The Virtual-Representational-Abstract Framework to Support Students With Disabilities in Mathematics

Abstract

The virtual-representational-abstract (VRA) framework adapts an evidence-based practice, the concrete-representational-abstract (CRA) framework, while trying to meet students and teachers where they are in terms of technology interest and use in education. This article discusses the VRA framework and the advantages of virtual manipulatives, and explains how a teacher can implement the VRA framework. Although more research is needed on the VRA framework, the VRA provides teachers an option as an instructional practice or intervention to support students struggling in mathematics.

Keywords

mathematics disabilities technology instruction intervention academic content area

The virtual-representational-abstract (VRA) is a framework for mathematics instruction. It is an adaptation of the concrete-representational-abstract (CRA) framework, which is also referred to as concrete-semiconcrete-abstract (CSA) framework (Flores, Burton, & Hinton, 2018). The CRA is an evidence-based practice for students with learning disabilities and often recommended as an instructional practice for students with disabilities and students receiving intensive intervention services within a response-to-intervention (RTI) framework (Agrawal & Morin, 2016; Bouck, Satsangi, & Park, 2017; National Center on Intensive Intervention, 2016; Powell, 2015). The CRA framework is taught via explicit instruction, an evidence-based practice itself (Doabler & Fien, 2013; Flores et al., 2018; National Center on Intensive Intervention, 2016).

Explicit Instruction

Explicit instruction within the CRA framework involves the teacher first modeling the mathematics for a few problems, then providing prompts and cues as needed while the student works to solve a few mathematical problems, and finally having the student independently solve some problems (Doabler & Fien, 2013). Throughout the modeling stages of explicit instruction, the teacher not only demonstrates how to solve the problems but also uses think-alouds (i.e., verbally narrates the steps; Agrawal & Morin, 2016). The teacher uses clear and consistent mathematical language during his or her think-aloud. The guiding phase, in which prompts or cues are given for when a student is stuck and/or makes a mistake, allows the teacher to provide immediate feedback to the student. The independent portion, in which students solve the problems without any assistance or prompts from the teacher, serves as a means of evaluating student understanding (Doabler & Fien, 2013).

Concrete-Representational-Abstract

The CRA framework is a graduated sequence in which instruction moves from a concrete phase to a representational phase and finally to an abstract phase (see Figures 1 & 2). In the concrete phase, students use concrete manipulatives (e.g., base 10 blocks, fraction tiles) to solve problems during all stages of explicit instruction: (a) modeling, (b) guiding, and (c) independence. In the representational phase, students use drawings or pictorial representations to solve the problems (e.g., lines and dots to represent the tens blocks and ones blocks, respectively, of base 10 blocks, or rectangles equally divided into the respective number of pieces to represent fraction tiles). Finally, in the abstract phase, students use just mathematical strategies to solve problems (Agrawal & Morin, 2016; Doabler & Fien, 2013). Students transition between phases in a systematic way, such as by obtaining 80% correct for at least three independent sessions or achieving a particular fluency rate for a minimum number of sessions (Flores, 2009; Mancl, Miller, & Kennedy, 2012).

Figure 1.

The concrete and virtual phases of CRA and VRA frameworks. The virtual manipulative is the Fraction Tiles by Brainingcamp (2017c).

Figure 2.

The representational and abstract phases of CRA and VRA frameworks.

As noted, the CRA is an evidence-based practice for students with learning disabilities (LD) when considering mathematics involving basic operations (e.g., subtraction, multiplication), meaning a sufficient number of high-quality articles demonstrate its efficacy (Bouck, Satsangi, & Park, 2017). A recent systematic review, highlights the long history of research regarding the CRA framework, suggesting its effectiveness for students with disabilities (Bouck & Park, 2018). In the existing literature on the CRA framework, which spans basic operations, fractions, place value, algebra, and geometry, the findings were overwhelming positive regarding the functional relation between use of the CRA framework and students’ mathematical performance or a positive change in students’ mathematical scores from pretest to posttest (Bouck & Park, 2018; Bouck, Satsangi, & Park, 2017).

Virtual-Representational-Abstract

Virtual-representational-abstract (VRA) is an emerging framework for mathematical instruction for students with disabilities (see Figures 1 & 2). As stated, the VRA is an adaptation of the CRA framework. In the VRA framework, virtual manipulatives are used in place of concrete manipulatives; the remaining phases—representational and abstract—are the same as in the CRA framework. Using the VRA framework, students first solve problems with virtual manipulatives. Next, students transition to solving problems using pictorial representations or drawings and, finally, to solving the problems abstractly with mathematical strategies (Bouck, Bassette, et al., 2017). Similar to the CRA framework, each phase of the VRA framework is taught via explicit instruction, with the modeling, guiding, and independent stages, and a predetermined criterion is set for transitioning from one phase to the next such as an accuracy percentage or fluency rate (Bouck, Bassette, et al., 2017).

