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=j@@pjTT PVirtual Manipulatives in Mathematics Education
David Young, 27 April 2006
Abstract
Physical manipulatives are a well-established tool for teaching mathematics that are supported by a strong research base. In recent years, online technologies such as Java and Flash applets have provided a method for creating and disseminating a new type of web-based manipulatives, called virtual manipulatives. Although the existing literature on virtual manipulatives has proposed a number of compelling benefits of their use, very little direct research on virtual them. This paper provides a summary of the current literature on virtual manipulatives as well as a survey of existing collections of virtual manipulatives on the web. In addition, two very important educational areas that have not received adequate discussion in the existing virtual manipulative literature are discussed. Namely, online education and secondary education. Finally, suggestions for future research are made, in which it is suggested that classroom studies on the effectiveness of virtual manipulatives should be abandoned in favor of studies that will be more likely to shed light on the benefits and appropriate uses of virtual manipulatives.
Introduction: Defining Virtual Manipulatives
Traditionally, the term manipulative has been used to describe physical objects with which students interact with to promote learning. In mathematics education, the use of physical manipulatives has been regarded as an effective instructional strategy for some time and is supported by a strong research base (Marzano 19998; Sowell 1989). Virtual manipulatives, are simply online versions of physical manipulatives. This paper will adopt the specific definition of virtual manipulatives provided by Moyer, Bolyard, & Spikell (2002): "... an interactive, Web-based visual representation of a dynamic object that presents opportunities for constructing mathematical knowledge" (p. 373). The key elements of this definition are that the virtual manipulatives must be web-based (not just computer-based), and that it must be manipulable by the user. Based on the definition above , most virtual manipulatives take the form of Java or Flash applets.
It is important to note that the term "virtual manipulative" is not universally used by developers or teachers, especially for topics beyond elementary education. It is common to find applets that fit this definition labeled simply as interactive math applets. It is also common to find individuals or groups who give original names to their virtual manipulatives, like Mathlets (JOMA Web Site 2006), Widgets (Wazzu Widgets Web Site 2006), or Gizmos (Explore Learning Web Site 2006).
Proposed Benefits of Virtual Manipulatives
Many authors have documented the perceived benefits of virtual manipulatives. A key aspect of these benefits is their availability online (Clements & McMillen 1996; Dorward 2002; Heath 2002; Leathrum 2001; Moyer & Bolyard 2002). Moyer et al (2002) point out that, "... the advantage of many emergent virtual manipulatives is that they are on the web, thereby allowing free access for schools that are online and constant availability for busy teachers and students who have limited time to get these sites during class" (p. 375). This ease of access is also seen as an ease of management (Dorward 2002).
Some authors have proposed that the applet technology on which the virtual manipulatives are based (usually Java or Flash) can in some cases provide advantages over more powerful programs with large learning curves (Gadanidis, Gadanidis, & Schindler 2003; Roschelle, DiGiano, & Chung 2000) . As Heath (2002) states, " Instructors can focus on concepts, modeling, and problem solving instead of teaching the syntax of MathCad or the keystrokes of the TI..." (p. 44). Other benefits of the applet platform are that a large number of developers are able to create and disseminate them and that applets can have a strong focus on specific concepts (Leathrum 2001). Furthermore, virtual manipulatives are capable of doing things that are simply not possible with physical manipulatives, pencil and paper, or other tools (Clements & McMillen 1996; Crawford & Brown 2003; Forster 2006; Keller, Wasburn-Moses & Hart 2002; Reimer & Moyer 2005)
From an instructional standpoint, virtual manipulatives provide students with instantaneous, corrective feedback (Clements & McMillen 1996; Crawford & Brown 2003; Durmus & Karakirik 2006; Reimer & Moyer 2005; Suh & Moyer 2005). Many authors have contended that this ability makes virtual manipulatives well-suited to inquiry-based learning and problem solving (Clements & McMillen 1996; Durmus & Karakirik 2006; Jacobs 2005). In their study of fifth graders using a fraction applet, Suh & Moyer found that, "..the applets allowed students to experiment and test hypotheses in a safe environment. The guided format features of the applets allowed guessing and trial-and-error, and at the same time, would not accept and incorrect response" (p. 10).
