Using Virtual Manipulatives to Support Teaching and Learning Mathematics

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1 Using Virtual Manipulatives to Support Teaching and Learning Mathematics Joel Duffin Abstract The National Library of Virtual Manipulatives (NLVM) is a free website containing over 110 interactive online tools (virtual manipulatives) and activities for K-12 mathematics such as geoboards, pattern blocks, graphing tools, and logic puzzles. The focus of the NLVM National Science Foundation (NSF) project was to develop flexible, focused, interactive tools, around which rich math learning activities could be developed. As part of a second NSF grant called the enlvm, interdisciplinary teams including classroom teachers extended the NLVM by developing and field testing interactive online math lesson called emodules. Each emodule provides materials for 1-5 days worth of instruction that use NLVM and other web resources, are aligned with state and national standards, and include online assessments. The enlvm also developed a web-application that allows teachers to organize, adapt and create lessons, track student performance, and collaborate with other teachers. This paper describes the NLVM, enlvm, research that has been conducted, and future directions for NLVM research. Conference Themes Research agenda, dynamic mathematics, resource base. Keywords NLVM, virtual manipulatives, mathlets, interactive online mathematics, authoring tools, evaluation, research, theory. I. INTRODUCTION esearchers at Utah State University began using and Rbuilding interactive tools for teaching and learning mathematics early in the 1990s, using Geometer's Sketchpad as part of summer workshops for teachers and developing CD-based tools to support teaching College Algebra. In 1999, researchers obtained a grant from the National Science Foundation to develop the National Library of Virtual Manipulatives (NLVM), a collection of interactive tools for K-12 mathematics freely available on the internet ( Early efforts focused on the development of math-rich Java applets that could be used on multiple browsers and operating systems, a challenge at that time because the support for Java was very uneven across platforms. The project organized tools into content strands and grade bands identified in the NCTM standards. In addition to tools, the NLVM site includes background information for teachers, help files explaining how to use the tools, and activities illustrating how the tools could be used. Along side of the development of the NLVM, researchers collaborated with publishers and authors to create companion Joel Duffin, is with Utah State University, Logan, UT USA ( He is CEO of Models for Learning, Inc a company whose mission is to prepare youth for careers in Science, Technology, Engineering and Mathematics. CDs and websites, that used NLVM tools, to accompany math education and liberal arts textbooks. Figure 1 shows an example of an NLVM activity that uses pattern blocks to teach rotation transformations: Figure 1 NLVM virtual manipulative II. VIRTUAL MANIPULATIVES Initial tools created by NLVM researchers included software counterparts to physical or concrete manipulatives commonly used to teach mathematics in elementary school classrooms, such as base blocks, geoboards, and pattern blocks. While additional tools were created that do not have physical counterparts such as graphing tools, the name virtual manipulatives was retained because it emphasizes interactivity. Other names that have been applied to similar tools include math tools and mathlets. Strengths of virtual manipulatives include: Constraints Constraints can be imposed that help focus student attention on mathematical rules. For example, in base blocks, ten blocks can only be allowed in the tens column. Seeding Virtual manipulatives can be seeded or configured for use in specific activities, increasing focus and saving start up time. Dynamic visualization Visual representations of mathematical concepts can be interacted with, allowing the learner to explore relationships. Multiple, linked representations Representations can be linked to help draw attention to the relationships between the representations and deepen understanding.