Each phase stands on its own and is different from the others. The virtual phase differs from the representation phase, as with virtual manipulatives the representation (i.e., base 10 blocks, fraction tiles, algebra tiles) already exists and students are not creating it, such as with the representational stage when they are drawing. The virtual manipulatives can be manipulated, including being rotated and also appear as three-dimensional objects (e.g., base 10 blocks), which is not the case in the representational phase. The abstract phase differs from both phases in that just the mathematical strategies are emphasized, such as counting up for subtraction, counting on for addition, or repeated addition for single-digit multiplication. The transition across phases is evident through the use of explicit instruction, particularly the modeling phase. During the modeling portion of each phase, the teacher or interventionist discusses the tools the student be will using the solve the problem—a virtual manipulative, a drawing, or just mathematical strategies. In the virtual phase students are explicitly told to use the app, in the representational phase students are explicitly told to draw pictures, and in the abstract phase students are explicitly told they are going to try to solve the mathematical problem just using mathematical strategies.

Research

At this time, an insufficient number of high-quality research studies exist to suggest the VRA is an evidence-based practice. Yet, the emerging research is positive. Bouck, Park, Shurr, Bassette, and Whorley (2018) explored the VRA framework to support place value (e.g., what is the value of the underlined number: 67), addition (e.g., 17 + 5), subtraction (e.g., 21 – 3), and multiplication (e.g., 3 × 4) for two middle school students with mild intellectual disability using a multiple probe across behaviors design. For the mathematical behaviors for both students, a functional relation existed between use of the VRA and each student’s mathematical accuracy. Similarly, Bouck, Bassette, et al. (2017) explored the VRA framework to support middle school students with disabilities in solving equivalent fractions problems (e.g., $\frac{1}{4} = \frac{}{12}$ ). In a multiple probe across participants design, the authors found a functional relation between the VRA framework and students’ mathematical performance relative to solving for equivalent fractions.

Virtual Manipulatives

As noted, the VRA framework differs from the CRA framework in one dimension, the use of virtual manipulatives as opposed to concrete manipulatives. Virtual manipulatives are typically digital versions of concrete manipulatives; concrete manipulatives being physical objects students manipulate to help them understand and explore mathematical problems or ideas (Bouck & Flanagan, 2010). Examples of concrete manipulatives include for-purchase products, such as base 10 blocks, fraction blocks, algebra tiles, and cubes, as well as everyday items such as beans, cups, straws, plates (Bouck & Flanagan, 2010). Virtual manipulatives are interactive digital objects students can manipulate to also explore and understand mathematical ideas (Moyer, Bolyard, & Spikell, 2002). Virtual manipulatives refer to both online (i.e., web-based) digital representations of concrete manipulatives and app-based digital versions of concrete manipulatives (i.e., apps for mobile devices). A variety of online and app-based virtual manipulatives exist. See Table 1 for examples of virtual manipulatives.

Table 1.

Online and App-Based Virtual Manipulatives Examples.

Virtual manipulatives	Description
National Library of Virtual Manipulatives, http://nlvm.usu.edu/	• Free • Web-based • Covers grades PK–12 and the mathematic areas of numbers and operations, algebra, geometry, measurement, and data analysis and probability
Illuminations from the National Council of Teachers of Mathematics, https://illuminations.nctm.org/content.aspx?id=3855	• Free • Web-based and apps • Games and manipulatives cover grades PK–12
Brainingcamp, http://brainingcamp.com/manipulatives/index.html	• Individual apps typically $0.99, bundled apps $9.99, and a class account for $95 per year • Web-based and apps, including for Chromebooks • Includes base 10 blocks, fractions, number line, pattern blocks, algebra tiles, and geoboards
The Math Learning Center, https://www.mathlearningcenter.org/resources/apps	• Free • Web-based and apps • Includes fractions, geoboards, number frames, money, number lines, base 10 blocks, pattern blocks

The rationale for the VRA framework lies with the advantages and benefits of virtual manipulatives as compared with concrete manipulatives. One advantage of virtual manipulatives is their potential to minimize stigmatization, particularly as students age (Satsangi & Bouck, 2015). Manipulatives are commonly used in elementary-aged mathematics instruction but are less common in secondary classrooms. When an older student needs a manipulative for earlier mathematical content (e.g., double- or triple-digit subtraction or equivalent fractions), she or he may feel embarrassed to use a concrete manipulative. Virtual manipulatives, particularly app-based manipulatives, are more discreet and involve students using a tablet or other mobile device. Hence, virtual manipulative may be more socially desirable and hence valued by peers (Bouck, Flanagan, Miller, & Bassette, 2012). Another benefit of virtual manipulatives, particularly those that are app based, is their portability (Satsangi & Miller, 2017). If the manipulative is portable, a student can use it in a variety of settings, including general education classes, special education classes, and even at home.