Another pedagogical benefit of virtual manipulatives is that they have the ability to provide multiple representations of a single concept at the same time (Clements & McMillen 1996; Heath 2002; Keller, Wasburn-Moses & Hart 2002; Moyer & Bolyard 2002; Suh & Moyer 2005). Reimer & Moyer (2005) argued that this ability provides an advantage over physical manipulatives, "Unlike physical manipulatives, electronic tools use graphics, numbers, and words on the computer screen to connect the iconic with the symbolic mode" (p. 7). It has also been proposed that this ability can promote transfer of knowledge from specific ideas to general knowledge (Clements & McMillen 1996; Durmus & Karakirik 2006; Jacobs 2005; Moyer & Bolyard 2002; Suh & Moyer 2005).
Another benefit that has been suggested for use of virtual manipulatives is that virtual manipulatives may be helpful for students with disabilities. (Miller, Brown, & Robinson 2002; Riley, Beard, & Strain 2004). In addition, several authors have contended that virtual manipulatives increase motivation and attention in students as well as teachers (Clements & McMillen 1996; Reimer & Moyer 2005; Leathrum 2001).
Existing Collections of Virtual Manipulatives
Hundreds of pages exist on the internet in which developers as individuals or groups have created and posted virtual manipulatives. It would be impractical to account for them all, but a simple internet search for "Math Java Applets" or "Virtual Manipulatives" will provide a number of resources and examples of virtual manipulatives.
In addition to the large assortment of individual collections, several groups have formed for the purpose of developing, collecting, and disseminating virtual manipulatives. Five of the most prominent of these groups are described below.
The National Library of Virtual Manipulatives is "an NSF funded project to develop interactive online learning units for 3-12" (NLVM Web site 2006). Many of the 200+ virtual manipulatives found at the National Library of Virtual Manipulatives are embedded into "eModules" which are fully developed lesson plans for implementation in to the classroom. The site allows teachers to create accounts for the purpose of developing virtual classrooms in which students can go through the activities laid out for them. The resources are developed by a variety of contributing authors and teachers are encouraged to perform "field tests" to provide feedback to the developers (NLVM Web site 2006).
The National Council of Teachers of Mathematics' (NCTM) Illuminations web site contains many internet resources, including virtual manipulatives, that are designed to "illuminate the vision for school mathematics set forth in Principles and Standards for School Mathematics" (Iluuminations Web site 2006). Their "Activities" section contain about 70 context-embedded virtual manipulatives on a variety of topics along with suggestions for use and exploration (Illuminations Web site 2006). In addition, NCTM provides an online publication, ON-Math, which includes articles on online resources for math education that sometimes contain discussion and examples of virtual manipulatives (ON-Math Web site 2006).
Shodor Education Foundation sponsors Project Interactivate, whose goals are "the creation, collection, evaluation, and dissemination of interactive Java-based courseware for exploration in science and mathematics" (Project Interactivate Web site 2006). Each of the 100+ virtual manipulatives found at Project Interactivate contain accompanying "what?", "how?", and "why?" sections which include background information on the math involved, instructions for how to use the applet, and a justification for why the topic is addressed, including correlation to standards. A section for teachers includes fully developed lessons for many of the available applets and "discussions" which outline a Socratic dialogue between student and mentor on each topic covered in the applets. In addition, the site contains several Java-based "Tools" such as calculators, converters, graph plotters etc. for student use and exploration (Project Interactivate Web site 2006).
The Educational Software Components of Tomorrow (ESCOT) project "is a test-bed for the integration of innovative technology in middle school mathematics. The project investigates replicable practices that produce predictably high quality digital learning resources." (ESCOT Web site 2006). With a focus on creating standards-based, reusable Java-Applets, ESCOT has created over 40 "Problems of the Week" for the Math Forum (www.mathforum.com), which are also available at the ESCOT web-page. Each problem is accompanied by instructions for use of the applet, a brief description of the math involved, and sample solutions. The problems are inquiry-based and often have several possible solutions and/or methods for solving.
