Journal of Special Education Technology
Assistive Technology
Associate Editor’s Column
Ashley A. Skylar, California State University–Northridge Virtual Manipulatives as an Assistive Technology Support for Students with High-Incidence Disabilities Guest Columnist: Kristin L. Sayeski, University of Nevada– Las Vegas
There is a long history of teachers using physical manipulatives, such as base-ten blocks or Cusienaire rods, to help students learn and understand mathematical principles. Indeed, research has demonstrated the power of providing hands-on, structured experiences with manipulatives to teach concepts such as place value and basic operations as well as more complex mathematical concepts (Moyer, 2001). Continuing this historical trend, advances in technology have given birth to a new generation of manipulatives—virtual manipulatives. Virtual manipulatives look similar to their three-dimensional counterparts and can be controlled and moved by the user in the virtual computer environment. This column provides information on how teachers can use virtual manipulatives as an assistive technology support for students with high-incidence disabilities. Students who struggle with mathematics represent a diverse group. Researchers have identified a variety of learner characteristics that contribute to difficulty learning math. Common characteristics include learned helplessness, passive learning (failure to connect prior learning with new information), memory problems, attention problems, strategy deficits (failure to use metacognitive or cognitive strategies), low academic achievement, and math anxiety (Allsopp, Kyger, & Lovin, 2007). Teachers can address these
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challenges head-on by selecting instructional strategies that have been designed to ameliorate learning deficits. One such set of instructional strategies includes virtual manipulatives. Virtual manipulatives are computer representations of physical mathematical manipulatives (base-ten blocks, color tiles, interlocking cubes, fraction circles, etc.). Several studies have demonstrated the power of using virtual manipulatives to teach a variety of topics and students of different age levels (Reimer & Moyer, 2005; Steen, Brooks, & Lyon, 2006; Suh & Moyer, 2005, 2007). Most virtual manipulative sites host Java-based or Flash applications that can be accessed directly from the Web site or downloaded for separate use. The applications tend to be colorful and engaging for users, such as a base-ten virtual manipulative (see Figure 1) from the National Library of Virtual Manipulatives (NLVM) (http://nlvm.usu.edu/). In addition, many Web sites provide information about the National Council for Teachers of Mathematics (NCTM) Standard(s) addressed in the simulation. Use of Virtual Manipulatives in the Classroom
Virtual manipulatives can be a “low-cost” assistive technology. In contrast to high-cost technologies— talking calculators, speech recognition software, scanners, text-to-speech software—virtual manipulatives are free and easily accessible to teachers who have computers and Internet access. Table 1 lists Web sites that offer virtual manipulatives for use in the classroom.
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Journal of Special Education Technology
Figure 1 Example of a Base-ten Virtual Manipulative
Source: National Library of Virtual Manipulatives (http:// nlvm.usu.edu/)
As with any technology, teachers need to plan for the effective use of virtual manipulatives. For example they must consider how they will support students before, during, and after the instructional activity as well as the different types of support needed for introducing a topic, practicing or applying a skill, or remediating a skill or concept (Zorfass, Follansbee, & Weagle, 2006). Depending upon the instructional goal, teachers can determine how the virtual manipulative is introduced, monitored, and supported. One strategy for introducing a new virtual manipulative is to provide a concrete model for students to refer to when working in the virtual environment. During the activity, instructional support may include working with a partner or documenting problem-solving thoughts in a reflection notebook. Upon completion of the experience, teachers should assess what students learned and determine if that learning is generalizable to other situations. Frequent opportunities to use virtual manipulatives increase students’ comfort and success with the material.
Table 1 Virtual Manipulative Web Sites Website Name/URL
URL and Description
National Library of Virtual Manipulatives (NLVM)
URL: http://nlvm.usu.edu/en/nav/vlibrary.html The NLVM is funded by the National Science Foundation. The Web site hosts a variety of Java-based mathematical tools designed to address a range of math concepts from pre-K to high school. The Web site is organized by strand (e.g., numbers & operations, algebra, geometry, measurement, and data analysis & probability) and grade level.
