., the blocks could connect like unifex cubes to each other). For example, if the problem was $\frac{1}{3} + \frac{1}{4}$ , the student was trained to connect (i.e., hook together) one one-third block and one one-fourth block, below and touching the one block. Next, the student was trained to find a common denominator using the fraction blocks. During the think aloud portion of the explicit instruction, the researcher indicated the common denominator was one in which both denominators were factors. After finding a common denominator, the student was trained to find the equivalent fraction for one or two of the fractions with the common denominator. Finally, the student was taught to count the number blocks to determine the numerator and subsequently write down the answer on the paper.

The app-based manipulative training was similar. Students were taught to bring out the one tile (i.e., the tile representing one whole), and then the correct subsequent fraction tiles. For example, if the problem was $\frac{1}{4} + \frac{3}{12}$ , the student was taught to drag out one one-fourth tile and then three one-twelfth tiles. The students were taught to connect the one one-fourth tile and the three one-twelfth tiles to each other as well as place them directly, and touching, underneath the one block. Next, the students were trained to find the common denominator by using the fraction tiles. Similar to the concrete phase, the researcher used think alouds during the modeling portion and told students that the common denominator was one in which both denominators were factors. After determining the common denominator, the student was trained to find the equivalent fraction with the common denominator for either one or both of the fractions, as appropriate. Finally, the student added the number of tiles to determine the numerator and wrote down the answer on the paper.

Intervention

Each student began the intervention phase, following successful training with both types of manipulatives. During intervention, students alternated between three conditions: concrete fraction blocks, app-based fraction tiles, and no manipulatives (i.e., extended baseline). Each condition consisted of five sessions, and no condition occurred for more than two consecutive sessions and the order of sessions for each student was arranged randomly. In each session, students solved five addition of fractions with unlike denominators problems. For both manipulative conditions, the researcher implemented the system of least prompts if students did not initiate a step of the task analysis within 10 s; note use, of lack thereof, of the system was least prompts was used as a measure of task independence. The system of least prompts, after independent, began with a gesture (i.e., using one’s hands to make a motion, such as gesturing or referencing with one’s hand to the manipulative or particular fraction manipulative, e.g., $\frac{1}{3}$ , an indirect verbal prompt (i.e., “what do you do next?”), a direct verbal prompt (e.g., “you need to find a common denominator”), and finally modeling (e.g., showing how to move a digital fraction tile; Doyle, Wolery, Ault, & Gast, 1988). Subsequent, more intrusive prompts (i.e., the increasing level of gesture, then indirect verbal, then direct verbal, and finally modeling) were used if the student failed to respond to a previous prompt and initiate the step within 10 s.

Concrete fraction blocks

In the concrete fraction blocks intervention condition, a student was given a probe, two sets of identical concrete fraction tower manipulatives, and a pencil. During concrete sessions, the blocks were placed in front of the student for their use. Students were expected to set up the one block (i.e., one whole) for each problem, set up the problem using the correct concrete fraction tower pieces, find the common denominator, use the concrete fraction tower blocks to find the equivalent fraction(s), and solve the five addition of fractions with unlike denominators problems. Students wrote the correct answers on the probe.

App-based fraction tiles

In the app-based fraction tiles intervention condition, students used the Fraction Tiles app (Brainingcamp, 2017) to solve the five addition of fractions with unlike denominators problems. At the start of each session, the researcher gave the student the iPad with the app open, the probe sheet with the five problems, and a pencil. To use the app, students touched the fraction piece they wanted (e.g., $\frac{1}{3}$ ) and dragged out the needed number (i.e., 2 one thirds). Similar to the concrete fraction block condition, students were expected to set up the digital one block (i.e., one whole) for each problem, set up the problem using the correct digital fraction tile pieces, find the common denominator, use the digital fraction pieces to find the equivalent fraction(s), and solve the five addition of fractions with unlike denominators problems. Students wrote the correct answers on the probe sheet.

No manipulative

The third condition in intervention was an extended baseline in which no manipulative was provided to solve the five addition of fractions with unlike denominators problems. At the start of each no manipulative condition session, researchers provided the student with the five-problem probe sheet and a pencil to solve independently.