Virtual manipulatives also support greater student independence, as many have inherent prompts, features, and scaffolds that decrease the need for a teacher to provide them (Bouck, Chamberlain, & Park, 2017; Shinn et al., 2017). As an example, one app-based manipulative, Base 10 Blocks by Brainingcamp (2017a), does not allow students to regroup from the subtrahend (i.e., the number one is subtracting) instead of the minuend (i.e., the number one starts with) when solving subtraction problems. With concrete manipulatives, a teacher or another individual would need to prompt a student not to regroup from the subtrahend or else the student would continue to make the mistake. Related, virtual manipulatives can decrease students’ cognitive load as compared with concrete manipulatives (Suh & Moyer, 2007). As an example, many virtual manipulatives keep the mathematics problem on the same screen as the manipulative (Bouck, Working, & Bone, 2017). Finally, some virtual manipulatives allow teachers to monitor students’ mathematical progress within the tool itself (Shinn et al., 2017).

Another benefit with virtual manipulatives is, in theory, less storage. While an unlimited number of tiles or blocks exist within an app or online manipulative, with concrete manipulatives, large numbers can quickly become unwieldly as well as consume limited space in the classroom. A potential challenge with virtual manipulatives, however, is students must possess sufficient fine motor skills to operate a touchscreen or a mouse. It is worth noting that to be independent in the CRA framework, students would also need fine motor skills to manipulative concrete manipulatives as well as to draw pictorial representations. Cost is another area of comparison between the two types of manipulatives. While concrete manipulatives can be expensive to purchase in classroom sets, apps or online virtual manipulatives can be free or relatively inexpensive. Nonetheless, virtual manipulatives carry the hidden cost of securing the appropriate hardware or access to the Internet to successfully use the tool.

Existing research that compares the use of virtual manipulative to the use of concrete manipulatives for students with disabilities also suggests a general preference for virtual manipulative for students with disabilities. Bouck, Satsangi, Doughty, and Courtney (2014) reported elementary-aged students with autism preferred to use the web-based virtual base 10 blocks manipulative as opposed concrete manipulatives. Several researchers (Bouck, Chamberlain, & Park, 2017; Bouck, Shurr, Bassette, Park, Kerr, & Whorley, 2018; Satsangi, Bouck, Taber-Doughty, Bofferding, & Roberts, 2016) have reported the majority of the secondary students with learning disabilities or mild intellectual disability preferred the virtual form of the manipulative (e.g., algebra balance scale, base 10 blocks, and fraction tiles, respectively) to the concrete. In these studies, the virtual manipulatives as compared to the concrete manipulatives were equally, if not more, effective for the students with disabilities in terms of their mathematical accuracy and/or independence in solving the problems.

Implementation

Although the limited existing literature on the VRA framework focuses on middle school students, the VRA framework can be implemented with almost every age group, including elementary and secondary students. As previously suggested, the benefits of the VRA as compared to CRA may be even more salient for secondary students, given the potential for concrete manipulatives to be viewed as stigmatizing to this population (Bouck, Chamberlain, & Park, 2017; Satsangi & Bouck, 2015). The VRA framework can also be used, in theory, with any high-quality virtual manipulative app or website (see Table 1). While a discussion of how to select a virtual manipulative is not included in this article, information regarding app selection for virtual manipulatives is available (Bouck, Working, & Bone, 2017). To date, the research on the VRA has explored apps by Brainingcamp (2017b), and current research comparing concrete and virtual manipulatives have used Brainingcamp apps and the National Library of Virtual Manipulatives.