Visual Fractions
URL: http://www.visualfractions.com/ This Web site provides a collection of fraction virtual manipulatives created by Richard E. Rand. Instructions, support explanations, model problems, and summary reports are provided. Topics range from identifying fractions to dividing and multiplying fractions.
Base-Ten Blocks
URL: http://www.arcytech.org/java/b10blocks/b10blocks.html This bare-bones Java-application Web site is maintained by Archytech. Students can use the application to solve teacher-generated problems. Tools such as the hammer will allow students to “break” tens or hundreds units into smaller pieces to be exchanged. Glue is used to join smaller units to make tens or hundreds blocks.
NCTM Illuminations
URL: http://illuminations.nctm.org/ This Web site lists a collection of resources sponsored by the National Council of Teachers of Mathematics called Illuminations. Under the “Activities” section, teachers can find interactive virtual manipulatives that range from teddy-bear counters to algebra balances.
Project Interactive
URL: http://www.shodor.org/interactivate/ The Web site contains a collection of Java-based courseware for mathematics and science. The Web site has sample lessons and information for teachers in addition to activities and tools for students. It is sponsored by the Shodor Foundation and covers a wide range of topics and concepts.
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Journal of Special Education Technology Using Virtual Manipulatives for Specific Mathematical Applications
Teachers interested in using virtual manipulatives in their classrooms can begin by either identifying the specific math standards they need to address or by searching for a specific virtual manipulative that corresponds to a physical manipulative already in use in their classroom. Virtual manipulative Web sites such as NLVM or Illuminations have search tools that facilitate either approach. Hundreds of virtual manipulatives exist for students in pre-K through elementary grades. Some of the most popular applications mirror common physical manipulatives found in elementary classrooms. For example, teachers can find virtual teddy bear counters, pattern blocks, color tiles, base-ten blocks, positive and negative integer chips, fraction circles and bars, geoboards, rectangle division
and multiplication, analog and digital clocks, and more. An example of one application is “Equivalent Fractions” found on the NCTM Illuminations Web site. If you have a browser available, type in http://illuminations. nctm.org/ActivityDetail.aspx?id=80, and allow the Java application to load. What you will see is a color example similar to the one presented in Figure 2 (you may not get precisely the same example as Figure 2), which provides a richer illustration of an equivalent fractions virtual manipulative. In Figure 2, the box on the left side contains a visual representation of a fraction such as ½; 1 out of 2 rows are colored in. The students are prompted to create an equivalent fraction by deciding (a) how many rows and/or columns to put in the box in the center and the box on the right, and (b) how many cells to click on to color inside each box. A number line below the boxes shows students the fraction they are creating as the box is filled. Students can keep score to track their responses.
Figure 2 Example of an Equivalent Fraction Virtual Manipulative
Source: NCTM Illuminations Web site (http://illuminations.nctm.org/ActivityDetail.aspx?id=80)
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Journal of Special Education Technology From this example, students are able to see multiple representations (the “box” fraction and the symbolic notation fraction) of fractions as they explore the concept of creating equivalent fractions. From an instructional standpoint, a lesson such as this can be used prior to introducing a symbol algorithm or after students have mastered reducing fractions with symbols. This type of virtual manipulative will provide students with the schema to make connections to the process of reducing or expanding fractions.
A wide range of applications exist for older students as well. One popular application is “Algebra Balance Scales” (see Figure 3) from the National Library of Virtual Manipulatives (http://nlvm.usu.edu/). In this application, two equations are set up as balances on a scale. Students solve simple linear equations by manipulating unit blocks (representing ones) and X-boxes (representing the unknown). Students can perform any arithmetic operation to solve for X as long as the operation is done to both sides of the balance. The goal is to get a single X-box on one side; the number of unit boxes left are the equivalent to X. Figure 3 illustrates how students would manipulate
Figure 3 Example of an “Algebra Balance Scale” Virtual Manipulative
Source: National Library of Virtual Manipulatives (http://nlvm.usu.edu/))
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Journal of Special Education Technology the algebra scales to set up 3x + 2 = x + 6. Students then systematically remove units to solve for X. Virtual Manipulatives as an Assistive Technology for Students with Learning Difficulties
An assistive technology like virtual manipulatives can address a range of learner difficulties. Many manipulative Web environments direct students to actively engage with the material, provide guiding questions, and create multiple opportunities for success. Students with a history of struggling with mathematics will be able to use the virtual manipulatives to verify their thinking and see immediate success. Conversely, students who are struggling with a concept can request a model demonstration, obtain immediate feedback on incorrect answers, or request additional instruction or explanations. Table 2 lists the different types of scaffolding or support available to students in many of the virtual manipulative Web environments.
