Abstract concepts are essential to understanding and performing mathematics. They are also a source of difficulty for many students who struggle with mathematics, many of whom find even basic mathematics concepts difficult to understand. A popular approach to help students understand abstract concepts is the use of manipulatives. Manipulatives enable students and teachers to represent concretely the abstract concepts that they are learning in mathematics class and to link these concepts to prior knowledge. Although they are used primarily in the elementary grades, they offer a useful means to introduce new concepts to all students.
For the most part, classroom use of manipulatives has involved concrete, or physical, manipulatives. However, with the advent of the World Wide Web, there is a new category of manipulatives—virtual. Virtual manipulatives are basically digital “objects” that resemble physical objects and can be manipulated, usually with a mouse, in the same ways as their authentic counterparts. Virtual versions of concrete manipulatives typically used in mathematics education, such as Base 10 Blocks, Cuisenaire Rods, and Tangrams, are available at no cost online. Many available virtual manipulatives are paired with structured activities or suggestions to aid implementation in the classroom.
As virtual manipulatives are relatively new, there is limited research on their effectiveness. However, research into concrete manipulatives provides insight into their possible uses and benefits to learning. This Research in Brief article discusses some of this research, focusing on virtual manipulatives, and provides guidance in how to realize the potential of virtual manipulatives to support mathematics learning. The article comprises four sections: an overview of how manipulatives can support mathematics learning, guidance in choosing virtual manipulatives, an extensive list of resources, and a brief review of the research base addressing instructional use of manipulatives in mathematics.
Overview of Using Manipulatives to Support Mathematics Learning
Manipulatives can help students understand abstract concepts in mathematics
Concrete manipulatives help students with disabilities improve their understanding of the abstract symbolic language of mathematics. Concrete manipulatives can also be used to clarify misconceptions and build connections between mathematical concepts and representations, fostering more precise and richer understandings. Students who struggle in mathematics often have trouble connecting visual and symbolic representations; virtual manipulatives can make such connections explicit to students. For example, Pan Balance – Numbers is a manipulative that is based on the balance pans that are used with younger children to demonstrate the concept of equality. With this virtual manipulative, students enter a different number expression (e.g., 6+8 and 7+7) on each side of the balance. If the two expressions are equivalent, the pans are shown balanced (visual representation), and the equation is presented in an on-screen window (symbolic representation).
Virtual manipulatives may lead to more complex, richer understandings of concepts
Although findings on this point are limited, classroom-based action research shows that student use of virtual manipulatives is sometimes more complex than their use of concrete manipulatives. This is due in part to the availability of unlimited objects—versus a finite set. Also, virtual objects can be altered in ways that concrete ones cannot, for example, the size, shape, and color of a block can be changed. Thus, in many instances, students can create more examples using virtual versus physical objects. Base 10 Blocks offers an excellent illustration of these points. This virtual manipulative provides students with an unlimited set of blocks for demonstrating place value. Additionally, sets of 10 can easily be joined together; for example, by “lassoing” 10 unit blocks they become a rod. A virtual hammer is available to break the rod back into 10 unit blocks. Because the time students would spend aligning and trading blocks is eliminated, there is more time for exploration.
Research suggests that students may also develop more complex understandings of concepts when using virtual manipulatives (Moyer, Niezgoda, & Stanley, 2005). Virtual manipulatives may be particularly helpful to students with language difficulties, including English Language Learners. These students often have trouble explaining what they are learning in math. With virtual manipulatives, they may be better able to clarify their thinking and demonstrate it to others. For example, with Base 10 Blocks, students can use the place-value layout to demonstrate their understanding. An added benefit of this manipulative is that the student can select English, French, or Spanish for the display.
Students need guidance in understanding the concepts that manipulatives represent
Manipulatives by themselves have no inherent meaning. It is important for teachers to make this meaning explicit and help students build connections between the concrete materials and the abstract symbols that they represent. This holds true for both concrete and virtual manipulatives, but virtual manipulatives often have this type of structure built in. Many virtual manipulative activities give students hints and feedback, something that the more traditional concrete manipulatives cannot do without teacher assistance. For example, using Tangrams students can virtually copy a design made from pattern blocks, and when a block is near a correct location, it will snap into place. This virtual manipulative also includes a hint function that will show the correct location of all the blocks. Additionally, virtual manipulatives often provide explicit connections between visual and symbolic representations, a feature which was found to benefit learning (Suh & Moyer, 2007).