To implement the VRA framework, a teacher must first determine the mathematical strengths and challenges of the students she is targeting for the intervention. Although the VRA framework can be implemented at a whole-class level, it is most advantageous at the small-group or individual level. The VRA framework, like the CRA framework, has limited research exploring the implementation of the framework outside of small-group or one-one-one instruction. Consistent with Tier 2 or Tier 3 instructional practices within RTI, once appropriate one-on-one pairings or small groups exist based on commonalities of struggles (e.g., single-digit addition with regroup, triple-digit subtraction with regrouping, fraction equivalence), the teacher can begin instruction. With a whole-class or small-group implementation, the teacher needs to differentiate the phase for each student, allowing each student to move at his or her appropriate pace from the virtual to the representational to the abstract phase. In other words, the transition from each phase should be unique for each student based on mastery within the phases.

Each lesson for the VRA begins similarly, with the teacher providing an advanced organizer, which involves the teacher explaining what the students will be learning, why it is important, and expectations (Flores et al., 2018). Using the example of subtraction with regrouping, a teacher’s advanced organizer might begin with the following:

Today, we are going to work on subtraction with regrouping. Subtraction is about finding the difference between two amounts. Sometimes with subtraction we are comparing two amounts and figuring out a change, sometimes we are taking away one amount from another and thus determining the difference between those two amounts, and sometimes we are trying to find a part of a whole. Subtraction, like addition, is central to knowing, doing, and using mathematics to answer questions in our everyday life. To solve these subtraction problems, we are going to need to understand numbers and what we mean by the tens and ones places. We are also going to need to remember how to count on from a number. For today’s lesson, I expect you to pay attention and try your hardest.

The teacher then uses explicit instruction, transitioning from modeling two or three problems to guiding students through the problems as needed. Finally, students solve a set number of problems independently. Figure 3 provides an example think-aloud (i.e., verbal narration) of how to solve an addition with regrouping problem. The teacher models and guides in the same way during the representational and abstract phases. Figure 4 provides another example of an implementation model, considering more advanced mathematical skills.

Figure 3.

Think-aloud example of the VRA framework with explicit instruction: Base 10 blocks.

Figure 4.

Think-aloud example of the VRA framework with explicit instruction: Algebra tiles.

The teacher transitions each student from one phase (e.g., virtual) to the next (e.g., representational) when the criterion is met for transition from one phase to the next (e.g., 80% accuracy across three sessions or X number of digits correct on a timed independent probe). When the transition occurs, the teacher needs to be explicit about the tool being using (e.g., an app, the students drawing, or just mathematical strategies) during the think-aloud and their own modeling of how to solve the problems. As noted, the research to date on the VRA has focused on individual administration of the VRA framework. To administer in a small group or a whole class, the teacher would need to demonstrate solving the problem with each phase in which all included students are currently operating (e.g., virtual, representational abstract), and then be explicit to each student the tool she or he is individually using.

Conclusion

The VRA framework capitalizes on the CRA evidence-based practice (Bouck, Satsangi, & Park, 2017), while simultaneously addressing student interest in technology and its increasing role and use in education. Virtual manipulatives are often available for free or at a low cost. They are engaging and shown to be just as effective in terms of independence and accuracy as concrete manipulatives, while offering the potential for less stigmatization, increased independence, and inherent scaffolding (Bouck, Chamberlain, & Park, 2017; Bouck & Flanagan, 2010; Satsangi & Bouck, 2015; Satsangi & Miller, 2017; Shinn et al., 2017). Although more research is needed on the VRA framework, the VRA provides teachers an additional option as an instructional practice or intervention to support students struggling in mathematics.

Footnotes

Authors’ Note

Neither author is affiliated with any company or product discussed in this column.

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.

References

Agrawal

Morin

L. L.

(2016). Evidence-based practices: Applications of concrete representational abstract framework across math concepts for students with disabilities. Learning Disabilities Research & Practice, 31, 34–44. doi:10.111/ldrp.12093

Bouck

E. C.

Bassette

Shurr

Park

Kerr

Whorley

(2017). Teaching equivalent fractions to secondary students with disabilities via the virtual-representational-abstract instructional sequence. Journal of Special Education Technology. Advance online publication . doi:10.1177/0162643417727291

Bouck

E. C.

Chamberlain

Park

(2017). Concrete and app-based manipulatives to support students with disabilities with subtraction. Education and Training in Autism and Developmental Disabilities, 52, 317–331.

Bouck

E. C.

Flanagan

S. M.

(2010). Virtual manipulatives: What are they and how teachers can use them? Intervention in School and Clinic, 45, 186–191.

Bouck

Flanagan

Miller

Bassette

(2012) Technology in action. Journal of Special Education Technology, 27(4), 47–57.

Bouck

E. C.

Park

(2018). A systematic review of the literature on mathematics manipulatives to support students with disabilities. Education and Treatment of Children, 41, 65–106.