The Journal of Online Mathematics and its Applications (JOMA) contains "Mathlets" in each issue that primarily address undergraduate mathematics topics. Roby (2001) defines a Mathlet as. "a small, interactive, platform-independent tool for teaching math..." (p. 3). In general this definition is compatible with the definition of virtual manipulatives given above. The featured "Mathlets" are either contributed by their authors, or by reviewers who have discovered them and want to feature them. In addition, JOMA contains a series of articles on the development of "Mathlets" including the standards with which they hold applets to in order to determine if they are of good enough quality to be featured.
It is also important to recognize the existence of virtual manipulatives that are available on a for-pay basis from a variety of sites. An example would be Explore Learning (www.explorelearning.com) who provides math (and science) virtual manipulatives on a variety of topics based on a subscription fee. Each of their virtual manipulatives are accompanied by an "exploration guide" and quiz and are available to view for free for 5 minutes at a time and a 30-day free trial is available (Explore Learning Web site 2006).
Existing Research on Virtual Manipulatives
Very little formal research has been conducted on the effectiveness of virtual manipulatives. Of the seven research studies addressing virtual manipulatives found for this review, three of them were classroom studies in which two showed some evidence of benefits from using virtual manipulatives and one showed no difference in using them as opposed to physical manipulatives or no manipulatives at all. Three studies were based on teachers, in which is was found that teachers had a wide variety of attitudes towards virtual manipulatives and little familiarity with them. The remaining study purported to show that Java applets featured capabilities that made them superior to graphing calculators in a particular situation. A brief summary of each of these studies follows.
Dorward (2002) conducted an informal study on the effectiveness of virtual manipulatives in which three groups of students were taught the same topics from three different teachers. One group was taught with physical manipulatives, on with virtual manipulatives and one with no manipulatives. Results on a unit test did not show any differences in student achievement between groups. However, the teachers themselves left the study with a firm belief that the virtual manipulatives were of great benefit to the students.
Reimer and Moyer ( 2005) studied a small group of third-graders learning about fractions with the use of virtual manipulatives. They concluded that, "Student interviews and attitude surveys indicated that the virtual manipulatives (1) helped students in this class learn more about fractions by providing immediate and specific feedback, (2) were easier and faster to use than paper-and-pencil methods, and (3) enhanced students' enjoyment while learning mathematics" (Reimer & Moyer 2005, p. 5-6). However, the authors do admit that the small class size and specific demographics fail to make the findings applicable to a broader population.
Suh and Moyer (2005) conducted a similar study of fifth-graders using virtual manipulatives in the classroom for learning about fractions. The authors concluded that the study showed that one applet, "supported student learning in three important areas: (1) It allowed discover learning, (2) It allowed students to make conjectures, and (3) It encouraged students to see mathematical relationshops" (p. 5). Similarly, they found that a second applet, "supported student learning in two important areas: (1) It linked symbolic and iconic modes, and (2) It helped to deter a common fraction error pattern" (pg. 7). As with the last study, the specific size and demographics of the class prevent any conclusions from being applied to a larger population.
Keller, Wasburn-Moses, & Hart (2006) studied the use of a Java applets for visualizations of 3-D objects in middle and secondary education. In their study, they looked at the effects of use on both students and teachers. Based on the data they had collected by the time of publishing they concluded that use of the applets "improve students spatial visualization skills, as indicated by improved ability to create isometric drawings, connect isometric drawings with other 2-D representations of 3-D objects, and translate among these representations" (p.1) and "enhance future teachers pedagogical content knowledge, as indicated by growth in their own spatial visualization skills and increased awareness of the teaching and learning issues related to isometric drawings" (p. 1). Their study highlights an important theme in the literature on physical and virtual manipulatives: that teachers play a significant role in the effectiveness of virtual manipulatives.
Gadanidis, Gadanidis, and Schindler (2003) studied the effects of the availability of Java-applets on lesson planning in pre-service teachers. In this study, they found that, "...use of the applets was one of several factors mediating pre-service teacher thinking, and typically not the major factor" (p. 323). Many of the pre-service teachers in the study expressed a lack of experience or familiarity with the applets. In addition, many expressed concerns about students playing with the applets without learning, classroom management problems, and the possible superiority of hands-on activities.