Best treatment

After 15 intervention sessions, each student completed three sessions in the best treatment phase. The best treatment (i.e., concrete or app-based fraction manipulatives) for each student was determined by calculating the percent of nonoverlapping data (PND; Gast & Spriggs, 2014). The researchers first determined PND for the dependent variable of accuracy; if there was no difference in accuracy, researchers calculated PND for independence. To calculate PND, the researchers determined when the number of sessions in one intervention condition (e.g., app-based fraction tiles) was superior to the other (e.g., concrete fraction blocks) and then divided by the number of comparison session (five; Wolery et al., 2014). For all three students, best treatment was calculated on independence data. During the three-session best treatment phase sessions, students were given the probe sheet, a pencil, and the manipulative calculated as best treatment. The procedures during best treatment mirrored those during intervention (e.g., manipulatives provided in conjunction with system of least prompts) and the same data were collected (i.e., accuracy, task completion time, and independence).

Generalization

Following the best treatment phase, each student completed two generalization sessions 2 weeks after his or her last best treatment session. The generalization phase used the same procedures as baseline.

The IOA and Treatment Fidelity

The IOA data were collected for each student for one baseline session, two intervention sessions in each condition, one best treatment session, and one generalization session. The IOA collected for all participants for all phases of the study for the dependent variables of accuracy. The IOA for independence was not collected during baseline, the extended baseline during intervention, or generalization, as no prompts were offered during these phases. For IOA accuracy data collection, a second researcher examined the first researcher’s assessment of correct or incorrect for each problem in a session; IOA was calculated by dividing the number of agreements by the sum of agreements plus disagreements. For independence IOA, a second researcher sat with the first researcher and the student and collected the same data (i.e., the system of least prompts) via a recording sheet. To calculate IOA for independence, researchers summed the number of agreements for each session on the data sheet and divided that number by the total possible times for agreements (i.e., agreements plus disagreements). For each student for all phases, IOA was 100% for accuracy. For Kelly, IOA for independence for concrete manipulatives was 97.5%, 95.7% for virtual manipulatives, and 100% for best treatment. Independence IOA for Miles was 95.2% for concrete manipulatives, 96.8% for virtual manipulatives, and 100% for best treatment. Doug’s IOA for independence was 95.7% for concrete manipulatives, 98.9% for virtual manipulatives, and 100% for best treatment.

Researchers recorded treatment fidelity data for 40% of intervention sessions for each condition and 33.3% of best treatment sessions. To assess treatment fidelity, researchers monitored if students were provided a probe sheet and pencil, the appropriate type of manipulatives dependent on condition, if students used the manipulative provided during the session, and if the researcher implemented the system of least prompts within 10 s. For each student for all phases and conditions, treatment fidelity was 100%.

Social Validity

Researchers conducted social validity interviews with each student at the conclusion of the intervention. The students were asked questions regarding which type of manipulative they liked the best, as well as if they preferred to solve the mathematics without any manipulatives.

Data Analysis

Researchers analyzed the task independence, accuracy, and task completion time data by conducting visual analysis of the graphed data. Researchers also determined the level, trend, and effect sizes for the dependent variables data (Gast & Spriggs, 2014). To calculate the level, the researchers found the median for each dependent variable for each student for each phase as well as condition of a phase. Then the researchers computed the stability envelope, which is a 25% interval around the median; data were stable if 80% of the data for that variable for that phase (or condition within a phase) fell within the stability envelope (Gast & Spriggs, 2014). To calculate the trend, researchers opted for the split-middle method, which involved finding the middle, mid-rate, and mid-date for each dependent variable for each phase or condition within a phase and then drawing a line between the mid-rate and mid-date (White & Haring, 1980). The researchers then evaluated the data in terms of acceleration, deceleration, or zero-celeration (Gast & Spriggs, 2014). To calculate an effect size, the researchers used Tau-U for accuracy and task independence data (Parker, Vannest, Davis, & Sauber, 2011). Researchers used the Tau-U Web-based calculator (see http://www.singlecaseresearch.org/calculators/tau-u; Vannest, Parker, & Gonen, 2011) and determined whether Tau-U scores were less than 65% (small effect), between 66% and 92% (medium effect), or greater than 93% (large effect; Parker, Vannest, & Brown, 2009).

Results

Kelly

During baseline, Kelly answered zero problems correctly for any session; her baseline data were stable with a zero-celeration trend (see Figure 2 and Table 1). In intervention, Kelly answered 100% of the problems correctly with both the concrete and app-based manipulatives; both conditions were stable with a zero-celeration trend. During the no manipulative condition, Kelly’s data were variable (range 0–3; µ = 2), with a zero-celeration trend. During the best treatment phase, Kelly was 100% accurate for each session with the app-based manipulatives. Her accuracy during generalization was higher than during baseline or the no manipulative condition during intervention (µ = 4.5).