Bouck

E. C.

Park

Shurr

Bassette

Whorley

(2018). Using the virtual-representational-abstract approach to support students with intellectual disability in mathematics. Focus on Autism and Other Developmental Disabilities. Advance online publication. doi: 10.1177/1088357618755696

Bouck

E. C.

Satsangi

Doughty

T. T.

Courtney

W. T.

(2014). Virtual and concrete manipulatives: A comparison of approaches for solving mathematics problems for students with autism spectrum disorder. Journal of Autism and Developmental Disabilities, 44, 180–193.

Bouck

E. C.

Satsangi

Park

(2017). The concrete-representational-abstract approach for students with learning disabilities: An evidence-based practice synthesis. Remedial and Special Education. Advance online publication. doi:10.1177/0741932517721712

10.

Bouck

E. C.

Shurr

Bassette

Park

Whorley

(2018). Adding it up: Comparing concrete and app-based manipulatives to support students with disabilities with adding fractions. Journal of Special Education Technology, Advance online publication. doi: 10.1177/0162643418759341

11.

Bouck

E. C.

Working

Bone

(2017). Manipulative apps to support students with disabilities in mathematics. Intervention in School and Clinic. Advance online publication. doi:10.1177/1053451217702115

12.

Brainingcamp. (2017a). Base Ten Blocks Manipulative (Version 1.1.0) [Mobile application software]. Retrieved from https://www.brainingcamp.com/manipulatives/base-ten-blocks/

13.

Brainingcamp. (2017b). Brainingcamp. Retrieved from https://www.brainingcamp.com/

14.

Brainingcamp. (2017c). Fraction Tiles Maniplatives. Retrieved from, https://itunes.apple.com/us/app/fraction-tiles-manipulative/id1111230448?mt=8

15.

Doabler

C. T.

Fien

(2013). Explicit mathematics instruction: What teachers can do for teaching students with mathematics difficulties? Intervention for School and Clinic, 48, 276–285. doi:10.1177/1053451212473141

16.

Flores

M. M.

(2009). Teaching subtraction with regrouping to students experiencing difficulty in mathematics. Preventing School Failure, 53, 145–152. doi:10.3200/PSFL.53.3.145-152

17.

Flores

Burton

Hinton

(2018). Making mathematics accessible for elementary students who struggle: Using CRA/CSA for interventions. San Diego, CA: Plural.

18.

Mancl

D. B.

Miller

S. P.

Kennedy

(2012). Using the concrete-representational-abstract sequence with integrated strategy instruction to teach subtraction with regrouping to students with learning disabilities. Learning Disabilities Research & Practice, 27, 152–166. doi:10.1111/j.1540-5826.2012.00363.x

19.

Moyer

P. S.

Bolyard

J. J.

Spikell

M. A.

(2002). What are virtual manipulatives? Teaching Children Mathematics, 8, 372–377.

20.

National Center on Intensive Intervention. (2016). Principles for designing intervention in mathematics. Washington, DC: Office of Special Education, U.S. Department of Education. Retrieved from http://www.intensiveintervention.org/sites/default/files/Princip_Effect_Math_508.pdf

21.

Powell

S. R.

(2015). Connecting evidence-based practice with implementation opportunities in special education mathematics preparation. Intervention in School and Clinic, 51, 90–96. doi:10.1177/1053451215579269

22.

Satsangi

Bouck

E. C.

(2015). Using virtual manipulative instruction to teach the concepts of area and perimeter to secondary students with learning disabilities. Learning Disability Quarterly, 38, 174–186.

23.

Satsangi

Bouck

E. C.

Taber-Doughty

Bofferding

Roberts

C. A.

(2016). Comparing the effectiveness of virtual and concrete manipulatives to learn algebra for secondary students with learning disabilities. Learning Disability Quarterly, 39, 240–253.

24.

Satsangi

Miller

(2017). The case for adopting virtual manipulatives in mathematics education for students with disabilities. Preventing School Failure: Alternative Education for Children and Youth. Advance online publication. doi:10.1080/1045988X.2016.1275505

25.

Shinn

Bryant

D. P.

Bryant

B. R.

McKenna

J. W.

Hou

M. W.

(2017). Virtual manipulatives: Tools for teaching mathematics to students with learning disabilities. Intervention in School and Clinic, 52, 148–153. doi:10.1177/1053451216644830

26.

Suh

J. M.

Moyer

P. S.

(2007). Developing students’ representational fluency using virtual and physical algebra balances. Journal of Computers in Mathematics Science and Teaching, 26, 155–173.