Manipulative Apps to Support Students With Disabilities in Mathematics

Abstract

Understanding mathematical concepts is important for all students, although often challenging for many students with disabilities. Historically, educators have used concrete manipulatives to support and build conceptual understanding. Mobile devices provide a valuable option to support students with disabilities in mathematics through app-based manipulatives. Although research is limited on app-based manipulatives, the emerging literature with virtual (i.e., digital) manipulatives more generally suggests student preference for virtual manipulatives without a loss of understanding. This column provides educators with information about app-based manipulatives and how to use them, with the goal of helping teachers make informed decisions about app-based manipulatives to support students.

Keywords

mathematics skills technology instruction assistive manipulatives apps

Manipulatives are a common instructional tool in general education mathematics classrooms (Carbonneau, Marley, & Selig, 2013). Manipulatives are also an accommodation provided to students with disabilities in mathematics and are perceived to be a benefit for this population in learning mathematics (Lai & Berkeley, 2012; Maccini & Gagnon, 2000). Typically in mathematics, manipulatives refer to concrete manipulatives. Concrete manipulatives are physical objects students can manipulate to develop a conceptual understanding of the mathematics being taught, such as base-10 blocks, tiles, and fraction circles (Bouck & Flanagan, 2010). However, over the past 2 decades, manipulatives increasingly turned digital (Bouck & Flanagan, 2010). Virtual manipulatives are digital manipulatives students can manipulate on a computer or similar device (e.g., mobile device) to develop conceptual understanding of mathematics (e.g., virtual base-10 blocks, virtual tiles); virtual manipulatives are often digital representations of concrete manipulatives. A typical website for free virtual manipulatives is the National Library of Virtual Manipulatives (Utah State University, 1999–2016).

The Internet, while providing flexibility for students with disabilities and typical peers to access virtual manipulatives at home, is also a limitation. If a school experiences connectivity problems, including staying connected or Internet speed, leading to inhibited Internet access, web-based manipulatives are not accessible. Another limitation with web-based manipulatives is that many use either Java or Adobe Flash and require an installed and enabled plugin. This can cause problems depending on the device accessing the manipulatives. If the computer or device blocks out-of-date plug-ins, the manipulative will not be accessible. Further, many mobile devices, such as Chromebooks and iPads, do not support Java applications, and iPads are unable to use Flash-based web content (Haßler, Major, & Hennessy, 2016). Finally, reliability can become compromised due to the transient nature of web-hosted software. Depending on where the resource is hosted, access to the manipulatives may be affected by a suddenly discontinued website, new login requirement, or change from free to fee-based.

App-Based Manipulatives

Advancements in technology are providing alternatives to web-based virtual manipulatives. App-based manipulatives, akin to virtual manipulatives but existing on a mobile device as an app and hence requiring no Internet, provide another solution and option within the mathematics technology to support students with disabilities. App-based manipulatives installed on a device such as an iPad or Android tablet are standalone programs that do not require plugins or often even the Internet. Apps are upgraded more frequently but often continue to work without a software update; most apps do not pose a security risk to a device. Another positive feature is that app-based manipulatives can be installed in advance. A more practical reason for the use of app-based manipulatives in mathematics is that apps allow student interaction with the mobile device. The act of touching a virtual object engages imagery processing and mental simulation of the manipulative’s behavior, giving students an experience very similar to real-life interactions (Brasel & Gips, 2015).

Evaluation of Apps

App-based manipulatives offer teachers convenient and flexible access to digital manipulatives, provided they have mobile devices, but also present the challenge of how one selects an app. iTunes and Google Play are filled with apps for all purposes, including education, mathematics more specifically, and mathematics manipulatives in particular. It becomes important for educators to carefully select appropriate manipulative apps from the multitude that exist. See Table 1 for examples of manipulative apps. Educators can gain support in evaluating apps for implementation by using a rubric (Bouck, Satsangi, & Flanagan, 2016). A variety of different rubrics exist to support educators in evaluating apps for implementation for students with disabilities, including one explicitly for students with learning disabilities (Ok, Kim, Kang, & Bryant, 2016) and ones more generally for students with disabilities across multiple ages, categories, and app purposes (Tammaro & Jerome, 2012; Walker, 2010; Weng & Doughty, 2014; for a detailed examination of the different rubrics, please see Bouck et al., 2016). While these rubrics can be applied to evaluating and selecting app-based manipulatives, especially the one by Ok et al. (2016), educators may want to evaluate app-based manipulatives further.

Table 1.

App-Based Mathematics Manipulatives.