Figure 2.

Accuracy (of five) for adding fractions.

Table 1.

Participant Accuracy, Independence, and Task Complete Time Data Across Phases.

				Intervention
Participant	DV	Data	Baseline	Concrete Manipulative	App-Based Manipulative	No Manipulative	Best Treatment	Generalization
Kelly	Accuracy	µ	0	5	5	2	5	4.5
		Level	Stable	Stable	Stable	Stable	Stable	Stable
		Trend	Zero-celeration	Zero-celeration	Zero-celeration	Zero-celeration	Zero-celeration	Acceleration
	Independence	µ	—	93.4%	95.9%	—	100%	—
		Level	—	Stable	Stable	—	Stable	—
		Trend	—	Deceleration	Deceleration	—	Zero-celeration	—
	Time	µ	1:02	4:30	5:36	3:34	3:43	3:52
		Level	Variable	Variable	Variable	Variable	Stable	Stable
		Trend	Acceleration	Deceleration	Acceleration	Deceleration	Acceleration	Slight deceleration
Miles	Accuracy	µ	0	5	5	1	5	1
		Level	Stable	Stable	Stable	Variable	Stable	Stable
		Trend	Zero-celeration	Zero-celeration	Zero-celeration	Acceleration	Zero-celeration	Zero-celeration
	Independence	µ	—	98.0%	96.9%	—	99.0%	—
		Level	—	Stable	Stable	—	Stable	—
		Trend	—	Acceleration	Zero-celeration	—	Zero-celeration	—
	Time	µ	0:50	5:46	5:31	4:48	3:39	1:43
		Level	Stable	Variable	Stable	Stable	Stable	Stable
		Trend	Acceleration	Deceleration	Slight deceleration	Acceleration	Slight acceleration	Slight deceleration
Doug	Accuracy	µ	0	5	5	4.4	5	4
		Level	Stable	Stable	Stable	Stable	Stable	Stable
		Trend	Zero-celeration	Zero-celeration	Zero-celeration	Acceleration	Zero-celeration	Zero-celeration
	Independence	µ	—	95.4	96.1	—	99.1	—
		Level	—	Stable	Stable	—	Stable	—
		Trend	—	Zero-celeration	Deceleration	—	Acceleration	—
	Time	µ	0:43	10:28	8:48	2:55	5:53	2:56
		Level	Stable	Variable	Stable	Variable	Variable	Variable
		Trend	Zero-celeration	Acceleration	Acceleration	Acceleration	Deceleration	Deceleration

Kelly needed few prompts and was basically independent in using both manipulatives during intervention (see Figure 3). With concrete manipulatives, her average for independence was 93.4% and for app-based manipulatives it was 95.9%. The independence data for both conditions were stable but with decelerating trends. The PND for app-based manipulatives to concrete manipulatives was 60% (concrete to app-based 20%); best treatment was app-based manipulatives. Kelly was 100% independent in each of her best treatment sessions.

Figure 3.

Independence (completion of task analysis steps without prompting) data.

Kelly completed the baseline sessions quickly, averaging 1:02 per session. Kelly’s task completion time increased during the no manipulative extended baseline condition during intervention (µ = 3:43). During the concrete manipulatives and virtual manipulatives sessions, her average task completion time was 4:30 and 5:36, respectively. The data for concrete manipulatives had a decelerating trend and were variable; the data for app-based manipulatives had an accelerating trend and were also variable. Kelly became more efficient during the best treatment app-based manipulative sessions (µ = 3:43) and her generalization task completion times were similar to the no manipulative intervention condition (µ = 3:52).

Miles

Miles answered zero problems correctly for any baseline session; his baseline data were stable with a zero-celeration trend (see Figure 2 and Table 1). During intervention with both concrete and app-based manipulatives, Miles answered 100% of the problems correctly resulting in both conditions having stable and zero-celeration data. Miles averaged one answer correctly per session during the no manipulative intervention condition (range 0–4). The no manipulative accuracy data were variable with an accelerating trend. During his best treatment phase—concrete manipulatives—Miles was 100% accurate for each session. His accuracy during generalization was low; one answer correct for each of the two sessions.

Miles needed few prompts and was increasingly independent as the sessions progressed during intervention (see Figure 3). His average independence was 98.0% for concrete manipulatives and 96.9% for app-based manipulatives. The data for both were stable; concrete manipulatives had an accelerating trend and app-based manipulatives a zero-celeration trend. The PND for concrete manipulatives to app-based manipulatives was 40% (app-based to concrete was 0%; they were equal 60%). During the concrete manipulative best treatment sessions, Miles only needed one prompt on one session; the other two were 100%.