Crawford and Brown (2003) explored teacher's rationale for choosing virtual manipulatives over alternative forms of instruction. They provided a group of in-service teachers with Roblyer and Edwards' (2000) "Elements of a Rationale for Using Technology in Education" to use as a guide as they were given time to explore the virtual manipulatives available at the National Library of Virtual Manipulatives. The teachers then shared their ideas and thoughts on the virtual manipulatives through an online survey developed by the researchers. The results showed a variety of teacher thoughts in which most teachers thought they could be beneficial. However, many teachers also mentioned concerns about classroom management and superiority of other methods.
Forster (2006) studied a year 12 course at a school in Western Australia in which graphing calculators and Java applets were use to teach descriptive statistics. He concluded that the Java applets provided effective methods for learning the concepts that were not possible with their graphing calculators. However, use of the calculators was a required skill for students on a tertiary entrance exam (p. 150). His final conclusion was related to technology integration in general and not virtual manipulatives in particular.
Support for Virtual Manipulatives Based on Other Research
Any conclusions to be drawn from the above studies are antidotal and in themselves are not capable of justifying the use of virtual manipulatives. This fact prompted Reimer and Moyer (2005) to conclude that, "the amount of research on high-quality dynamic virtual manipulatives is so limited that a judgment about their potential uses in mathematics instruction is entirely speculative" (p. 8). However, several authors have attempted to justify the use of virtual manipulatives without the use of original research.
Physical manipulatives have been considered effective teaching tools for some time and are supported by a strong research base (Marzano 19998; Sowell 1989). This forces consideration of whether the research base supporting physical manipulatives can be directly transferred to the support of virtual manipulatives. Two arguments have been made, based on existing research, that assert that this research base does transfer to virtual manipulatives.
Clements and McMillen (1996) cited the work of Piaget and Holt to argue that virtual manipulatives are no less concrete than physical manipulatives, in that they are both simply symbolic representations of abstract concepts. Specifically, Clements and McMillen argue that the power of manipulatives lie in their concrete nature and that virtual manipulatives are no less concrete than physical manipulatives. The conclude that "The important point is that 'concrete' is quite literally, in the mind of the beholder" (Clements & McMillen 1996, p. 272). The logical consequence of this assertion is that the research base supporting physical manipulatives transfers to support the use of computer-based manipulatives.
In his defense of the validity of the virtual manipulatives found at ExploreLearning.com, Cholmsky (2003) cites the Clements and McMillen article mentioned above and also asserts that Marzano's (1998) meta-analysis of instructional methods that work supports the use of virtual manipulatives. The specific methods mentioned by Cholmsky from Marzano's work are the use of graphical/non-linguistic representations (Cholmsky 2003, p. 7), the use of physical manipulatives (p. 12), hypothesis testing (p. 17), and direct instruction preceding application (p. 19). In their previously mentioned study of fifth-graders using virtual manipulatives to learn about fractions, Reimer and Moyer (2005) cite the same study from Marzano, specifically regarding graphical/non-linguistic representations, to claim that virtual manipulatives can be an effective learning tool.
It is also important to note that many of the researchers mentioned above warn that use of virtual (or physical) manipulatives do not guarantee successful learning (Clements & McMillen 1996; Reimer & Moyer 2005). In addition, virtual manipulatives can be used in ways that do not promote the sort of authentic, inquiry-based learning that they are generally purported to adhere to (Clements & McMillen 1996; Reimer & Moyer 2005; Durmus & Karakirik 2006). In particular, Clements and McMillen (1996) warn that, "they [manipulatives] do not carry the meaning of the mathematical idea. They can even be used in a rote manner. Students may need concrete materials to build meaning initially, but they must reflect on their actions with the manipulatives to do so" (p. 271). Similarly, Reimer & Moyer (2005) point out that "The mere use of manipulatives does not guarantee that students understand concepts and procedures and be able to connect these concepts to abstract symbols without teachers making these connections explocit" (pp. 6-7). In these statements, the authors are essentially contending that the teacher carries more responsibility than the technology in regards to the effectiveness of virtual manipulatives.
Suggestions for Future Research
As is evidenced above, the literature on both physical and virtual manipulatives includes recognition that they are largely dependant on teacher's actions. As such, research should not focus on classroom studies seeking to ascribe direct teaching benefits to virtual manipulatives. Instead, research should focus on those areas that appear to be most influential to the success of virtual manipulatives, the teacher and the instructional design.