Students report that virtual manipulatives are easy to use and as engaging as concrete ones. Although virtual manipulatives provide some support for individual student use, as with physical manipulatives, students benefit from teacher guidance to help them use the manipulative correctly and connect to the underlying math. Most virtual manipulatives include activities and suggestions for teachers (and often parents), as well as ideas for student discussions and sharing. They can also be used with interactive whiteboards, so that the teacher can involve the whole class in an interactive lesson (Mildenhall, Swan, Northcote, & Marshall, 2008). Zorfass and her colleagues have developed an additional resource, an Instructional Planning Matrix (Zorfass, Follansbee, & Weagle, 2006), which is designed to help teachers use Web-based applets, including virtual manipulatives. The matrix is included in an article available here.
Choosing a Virtual Manipulative
Although relatively new, virtual manipulatives can support learning in mathematics for all students, including those with disabilities. As with concrete manipulatives, they need to be integrated into the curriculum and not just used as an adjunct activity. Used wisely, they provide students with opportunities for guided exploration, helping them to build a solid understanding of mathematical concepts and demonstrate learning. Several factors should be considered when selecting virtual manipulatives:
- Can the level of difficulty be adjusted for different students?
- What type of feedback do they provide?
- Will teachers need to provide feedback and support?
- How clear are the instructions for use?
For more information on virtual manipulatives, see What Are Virtual Manipulatives, by Moyer, Bolyard, and Spikell (Moyer, Bolyard, & Spikell, 2002).
Pan Balance – Numbers
Pan Balance – Numbers is one of a series of virtual manipulatives available through the Illuminations website (see below) that support students in investigating the concept of equivalence.
Tangrams is a virtual manipulative available through the National Library of Virtual Manipulatives website (see below). It is based on the ancient Chinese tangram blocks. The blocks can be dragged, rotated, and flipped in order to copy designs.
For more information on virtual manipulatives, see What Are Virtual Manipulatives, by Moyer, Bolyard, and Spikell (Moyer, Bolyard, & Spikell, 2002), originally published in Teaching Children Mathematics.
Arcytech – Educational Java Programs
Arcytech – Educational Java Programs, designed by Jacobo Bulaevsky, includes interactive tools for several manipulatives commonly used in the elementary grades, including Cuisenaire rods, base 10 blocks, pattern blocks, and fraction bars. Each tool has instructions and suggested lessons.
Illuminations, developed through a partnership between the National Council of Teachers of Mathematics (NCTM) and MarcoPolo, has been designed as a companion to illuminate the NCTM standards for mathematics. It includes interactive tools that support exploration of math concepts. Tools are categorized by grade level.
An additional feature is Web Resources, a list of links to web resources that have been reviewed by an editorial board of experts. These resources can be searched by grade level and standard.
National Library of Virtual Manipulatives for Interactive Mathematics
Developed at Utah State University and funded by the National Science Foundation, the National Library of Virtual Manipulatives is a library of web-based interactive virtual manipulatives and concept tutorials. Manipulatives are sorted by grade level and math content area, and each manipulative includes instructions, suggested activities, lesson plans, and connection to relevant NCTM standards. The manipulatives include some that are commonly used in teaching (base 10 blocks, pattern blocks, and algebra blocks).
Project Interactivate is developed and maintained by Shodor Educational Foundation, a non-profit education and research organization that focuses on developing valid models to support understanding. It includes over 100 interactive tools and activities that allow students to explore mathematics. Tools and activities are categorized by math content (number and operations, geometry and measurement, function and algebra, and data analysis and probability). All include explanations of how to use them in teaching and why the activity is useful. Many are incorporated into lesson plans that are also available, and some of these include downloadable worksheets to use in lesson. Also included is a discussion section that models how to introduce or explains different concepts to students.