Miles completed his baseline sessions quickly (µ = 0:50 per session). On average, his task completion time was comparable between concrete manipulatives (µ = 5:46, range 3:26–7:55) and app-based manipulatives (µ = 5:31, range 3:39–9:42). His task completion time data for concrete manipulatives were variable with a decelerating trend. His task completion time data for app-based manipulatives were stable with a slight decelerating trend. Miles’s average task completion time during the no manipulative intervention condition was 4:47. During best treatment, Miles’s average task completion time was 3:41 and during generalization it was 1:43.

Doug

During baseline, Doug answered zero problems correctly; his baseline was stable with a zero-celeration trend (see Figure 2 and Table 1). During intervention, Doug answered 100% of the questions correctly with both the app-based and concrete manipulatives and an average of 4.4 (range 4–5) questions correctly during the no manipulative intervention condition. Data for all three conditions were stable; the trends for concrete and app-based manipulatives were zero-celeration and for no manipulatives was acceleration. During the best treatment phase, Doug answered all five questions correctly for each of the three sessions. Finally, during generalization, Doug answered four questions correctly for each of the two sessions.

Overall, Doug was independent in both conditions (µ = 95.4% for concrete manipulatives and µ = 96.1% for app-based manipulatives; see Figure 3). The independence data for both conditions were stable; the trend for concrete manipulatives was zero-celeration and deceleration for app-based manipulatives. The PND for concrete manipulatives to app-based manipulatives was 60% (20% for app-based to concrete); concrete manipulatives were determined to be the best treatment. During best treatment, Doug was independent (µ = 99.1%). Only for the first best treatment phase did Doug need one prompt; for the last two he was 100% independent.

During baseline, Doug completed sessions quickly (µ = 0:43). His average intervention task completion time was 10:29 in the concrete manipulatives condition, 8:48 in the app-based manipulatives condition, and 2:55 in the no manipulative condition. The data for both the concrete and no manipulatives condition were variable; the data were stable for the app-based manipulative condition. For all three conditions, the trend for data was acceleration. During best treatment, Doug average task completion time was 5:53 and during generalization it was 2:56.

Social Validity

All three students reported enjoying both the app-based manipulatives and the concrete blocks, although Kelly expressed a strong preference for the app-based manipulative. She felt the app-based manipulative was easier and she would choose that option if allowed; Kelly would smile when she was told the session would involve using the app-based manipulatives. Miles, too, would elect to use the app-based manipulative if allowed, but he felt both were fun. He indicated he preferred the app-based manipulatives as it was easier than taking apart blocks (i.e., “just hit clear” [on the iPad]). Doug did not indicate a strong preference, but as the study progressed and he moved from using both types of manipulatives to set up and solve the problem to using the manipulatives to check his answer after he had attempted to solve the problem abstractly. At the end of the study, Doug expressed a preference to use any type of manipulative a means of checking his accuracy.

Discussion

This study explored student use of app-based fraction tiles (i.e., delivered on an iPad) in comparison to concrete fraction blocks for solving problems involving addition of fractions with unlike denominators. Researchers compared the manipulatives in terms of student task independence in completing the problems with the manipulatives, accuracy, and task completion time. The results of the single case adapted alternating treatment design study with three middle school students—two with mild intellectual disability and one with learning disabilities—suggested both types of manipulatives were effective in terms of accuracy and differences were minimal with regard to task independence.

The results were idiosyncratic in terms of the most effective manipulative relative to independence: Two were more independent with the concrete manipulatives and one with the app-based manipulatives. In terms of accuracy, all three students were successful with both manipulatives and were more successful in both conditions as compared to baseline and extended baseline during intervention. The effectiveness of the app-based manipulative observed in this study are consistent with previous research (cf., Bouck, Chamberlain, & Park, 2017; Satsangi & Bouck, 2015; Satsangi, Bouck, Taber-Doughty, Bofferding, & Roberts, 2016). The results further support the use of virtual manipulatives as a tool for students with disabilities; yet extend the exploration of app-based manipulatives as opposed to online manipulatives as well as virtual manipulatives for supporting fractions. Previous research comparing virtual and concrete manipulatives involved basic operations, linear algebra, and area and perimeter (Bouck, Chamberlain, & Park, 2017; Satsangi & Bouck, 2015; Satsangi et al., 2016).