In the area of instructional design, Gadanidis et al (2004) have argued that more research should be done on the application of human-computer interface theories to the development of virtual manipulatives. The same group has suggested that applets may be more effective when embedded in a real-word context and that further research should be done to confirm or deny this hypothesis (Gadanidis et al 2004). Both of these suggestions are appropriate, based on the findings of the existing virtual manipulative literature described above.
In regards to teachers, several authors have suggested that more research be conducted on how to increase teachers' awareness of and abilities to effectively use virtual manipulatives for teaching (Crawford & Brown 2003; Gadanidis, Gadanidis, Schindler 2003; Reimer & Moyer 2005). In addition, Gadanidis et al's (2004) question about embedding virtual manipulatives in context is relevant to instructional design, but could also be relevant to teacher use as teachers could embed applets into a context that does not exist in the applet itself.
Suggestions for future Development and use of Virtual Manipulatives
Several authors have made important suggestions for the future development of virtual manipulates. Gadanidis, Kamran, and Hai-Ning (2004) have argued that development of virtual manipulatives should be based on sound pedagogy and take into consideration current research and theories on human-computer interaction. Additionally, it has been suggested that development of virtual manipulatives should be based on classroom testing and feedback (Gadanidis et al 2004; Keller, Wasburn-Moses & Hart 2002; Moyer, Bolyard, & Spikell 2002). Groups from ESCOT and JOMA have also made calls for a reusability in virtual manipulatives from a pedagogical and programming perspective (Roschelle, DiGiano, & Chung 2000; Roschelle, Hand, DiGiano 2001). Finally, Roby (2001) has suggested the use of an agreed upon set of standards to hold future applets to.
It is also clear from the literature described above that teachers need to be actively involved in directing student use of virtual manipulatives (Reimer & Moyer, 2005; Suh & Moyer 2005). In doing so, teachers should ensure that students are provided with opportunities to reflect and explore (Clements & McMillen 1996). In order to support teachers in the use of virtual manipulatives, several authors have asserted that teachers need more training (Gadanidis, Gadanidis, & Schindler 2003; Suh & Moyer 2005) and that teachers need to communicate experiences and results with colleagues and developers (Leathrum 2001; Roby 2001; Moyer, Bolyard, & Spikell 2002).
What's Missing
While remaining relatively small, the existing literature base on virtual manipulatives provides a compelling rationale for the effectiveness of virtual manipulatives and several appropriate suggestions in regards for their future development, use and research. However, the existing literature has not adequately addressed two primary areas of education: online (or virtual) schools and secondary education.
Online schools are in a unique position to take advantage of virtual manipulatives as they are entirely online resources. Virtual manipulatives have the potential to provide online schools a way to implement manipulatives in mathematics education where the use of physical manipulatives is impractical at best. As is evidenced above, several authors have suggested that it is imperative that teachers play a crucial role in the use of virtual manipulatives in the classroom. However, the specific actions and methods used by teachers in an online environment often differ from those in the face-to-face classroom. Because of this, more literature and research should focus on the use of virtual manipulatives in online education. In addition, researchers looking to study the use of virtual manipulatives could receive much larger sample sizes and potentially controlled environments in an online classroom than in a face-to-face classroom.
The term "manipulative" has long carried a connotation of use for early elementary ages and the addition of the word "virtual" does not appear to have altered that perception. As such, there are currently collections of virtual manipulatives devoted specifically to elementary and middle school topics, with much fewer being devoted to secondary math topics. Similarly, almost all of the existing literature on virtual manipulatives is based on elementary or middle school education, with some focus on undergraduate education, but very little on secondary. In the areas of development and research, more virtual manipulatives need to be focused on secondary math topics.
Conclusion
Virtual Manipulatives provide a compelling and promising tool for teaching and learning mathematics. The existing literature on virtual manipulatives has effectively pointed out the many benefits that they may hold, while recognizing that their effectiveness is primarily reliant on teachers and instructional design. As such, further research on virtual manipulatives should focus on those two areas. In addition, more literature and research is required in the areas of virtual manipulative development and implementation in online and secondary education.
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