2 Hints and immediate feedback Software can make learning more efficient by giving hints when students request it and providing feedback when it recognizes mistakes. Simulations Learners can run simulations that otherwise would be prohibitive in a classroom setting. For example, a spinner can be spun 1000's of time in only a few seconds. Instructional sequences Sequences of instructional activities can be built into online materials that scaffold and focus student learning. Cost Many websites such as the NLVM provide virtual manipulatives for free on the internet. Saving The state of virtual manipulatives can be easily printed or saved so that it can be reviewed, modified, and discussed later. Maintenance In contrast to physical manipulatives, virtual manipulatives don't get lost or broken. Access Virtual manipulatives can be accessed anywhere there is internet access including in student homes. Significant research has been conducted on concrete manipulatives that demonstrate their value for deepening conceptual understanding [1]. Early research on virtual manipulatives attempted to build on research on concrete manipulatives and focused on comparing concrete with virtual manipulatives [2]. While research demonstrated benefits of virtual manipulatives, results vary widely depending on student preparation and stage in learning, instructional strategies, and targeted topics and learning outcomes. Additionally, results showed that benefits can be gained by using them together. III. THEORY FOR THE ADAPTATION OF INTERACTIVE ONLINE LEARNING ACTIVITIES (TADRIOLA) As a significant number of NLVM tools began to be developed, researchers turned their focus to studying classroom use of interactive math tools. Researchers developed a theoretical framework for understanding classroom use of interactive math technology and guiding the design of tools to help support effective, efficient, enjoyable use of that technology. A. Theory Development Based on pilot studies and a review of the literature, researchers proposed a theory, called TATSTAM, that described barriers to use of mathlets and tasks or activities teachers carry out while using mathlets, and prescribed guidelines for the design of tools to support effective use of mathlets [3]. TATSTAM research included the development of an authoring tool called TADRIOLA to support teacher use of interactive online math tools. The formative research methodology depicted in Figure 2 was used to evaluate and iteratively improve the theory. Researchers used focus group, user test, interview, and questionnaire techniques with over 40 K-12 math teachers to iteratively evaluate and refine TASTAM and TADRIOLA. Figure 2 Formative Research Methodology Used to Develop TATSTAM B. Barriers to Teacher Adoption of Mathlets Barriers identified by TASTAM include: Preparation time - Many teachers think that it takes too long to reuse and adapt mathlets. Delivery time - Teachers find that the curriculum is so full and that the pressure to cover all of the content that is tested makes it difficult to find time to include mathlets in their teaching. Insufficient skills - Many teachers think that they lack the technical skills needed to reuse and adapt mathlets. Technology availability - Many teachers think that they do not have adequate, reliable, and timely access to the hardware, software, and network connections needed to use mathlets in their teaching. Resource inflexibility - Many teachers think that most mathlet-based online materials cannot be easily modified in the ways that they want. Context switching - Many teachers think that having to switch back and forth between the different tools needed to reuse and adapt resources is complex and confusing. Grain size mismatch - Many teachers think that the size of online learning resources is often larger than is needed and it is difficult to use only the relevant parts. Search difficulties - Many teachers think that it is hard to find appropriate quality resources.

3 Resource unfamiliarity - Many teachers experience difficulty figuring out how to adapt online resources that they find to fit what they are teaching. C. Teacher Goals Influencing Use of Mathlets During TATSTAM research it became apparent that the goals that teachers were trying to accomplish greatly influenced their attitudes towards and use of technology in teaching. Goals identified by TASTSAM as being important to how teachers use technology in their teaching included: Motivate students - Maintain student interest and motivation for learning. Meet individual needs - Meet the needs of individual learners by providing self- paced learning opportunities. Cover important content - Cover content specified by state standards and that teachers feel is important. Use effective materials and technology - Use computers and other technologies and materials in ways and contexts where they have demonstrated effectiveness. Write activity instructions - Write instructions telling students how you want them to interact with mathlets to solve problems or complete activities. Ensure reliable access to materials - Place prepared online materials on systems that can be reliably accessed at times when they are to be used. Use resources - Use prepared online materials with students. Assess student performance - Easily assess work that students do while working with mathlets. Share resources - Share created or adapted courses, lessons, activities and mathlets with other teachers and specify the conditions of their reuse. As illustrated in Figure 3, these tasks should be considered in the context of the the goals a teacher is trying to accomplish. Manage preparation and delivery time - Balance the time requirements of reusing and adapting mathlets with other demands on teacher time. Effectively use class time. Figure 3 A model of teacher use of mathlets. Monitor student learning - Track student activities, performance, and learning. D. Model of Teacher Use of Mathlets TATSTAM identified the following tasks that teachers wanted to accomplish as part of using mathlets: Find reusable resources - Find online courses, lessons, activities, and mathlets appropriate for teaching chosen learning objectives. Understand found resources - Learn enough about a resource that has been found to see how to reuse and adapt it. Organize class materials - Select, group, and organize online class materials so that they can be easily accessed from a single place. Prepare lessons and online tests - Select, group, and sequence online activities into lessons designed to support target learning outcomes. Change the appearance and functionality of mathlets - Change the appearance and functionality of mathlets to make them appropriate for given activities. E. Design Guidelines for Interactive Online Learning Environments TATSTAM research produced the following guidelines for the design of authoring tools for mathlets: Preview without registering - Make it possible to preview functionality and available resources without registering. Variety of high quality resources - Provide access to high quality learning resources appropriate for teaching the content areas that the teachers who your tool is designed for cover. Search tool and multiple types of indexes - Make it easy to find appropriate resources by browsing and