MathTools, part of the Math Forum at Drexel University, includes a catalog of technology resources for math on the web. The catalog includes hundreds of tools, lessons, and activities that are categorized and searchable by grade level and content. Each has been submitted by a registered user (registration is free and open to all). Each includes a description, the technology type (e.g., Java applet, Flash), and ratings, reviews, and discussions from other registered users of the site. Registered users can save any activities to their own “My Math Tools” portion of the site. The activities from Arcytech, Project Interactivate, and The National Library of Virtual Manipulatives are included in the catalog. Another feature is the Research Area, which monitors and summarizes research on the uses of technology in math education.
Expand this section to read more about the research behind this article. [Hide]
Manipulatives can help students understand abstract concepts in mathematics
Research studies have evaluated the effectiveness of manipulatives as a tool in mathematics instruction. One line of research has studied the Concrete-Representational-Abstract (CRA) sequence of instruction, a form of explicit instruction that moves students from concrete manipulatives to pictorial representations of those manipulatives and finally to abstract concepts. Butler, Miller, Crehan, Babbitt, & Pierce (2003) compared the effectiveness of teaching fraction concepts to students with learning disabilities using a CRA approach versus a Representational-Abstract (RA) approach (starting with pictorial representations and moving to abstract concepts, with no concrete manipulatives). Fifty middle school students with mild to moderate disabilities were assigned to the CRA or the RA group. While both groups improved their understanding of fractions, the CRA group had overall higher scores than the RA group.
A study by Witzel, Mercer, & Miller (2003) also supports the effectiveness of a CRA approach for developing the basics mathematics skills of students with learning disabilities. Students were taught to solve algebraic equations using either a CRA approach or a traditional approach. The study involved 34 matched pairs of students in grades 6 and 7 who either had been diagnosed with learning disabilities or were categorized as at risk for learning problems. After a 4-week intervention, both groups showed improvement, but those taught with the CRA group significantly outperformed those who had received traditional instruction.
In another CRA study (Maccini & Hughes, 2000) six adolescents with learning disabilities used algebra tiles to represent algebra word problems during the concrete phase of instruction. The students were able to transition successfully to pictorial and ultimately symbolic representations of the problems.
Other studies have focused more specifically on concrete manipulatives. Marsh and Cook (1996) studied the use of Cuisenaire rods as a support for solving word problems with three third grade students with learning disabilities. The students were not only more successful at selecting the correct operation when using the manipulatives but continued to improve after the manipulatives were withdrawn. Cass, Cates, Smith, and Jackson (2003) used case-study methods to investigate the effectiveness of teaching perimeter and area concepts using manipulatives (geoboards). Study participants were three fourth grade students with learning disabilities, all of whom improved in their ability to solve these geometric problems.
Although research on virtual manipulatives is in its early stages, available research supports their value. Reimer and Moyer (2005) investigated the performance of 19 third grade students during a 2-week unit on fractions that used virtual manipulatives. Over half of the students improved their conceptual understanding of fractions on a teacher-designed measure. In another study of 19 second grade students, Moyer, Niezgoda, and Stanley (2005) observed that virtual base-10 blocks enabled students to demonstrate more sophisticated strategies and explanations of place value. Bolyard and Moyer-Packenham (2006) studied the use of virtual manipulatives with 99 sixth grade students learning addition and subtraction of integers.The students showed significant gains in achievement, and the researchers concluded that virtual manipulatives can support learning these concepts.
Two studies have compared the use of virtual manipulatives to more traditional materials. Suh and Moyer (2007) compared the use of concrete and virtual manipulatives in third grade students studying algebraic thinking. Both types of manipulatives were associated with higher achievement and increased flexibility in representing algebraic concepts. Steen, Brooks, and Lyon (2006) compared the academic achievement of a group of first grade students who used virtual manipulatives for practice in geometry instruction (treatment group) to another group who did not (control group). A total of 31 students were randomly assigned to either the treatment or control group. Achievement was measured by the Grade One and Grade Two assessments provided by the classroom textbook’s publisher. The treatment group improved significantly on both the Grade One and Grade Two tests, while the control group showed significant improvement only on the Grade One test. The treatment teacher also noted that her students showed increased motivation and increased time on task.
