How to apply virtual manipulatives to classroom instruction
"Manipulatives are objects used to help better understand abstract mathematical concepts or properties, and are found in two varieties, the physically represented form of concrete manipulatives, and the computer generated form of virtual manipulatives" (Bouck et al, 2013, p. 181). Representations of math concepts include concrete representations, visual or pictorial representations, abstract representations, and dynamic virtual representations (Moyer et al, 2008, p. 203). While many teachers are able to use concrete manipulatives in the classroom, many are not aware of the capabilities that virtual manipulatives have to offer and therefore they either do not use them at all in classroom instruction, or they use them inadequately. Virtual manipulatives are essentially applets that are placed on the internet for student use. Applets are "smaller, stand-alone versions of application programs" (Moyer et al, 2005, p 17). There are several ways in which to use virtual manipulatives such as (1) introducing or reviewing a mathematical concept or idea, (2) developing the understanding of mathematical concepts by visually representing abstract ideas, (3) scaffolding the learning of students, (4) and actively engaging students in the learning process (Bouck and Flanagan, 2010, p. 188). They are also most readily used for investigation, skill solidification, introduction of new material, games, and to extend learning for high achievers (Moyer et al, 2008, p. 209). When looking at the instructional process from a pedagogical point of view, it has been shown that the investigation and skill solidification portions of the instructional sequence are at the core (Moyer et al, 2008, p. 214). It is important when developing instruction through the use of virtual manipulatives that they be viewed as a cognitive tool. "Their characteristics of a cognitive tool are evident in the capability that allows users to act on the virtual manipulatives as representations of objects, with the consequences of the user's actions resulting in visual on-screen feedback from the virtual tool" (Moyer et al, 2008, p. 203). Emphasis is placed on the word tool because virtual manipulatives alone are not enough to support and enhance student understanding and application. Therefore, when selecting virtual manipulatives, it is important to consider several characteristics of each virtual manipulative in terms of the mathematical fidelity, cognitive fidelity, pedagogical fidelity, and externalized representations of each individual tool or applet (Zbiek, et al, 2007). Moyer, Salkind, and Bolyard (2008) describe mathematical fidelity as "the degree to which the mathematical object is faithful to the underlying mathematical properties of that object in the virtual environment; while the cognitive fidelity refers to how well the virtual tools reflect the user's cognitive actions and possible choices while using the tool in the virtual environment" (p. 204) It is Zbiek et al (2007) who describes pedagogical fidelity as "the extent to which teachers (as well as students) believe that a tool allows students to act mathematically in ways that correspond to the nature of mathematical learning that underlies a teachers practice" (p. 1187). The use of manipulatives in general requires thought and appropriate use. Durmus et al (2006) describes the selective criteria of effective manipulatives in six key points.
The materials should be used frequently in a way that is consistent with the goals.
They should be used in conjunction with other aids including textbooks, pictures and video.
They should be appropriate to the mathematical content and to the standards being addressed.
They should be used in conjunction with exploratory and inductive approaches.
It is important to use the simplest form and the simplest materials.
Manipulatives should be used with programs that record results symbolically.
These same techniques and criteria should be used when selecting and implementing virtual manipulatives as well. However, when it comes to virtual manipulatives the implementation process can be a bit more intensive. Since there is such a large array of virtual manipulatives available on the web it is is important to select the correct computer based program for your needs. A way to do this is to follow a basic set of guidelines when previewing virtual manipulatives. For virtual manipulative sites to be beneficial they must have uncomplicated actions that allow users to change, repeat and undo; they must allow students to save their work for review later; they need to dynamically link different representations and make valid connections between the pictures displayed and the symbols that are used; they need to allow for teachers and students to develop, pose, and engage in their own questions; and they must allow students to develop in the use of a virtual mathematical tool (Durmus et al, 2006). With this set of guidelines in mind an educator is able to select the best program for virtual manipulatives and select and use proper manipulatives in the learning environment. Clements and McMillen (1996) provide a check list for the proper use of virtual manipulatives (p. 77). It would be of great pedagogical benefit to use computer manipulatives for assessment as mirrors of students' thinking; guide students to alter and reflect on their actions, always predicting and explaining; create tasks that cause students to see conflicts or gaps in their thinking; have students work cooperatively in pairs; use one computer to focus and extend follow up discussions with the class; recognize that much information may have been introduced before moving to work on computers, including the purpose of the software, ways to operate the hardware and software, mathematics content and problem solving strategies; and finally use extensible programs for long periods across topics when possible (Clements and McMillen, 1996, p. 77). The implementation of virtual manipulatives is an extensive process that requires extra effort and research on the part of the instructor, but if done correctly, holds the ability to lower the burden of the educator, give every student the opportunity to explore on their own, and allow for a more conceptually based understanding.Students must not only be presented with a "learning with model" approach but also must employ a "learning to model" approach that allows for real-world problem understanding (Durmus et al, 2006).