Memory Problems
Students with memory problems often require a deeper understanding or mastery of a concept in order to facilitate storage and retrieval. Difficulties with memory do not necessarily reflect a difficulty with retaining information; more often, students did not learn the material to a degree of mastery that allows them to store and later retrieve the information (Lieberman, 2004). For example, a student who fails to connect the concept of multiplication with the concept of repeated addition will store “steps for multiplying” as different and separate knowledge of addition. In contrast, a student who makes connections across these operations will develop a schema for understanding the processes and more likely be successful in learning multiplication. Virtual manipulatives allow students to gain a deeper understanding of complex mathematical concepts and, therefore, facilitate memory retention. One example is a base-ten block application that allows students to move a 10-unit stick to the one’s column for a two-digit sub-
Table 2 Types of Scaffolds and Supports Available for Students in Virtual Manipulative Web Environments Instructions
Many virtual manipulative Web sites will have instructions that illustrate how the manipulative can be used to answer the problem and step-by-step directions for how to solve the problem.
Guiding Questions
As students work through a problem, some Web sites will provide guiding questions. For example, a problem using visual fraction manipulatives may prompt the student, “What part of the circle is shaded?” Or a division problem with base-ten blocks question may be, “How many are left?”
Explanations
If a student is unsure about a problem, some Web sites will provide an “Explain” button. The explanation can be a combination of images of the virtual manipulative, the corresponding symbolic equation, and/or descriptions of how the answer was derived.
Immediate Feedback
Unlike paper-and-pencil exercises, the virtual manipulative Web sites provide immediate feedback to students—“Correct,” “Way to Go!,” “Incorrect,” or “Try Again.” On some Web sites, additional information or hints are provided when students submit incorrect responses.
Multiple Practice Opportunities
Students have multiple opportunities for practice on virtual manipulative Web sites. Some Web sites will allow users to select level of difficulty, while others will move students from easy to more challenging problems.
Score Reports
Many Web sites will allow students to print out a score report for documentation of their Web experience.
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Journal of Special Education Technology traction problem requiring regrouping. The old “borrowing” method that taught students to simply draw a line through the number in the tens place, subtract one from that number, and add a “1” to the number in the one’s place failed to help students understand what was being borrowed. The physical act of moving and adding 10 units to the number in the one’s column reinforces the mathematical principle of regrouping. Failure to Use Strategies
Another common trait of students with high-incidence disabilities is their failure to generate and apply strategies (Mercer & Pullen, 2005). Many of the virtual manipulative applications are “self-correcting.” That is, students receive immediate feedback on incorrect responses. The feedback allows students to go back and approach the problem in a different way. Even though students can “guess and check” their way through, some sites only allow limited guesses before providing the correct answer and a model. Thus, students are rewarded for using correct strategies and receive immediate feedback on incorrect ones. Virtual manipulatives also provide opportunities to repeatedly apply problem-solving strategies. Such repetition reinforces application of the strategy. When teachers provide additional practice with the strategy (concrete and virtual), students are more likely to be able to apply the strategy in novel situations. Low Achievement
Lastly, low academic achievement often hinders students’ ability to make appropriate gains in mathematics. Certainly, students who do not have a firm understanding of basic mathematical concepts will not do well in problem solving. Although limited research on virtual manipulatives exists, research on the use of physical manipulatives provides support for the use of manipulatives for teaching fundamental math concepts (Mastropieri, Scruggs, & Shiah, 1991). The tremendous range of different virtual manipulatives provides teachers with multiple means of representing and teaching math concepts. Ultimately, the purpose of virtual manipulatives is to assist students in achieving a deeper understanding of mathematical principles. Manipulatives, virtual or physical, are not efficient for straight computational use. Nor are manipulatives a panacea for all students who have difficulties in mathematics. Effective use of virtual manipulatives should lead to students connecting concepts to