Virtual manipulatives may lead to more complex, richer understandings
In addition to their study of virtual base-10 blocks, Moyer, Niezgoda, and Stanley (2005) also reported on a project in a kindergarten class of 18 students. The students were engaged in a 3-day lesson on patterns. On the first day, they used wooden pattern blocks, on the second day they used Web-based virtual pattern blocks, and on the third day they drew patterns freehand on construction paper. When using the virtual blocks, most students created more complex patterns and used more total blocks. The researchers further noted that the second graders in the base-10 block study showed more sophisticated strategies after using the virtual manipulatives. In addition, they note that the English Language Learners were able to demonstrate their understanding of place-value concepts even though they could not explain them verbally.
Students need help understanding the concepts that manipulatives represent
Teachers play an important role in helping students understand the concepts that manipulatives represent. This was highlighted in a 1-year study of 10 middle school mathematics teachers and their use of manipulatives (Moyer, 2001). Teachers who were unable to represent mathematics concepts themselves were more likely to use manipulatives as a diversionary rather than instructional activity.
The CRA studies described above also demonstrate the importance of structure and guidance in linking concrete materials to abstract concepts. For example, students in Maccini and Hughes’ study (2000) were taught not only to use algebra tiles to represent word problems but also to use a structured strategy in solving them. In the Reimer and Moyer study (2005), students benefited from another important aspect of guidance: feedback. When students were interviewed regarding their impressions of the virtual manipulatives, an emergent theme was their appreciation for the immediate feedback possible with the computer-based manipulatives.
Bolyard, J. J. & Moyer-Packenham, P. S. (2006). The impact of virtual manipulatives on student achievement in integer addition and subtraction. Paper presented at the annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, November, 2006, Mérida, Yucatán, Mexico. Retrieved 11/20/08 from http://www.allacademic.com/meta/p115340_index.html
Butler, F. M., Miller, S. P., Crehan, K., Babbitt, B., & Pierce, T. (2003). Fraction instruction for students with mathematics disabilities: Comparing two teaching sequences. Learning Disabilities Research & Practice, 18(2), 99-111.
Cass, M., Cates, D., Smith, M., & Jackson, C. (2003). Effects of manipulative instruction on solving area and perimeter problems by students with learning disabilities. Learning Disabilities Research & Practice, 18(2), 112-120.
Maccini, P., & Hughes, C. A. (2000). Effects of a problem-solving strategy on the introductory algebra performance of secondary students with learning disabilities. Learning Disabilities Research & Practice, 15(1), 10-21.
Marsh, L. G., & Cooke, N. L. (1996). The effects of using manipulatives in teaching math problem solving to students with learning disabilities. Learning Disabilities Research & Practice, 11(1), 58-65.
Mildenhall, P., Swan, P., Northcote, M., & Marshall, L. (2008). Virtual manipulatives on the interactive whiteboard: A preliminary investigation. Australian Primary Mathematics Classroom, 13(1), 9-14.
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.
Moyer, P. S., Bolyard, J. J., & Spikell, M. A. (2002). What Are Virtual Manipulatives? Teaching Children Mathematics, v8(n6), p372.
Moyer, P. S., Niezgoda, D., & Stanley, J. (2005). Young children's use of virtual manipulatives and other forms of mathematical representations. In W. J. Masalaski & P. C. Elliott (Eds.), Technology-Supported Mathematics Learning Environments (pp. 17-34). Reston, VA: National Council of Teachers of Mathematics.
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. (2007). Developing students' representation fluency using virtual and physical algebra balances. Journal of Computers in Mathematics and Science Teaching, 26(2), 155-173.
Witzel, B. S., Mercer, C. D., & Miller, M. D. (2003). Teaching algebra to students with learning difficulties: An investigation of an explicit instruction model. Learning Disabilities: Research & Practice, 18(2), 121-131.