All three students performed well in terms of accuracy and were generally independent, with the level of prompting decreasing across intervention sessions in both treatment conditions. However, Miles and Kelly struggled in extended baseline sessions (i.e., no manipulative condition). In addition, not all students were able to generalize their success with solving fraction addition problems involving unlike denominators when a manipulative was not provided. Miles struggled to correctly solve the addition of fractions with unlike denominators problems when he did not have a manipulative. In contrast, Kelly and Doug both eventually internalized the process of solving such problems, as demonstrated by their success during generalization. Perhaps the lack of explicit instruction during the intervention was problematic for Miles. Although the researchers provided explicit instruction during the pretraining portion with regard to using each manipulative type, explicit instruction on how to determine the common denominator and equivalent fractions without a manipulative was not a part of the training (i.e., it was implicit).

Implications for Practice

One implication for practice based on these results is the value in providing middle schools students with mild intellectual disability and learning disabilities manipulatives when solving problems involving adding fractions with unlike denominators. All three students were more effective in solving the problems with the manipulatives than during baseline. Hence, this study supports the existing literature recommending the use of manipulatives for students with disabilities as well as all students in general (Lai & Berkeley, 2012; Maccini & Gagnon, 2000; Marley & Carbonneau, 2014). Another implication is that app-based manipulatives can be a tool to support students with mild intellectual disability and learning disabilities in solving problems involving adding fractions with unlike denominators. In other words, while concrete manipulatives are the more commonly used tool, app-based manipulatives should be considered a potential support or accommodation for students with disabilities. Given the potential for stigmatization of concrete manipulatives for secondary students (Bouck, Chamberlain, & Park, 2017; Satsangi & Bouck, 2015), app-based manipulatives should be considered as a legitimate and more age-appropriate option, particularly given the lack of decline in accuracy. A final implication involves the need to provide explicit instruction in mathematics for adding fractions with unlike denominator problems. While Doug was able to internalize the mathematics from the two trainings as well as repeated implicit exposure with both type of manipulatives during intervention, Kelly and Miles were not. Kelly and Miles struggled more with solving the addition of fractions with, unlike denominators, problems without a manipulative during intervention. Kelly was, however, successful in solving problems without a manipulative during generalization.

Limitations and Future Directions

This study possesses multiple limitations. One limitation involves the lack of data collection of the task independence dependent variable (i.e., prompting) during baseline, no manipulative (i.e., extended baseline), and generalization conditions. Another limitation involves the inclusion of students with different types of disabilities (i.e., two students with mild intellectual disability and one student with a learning disability); however, all students were initially served in the same special education class and presented similar mathematical struggles and needs. Any student who met the inclusion criteria was included, despite a primary focus on students with mild intellectual disability, as per the federal Individuals with Disabilities Education Act (IDEA) law, services are not to be based on a disability category (Yell, 2012). A third limitation of the study was that the authors implemented the system of least prompts in conjunction with the two types of manipulatives. The authors were unable to separate the impact of the system of least prompts separately from the manipulatives. Additionally, the authors used the same fractions across both manipulative conditions; future research may consider teaching separate problems for each manipulative time. A final major limitation is the small sample size, which negatively affects generalizability of the findings. Although a sample size of three is appropriate for the single-case research design, it is still a small N.

Future research should explore teaching students to add fractions with unlike denominators via the CRA or the VRA instructional sequences, both which employ explicit instruction involving modeling and think alouds of how to solve problems with tools and mathematics (Doabler & Fien, 2013). Future research should also compare different types of app-based or concrete fraction manipulatives. In this study, the concrete manipulative was connecting fraction blocks but other options include fraction strips, fraction tiles, fraction circles, and fraction squares. Similarly, this study used the app-based manipulative of fraction tiles by Brainingcamp (2017) but other app-based or even Web-based (i.e., online) options exist. Related, future research should also seek to explicitly compare online virtual manipulatives to app-based virtual manipulatives. Beyond replication, researchers should seek to expand upon the results found in this study to explore the impact of concrete and app-based fraction manipulatives for students with other types of disabilities as well as with different age groups. Finally, researchers should seek to conduct studies with larger sample sizes, including undertaking studies that involve group experimental design. Related, given the small sample size in this study—and many of the studies involving virtual manipulatives—researchers should look collectively at the body of work regarding virtual manipulatives for students with disabilities and analyze virtual manipulatives as an evidence-based practice.

Footnotes

Declaration of Conflicting Interests

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

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The first author received a seed grant from the IRTL of the College of Education at Michigan State University to fund this project.

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