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corresponding symbolic representations. Teachers must play an integral role in setting up instruction to facilitate this type of student learning. Research on mathematics has demonstrated the efficacy of moving students through a concrete to representational to abstract sequence of instruction (Anstrom, 2006; Miller & Mercer, 1993). That is, first set up experiences in which students use concrete manipulatives in order to solve a problem. Then, transfer students to representations of the concrete manipulative. Representations can be tallies, pictures, or stamped impressions—or in this case, virtual manipulatives. Finally, guide students in connecting the mathematical principle to the symbolic level of numbers and notational operations. Virtual manipulatives can provide an important bridge from a student’s experiences with the concrete manipulative to symbolic notation. Conclusion
Virtual manipulatives offer new opportunities as an assistive technology for teachers of students with highincidence disabilities. As the research base on the use of virtual manipulatives (for both students with and without disabilities) grows, more information on how teachers can set up effective and efficient instruction will become available. The technology continues to improve, and feedback from teachers and researchers results in improved applications that are delivered more efficiently and effectively. Thus, virtual manipulatives can facilitate and accelerate what we know to be best practice for students with high-incidence disabilities. References Allsopp, D. H., Kyger, M. M., & Lovin, L. H. (2007). Teaching mathematics meaningfully: Solutions for reaching struggling learners. Baltimore, MD: Paul H. Brookes. Anstrom, T. (2006). Achieving mathematical literacy: Interventions for students with learning disabilities. Washington, DC: American Institutes of Research. Lieberman, D. A. (2004). Learning and memory. Belmont, CA: Tomson-Wadsworth. Mastropieri, M. A., Scruggs, T. E., & Shiah, S. (1991). Mathematics instruction for learning disabled students: A review of research. Learning Disabilities Research & Practice, 6(2), 89-98. Mercer, C. D., & Pullen, P. C. (2005). Students with learning disabilities (6th ed.). Upper Saddle River, NJ: Pearson Prentice Hall. Miller, S. P., & Mercer, C. D. (1993). Using data to learn about concrete-semi-concrete-abstract instruction for students with math disabilities. Learning Disabilities Research & Practice, 8, 89–96.
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Journal of Special Education Technology Moyer, P. S. (2001). Are we having fun yet? How teachers use manipulatives to teach mathematics. Educational Studies in Mathematics: An International Journal, 47(2), 175-197. Reimer, K., & Moyer, P. S. (2005). Third-graders learn about fractions using virtual manipulatives: A classroom study. Journal of Computers in Mathematics and Science Teaching, 24(1), 5-25. Steen, K., Brooks, D., & Lyon, T. (2006). The impact of virtual manipulatives on first grade geometry instruction and learning. Journal of Computers in Mathematics and Science Teaching, 25(4), 373-391. Suh, J., & Moyer, P. S. (2005). Examining technology uses in the classroom: Developing fraction sense using virtual manipulative concept tutorials. Journal of Interactive Online Learning, 3(4), 1-22. Suh, J., & Moyer, P. S. (2007). Developing students’ representational fluency using virtual and physical algebra balances. Journal of Computers in Mathematics and Science Teaching, 26(2), 155-173. Zorfass, J., Follansbee, R., & Weagle, V. (2006, June). Integrating applets into middle grades math: Improving conceptual understanding for students with math difficulties. Technology in Action, 2(2), 1-8.
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Author Notes Kristin L. Sayeski is an assistant professor of special education at the University of Nevada–Las Vegas. Correspondence regarding this guest column can be sent to: Kristin L. Sayeski, CEB 118, The University of Nevada–Las Vegas, 4505 Maryland Parkway, Las Vegas, NV 89154-3014.Email to
[email protected] If you have an assistive technology topic or product that you would like to see covered or if you are interested in being a guest writer, please send your comments to: Ashley A. Skylar Department of Special Education California State University–Northridge Northridge, California 91330-8265 (818) 677-2526
[email protected]
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