App and Device Compatibility	Information/Details
Base-10 Blocks Manipulative (Brainingcamp, LLC, 2016); iPad; web-based	• $0.99 • Supports addition and subtraction ones through thousands and thousandths through ones as well as place value • Can write on app like a smartboard • The two numbers (e.g., minuend, subtrahend) are different colors • Shows problem and answer • Can be purchased as part of a 10-app bundle for $5.99; others include fractions, color tiles, and algebra tiles • Provides scaffolds that prevent students from ungrouping from the subtrahend or exchanging one blocks for a tens block without 10 ones
Geoboard, by The Math Learning Center (Clarity Innovations, 2015); iPad; iPhone; iPod Touch; Chrome-based; web-based; Windows operating systems	• Free • Supports geometry ideas, such as shapes, perimeter and area, angles, and congruence • Resembles concrete geoboard with digital pegs and rubber bands • Shapes made can be shaded different colors • Allows grid lines to be numbered or not numbered
Hands-On Math Tangrams (Ventura Educational Systems, 2016); iPad	• $2.99 • Supports geometry concepts such as attributes of shapes, angles and measurement, and the meaning of fractions • Includes the classic seven tangram pieces, which can be moved, rotated, and grouped • Offers open-ended discovery as well as directed teaching modes
Number Line, by the Math Learning Center (Clarity Innovations, 2016); iPad; Chrome-based; web-based	• Free • Supports number relationships with counting, comparing, adding, subtracting, multiplying, and dividing • Number lines can be blank, whole numbers, fractions, decimals, or negative numbers • Includes pencil and line tool for annotating • Allows numbers on the number line to be hidden and includes frames for highlighting numbers
Shapes–3D Geometry Learning (Setapp Sp. z o. o., 2015); iPad; iPhone; iPod Touch	• $3.00 • Supports geometric concepts of faces, edges, and vertices • Allows deconstructing of 3D shapes into basic 2D shapes, unfolding of 3D shapes, and transparency of faces • Allows for highlighting and coloring of vertices, edges, and faces • Includes 27 common 3D shapes
Virtual Manipulatives! (ABCya.com, LLC, 2014); iPad; some aspects are web-based	• Free • Supports building fractional pieces, fractions larger than one whole, and finding equivalent fractions • Includes fractional pieces with decimals and pieces with percent • Shows fractions, decimals, and percent in either bars, rectangles, or circles (students select) • Numbers can be shown or hidden, and pencil tool allows for annotation

For example, when evaluating app-based manipulatives, educators may want to look for ones that are similar to concrete manipulatives so students have some familiarity (e.g., base-10 blocks and base-10 block app; tiles and digital tiles for area and perimeter). In addition, educators may also want to look for app-based manipulatives that provide additional affordances beyond concrete manipulatives, such as features that scaffold student learning or use of manipulative (i.e., a base-10 block app that does not allow a student to regroup from the subtrahend). The scaffolding ideally includes the possibility that the support from the manipulative can be lessened as students become more proficient as the ultimate goal is for students to be able to engage with the mathematics independently. A benefit of app-based manipulatives is their ability to be used in multiple environments, including supporting students with disabilities at home. Apps that are intuitive and/or provide tutorial options or help features allow students and their parents to independently use the app without a teacher.

App-based manipulatives are relatively new, with a limited research base. The apps highlighted in this article, both in Table 1 and throughout the text as examples and in the vignettes, were selected because they were appropriate for various age levels and disabilities, highlight the flexibility and use of app-based manipulatives across multiple mathematical domains (e.g., numbers and operations, geometry, measurement), and are available for free or low cost. In addition, the highlighted apps are consistent with the authors’ recommendations for educator considerations for app-based manipulatives: (a) familiarity (i.e., similar to concrete-based manipulative), (b) scaffold student learning beyond concrete manipulatives (e.g., do not allow students to incorrectly use the app), (c) flexibility within the app (e.g., lessen support and/or multiple use), and (d) are intuitive and/or provide tutorials to support independent student use. For example, the Number Line app (Clarity Innovations, 2016) allows educators to select the most appropriate number line for students based on their mathematical level (e.g., whole numbers, negative numbers, blank), which allows students to transition from using the app to being independent without the app. Teachers can also make annotations on the app and then save them so students can access this support when the teacher is not present.

Using App-Based Manipulatives in Practice

Teachers can use app-based manipulatives in a variety of ways. App-based manipulatives can be used in whole class instruction in general education and special education settings. If teachers are working in a 1:1 environment or have tablets for their classrooms, app-based manipulatives can be used in place of concrete manipulatives. App-based manipulatives can also be used as an option along with concrete manipulatives that students can choose, which supports the implementation of universal design for learning (UDL) in classrooms (i.e., multiple means of engagement; Council for Exceptional Children, 2005). Related to UDL, app-based manipulatives provide an option for students for whom the fine motor skills of working with concrete manipulatives pose challenges. For teachers in less technology-rich settings, app-based manipulatives can be used in centers or stations in both general education and special education settings, again as an alternative to or an option with concrete manipulatives.

App-based manipulatives can also be used to explicitly support students with disabilities or struggling students. Teachers or interventionists can use app-based manipulatives to support students at Tier 2 and Tier 3 as an alternative to concrete manipulatives, or as another option, students can select with concrete manipulatives. Virtual manipulatives could also be used in place of concrete manipulatives in the evidence-based, explicit instruction concrete-representational-abstract (CRA) teaching approach (Agrawal & Morin, 2016). Although research does not exist specifically examining the replacement of app-based manipulatives for concrete in the CRA approach, previous research comparing virtual manipulatives and concrete manipulative indicate that students are (a) at least equally successful with virtual manipulatives, if not more so; (b) are more independent in solving problems with virtual manipulatives; and (c) report a preference for virtual manipulatives over concrete manipulatives (Bouck, Satsangi, Doughty, & Courtney, 2014; Satsangi, Bouck, Doughty, Bofferding, & Roberts, 2016).

Given that the research base on virtual manipulatives for students with disabilities and their typical peers is still emerging, it is important that teachers monitor student progress when using the apps. App-based manipulatives should not be used simply because they are the newest technology or because students prefer using an app on a tablet to concrete manipulatives but because app-based manipulatives offer an educational benefit. Hence, teachers should collect and monitor student performance when using app-based manipulatives to determine if students are actually improving in their mathematics performance with the tools.

Why Use App-Based Manipulatives

The literature finding student preference and success with virtual, or app-based manipulatives, is not surprising. Suh and Moyer (2007) suggested virtual manipulatives reduce the cognitive load for students, as compared to concrete manipulatives, given that some virtual manipulatives demonstrate the mathematics simultaneously with the problems (e.g., 34 + 28) and the objects. Bouck et al. (2014) and Satsangi et al. (2016) found different types of virtual manipulatives had inherent supports that benefited students. For example, in one study involving double- and triple-digit subtraction with regrouping, with the app-based manipulative Base-10 Blocks Manipulative by Brainingcamp LLC (2016), students were not allowed (by the app) to ungroup from the subtrahend rather than the minuend (see Figure 1; Bouck, Chamberlain, & Park, in press). With concrete manipulatives, students can make this error unless teacher corrected. App-based manipulatives may be less stigmatizing for students with disabilities, particularly secondary students with disabilities given that many concrete manipulatives are typically designed for younger children (Satsangi et al., 2016). However, it is age appropriate and socially desirable to use mobile devices at school or at home (Bouck et al., 2012).

Figure 1.

Using Base-10 Blocks Manipulative by Brainingcamp LLC (2016) to solve triple-digit subtraction (Used by permission).

App-based manipulatives integrate concrete physical actions with mathematical representations (Moyer-Packenham & Westenskow, 2013). As a student interacts with the manipulative by moving pieces on the screen, numbers change to represent, mathematically, what the action just did. This provides feedback to the student, allowing for the immediate linking between concrete actions and abstract mathematical concepts. For example, when using a geoboard app (e.g., Geoboard; Clarity Innovations, 2015), students can find the area of a shape, such as a rectangle, by counting the gridlines or turning the numbering option on and manipulating a side to show how it impacts calculations. In addition, the physical and haptic nature of moving manipulatives on a touchscreen allows for a student to interact in a natural way but with more efficiency (Moyer, 2001). For example, a student can move multiple components at once, such as snapping base-10 cubes together in the Base-10 Blocks Manipulative (Brainingcamp, LLC, 2016) before moving them. Students can also quickly reset the manipulatives, often via selecting an eraser tool or the reset button, such as with the Hands-On Math Tangrams (Ventura Educational Systems, 2016). Finally, students can snap together and pull apart pieces within virtual manipulatives, such as with the fraction tile pieces in the Virtual Manipulatives! Fraction app (ABCya.com, 2014) that snap together when moved in close proximity to another fraction tile. These features increase efficiency by allowing more time for learning while also negating problems with fine motor skills (Moyer-Packenham & Westenskow, 2013).

App-based manipulatives provide an easy way for teachers to differentiate within a lesson. Certain features within apps can be turned off or hidden, providing an effortless way for teachers to change the level of support manipulatives give each student. For example, teachers can choose to turn off negative numbers for the Number Line by the Math Learning Center (Clarity Innovations, 2016) if students are not ready for that concept or include decimals and/or fractions for students who need an additional challenge. Virtual Manipulatives! (ABCya.com, 2014) gives the option of using rectangles or circles to illustrate fractions, decimals, and percent. This is beneficial because it allows students to see the value in the way that is most clear to them for a particular problem. For instances, when solving a story problem about pizza, students would most likely choose the circular shape, and when solving one about a yard, they would choose the rectangle.

Vignettes of App-Based Manipulatives

As noted, app-based manipulatives can be used to support students in a variety of mathematical content areas. App-based manipulatives can also be used across a large range of grade levels from elementary through secondary. The following three vignettes provide information about a student and how an app was used to address his or her area of struggle within mathematics. Please see Note 1 regarding the situation under which each vignette was developed.

Olivia is a sixth-grade student with a learning disability in mathematics. She attends a pullout mathematics class taught by a special education teacher. Her latest KeyMath assessment suggested Olivia struggled with triple-digit subtraction; her teacher often commented that she was not independent in using concrete manipulatives despite their consistent presence in the classroom during instruction and independent work. With Olivia’s growing resistance to using concrete manipulatives in the special education setting, her teacher decided to use the app-based manipulative Base-10 Blocks Manipulative (Brainingcamp, LLC, 2016) as she had three tablets for her classroom. Olivia’s teacher began by first making sure Olivia knew how to use the app to assist in solving double- and triple-digit addition and subtraction problems with regrouping. When Olivia demonstrated that she could independently use the app, her teacher allowed Olivia to select if she wanted to use concrete or app-based manipulatives during her independent class work. Olivia selected to use the app-based manipulative, commenting that she liked that it would not allow her to ungroup from the wrong number and that she felt she able to solve problems faster with the technology. After monitoring Olivia’s class work, her teacher found that Olivia was consistently using the app-based manipulative on her class work, and her percentage of problems correct increased from 45% correct to 90% correct within two weeks.

Devin is a third-grade student with a learning disability in mathematics. He learned about equivalent fractions in class, and while he understands the role of unit fractions, he struggles to understand how two different fractions could be equal. Devin tried to use the set of paper-based fraction strips created during classroom instruction but struggled to organize all of the pieces and align them properly, resulting in inaccurate answers. This led Devin to become frustrated. While pushing into the general education classroom, the special education teacher brought an iPad with the Virtual Manipulatives! fraction app-based manipulative (ABCya.com, LLC, 2014). Devin’s teacher showed him how to use the fraction strips on the iPad. Devin was excited to be able to so quickly snap fractional pieces together to represent fractions and clearly see when fraction chains were the same size. Within the app, Devin’s teacher was able to annotate a few samples with the pencil feature, drawing on top of the fraction tiles to show how different size fraction tiles align and marking notes to help explain equivalence. These notes and annotations were saved as an image on the tablet so Devin could return to it later as a reference. Devin was also able to annotate his own work and save it for his teacher to review to determine where, if any, misconceptions were occurring.

Joel is a ninth-grade student enrolled in his first industrial technology class. The idea of taking an industrial technology class was very exciting to him, but even within the first few weeks, Joel already struggled with the mathematics component. More specifically, Joel struggles with visualizing how different shapes fit together to make larger projects. He is hesitant to ask for help because he does not want to drop a class he really enjoys. Mr. Kramer, the industrial technology teacher, gave Joel cutout prototypes of basic projects to help him practice but explained it might be better for him to try a virtual manipulative so that Joel has more control over the shapes used and how they could piece together. Joel’s school does not have an advanced building program on any of the school computers because they are too expensive, so with Mr. Kramer’s recommendation, Joel decided to download Shapes-3D Geometry Learning (Setapp Sp. z o. o., 2015) on his iPhone. This app allows Joel to take apart 3D shapes and move them around, enlarge the shapes to get a better view, and tells the number of faces, edges, and vertices. Joel was able to change the color of each side and rotate shapes, which helped when he was trying to calculate dimensions and cutouts for projects. Joel said he liked the app because he could practice his geometry skills on his phone without anyone realizing he was using manipulatives.

Conclusion

App-based manipulatives provide a valuable option to support students with disabilities in mathematics provided educators have access to mobile devices. The research supports the use of manipulatives in supporting students with disabilities and typical peers in using concrete manipulatives to understand and solve mathematical problems (Carbonneau et al., 2013). Further, theory and an emerging research base validate the use of digital manipulatives to support students in mathematics (Bouck et al., 2014; Satsangi et al., 2016; Suh & Moyer, 2008). However, additional research is needed on app-based manipulatives and their impact on the mathematical learning and performance of students with disabilities.

Footnotes

Declaration of Conflicting Interests

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

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

Notes

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