4 searching using classifications schemes that teachers are familiar with such as state standards. Adaptation tools - Provide features that allow teachers to modify resources that they have found. Activity list editor - Provide features for selecting and sequencing activities in a given lesson. Activity editor - Provide, on a single screen, which resembles what students see, tools for viewing and editing all aspects of a given activity. Activity preview tool - Allow teachers to quickly view exactly what an activity that they are editing will look like to students. Mathlet chooser - Provide indexes of available mathlets and can be used to choose ones to build activities around. Web resource finder tool - Provide a feature for finding resources on the web and including ones that have been found in an activity. Mathlet configuration tools - Make it possible for teachers to change the appearance and functionality of mathlets without programming. Instructions editor - A feature for adapting and writing instructions for activities. Assessment tools - Provide features that allow teachers to provide controls that students can use to submit or print their answers and work. Provide features that allow teachers to review, grade, and give feedback on student work. Class, lesson, and activity sharing tools - Features for sharing classes, lessons, and activities that have been created. Usability - Follow standard usability guidelines and user testing processes to insure all features are usable by teachers. Follow software conventions, use familiar terminology, and provide easily understood navigation controls. Offline versions - Provide versions that allow materials to be authored and used without being connected to the Internet. Support information - Provide online help information, user manuals, and training. IV. ENLVM In 2004, researchers obtained NSF funding for the enlvm project ( to extend the NLVM based on TATSTAM research. A. Interdisciplinary Team Approach to Development The enlvm worked with inter-disciplinary teams to develop interactive online lessons called emodules. Teams included classroom teachers, math educators, mathematicians, instructional designers, programmers, and graphic designers. The development process was iterative and involved creating design documents, paper and functional prototypes and conducting expert reviews, user tests and field tests. Teams that included teachers and math specialists who were also part of the Maine Laptop Project used a Lesson Study approach to develop the lessons. Each developed emodule included a lesson plan to guide teachers, one or more lessons consisting of a sequence of activities that utilized mathlets, and an online quiz. Indexes into emodules were organized according to courses in which the emodules could be used. B. Selecting Topics and Tools emodule topics were selected by taking into consideration: What are important topics to cover as identified by state and national standards? What topics are difficult to teach or learn using traditional methods? What topics can the unique characteristics of technology provide significant benefit? What technology tools already exist that can be leveraged to create emodules efficiently? C. Activity Authoring and Delivery In order to author, deliver, and allow teachers to track student performance, the enlvm developed a webapplication uniquely suited to support interactive online math tools. Key functionality included in the enlvm webapplication: Mathlet configuration Authors can change mathlet functionality, appearance, and text through web interfaces that do not require programming. Mathlet seeding Authors can seed mathlets with data sets and figures so that they will be initialized that way for a specific activity. Mathlet saving Learners can save work that they do with a mathlet so they can return to it later and teachers can review it. Inclusion of 3 rd party tools Authors can use frames to seemlessly include resources from anywhere on the web in lessons they create. enlvm researchers obtained permission for teachers to do this from major mathlet providers and made it easy to do. When 3 rd party tools are displayed attribution is automatically displayed by the enlvm delivery tool.

5 Instruction authoring Authors can use web-based rich text editors to write activity instructions and questions to accompany mathlets. Question and feedback authoring Authors can easily include controls such as text boxes and buttons for learners to respond to questions the authors have written. Authors can also specify a variety of types of feedback and that answers be recorded. D. Adaptation Tools The enlvm provides multiple ways for teachers to adapt existing lessons. Teachers can copy lessons into courses they set up and reorganize, combine, and modify lessons and activities. E. Assessment Tools In order to help teachers monitor student work using the enlvm, the system records student responses submitted using mathlets and forms. Teachers can view responses in real-time for specific activities and for entire lessons, both for individual students and the entire class at one time. This functionality can be used both to view student work on lesson activities and online quizzes. Teachers report using this functionality to check student progress and identify points that need clarification. F. Publishing and Updating emodules The publishing model of the enlvm web application is that a teacher or author can make changes to the live version of a course, but if they want to allow it to be copied by other teachers, they publish a snapshot of the course. Teachers can then copy the course into their private area where they can then adapt it as needed. When a course is published, the system keeps track of where it came from so that when the original version is updated, teachers who have made copies of the lesson will be notified at their next login, so that they can retrieve updates if they choose to. V. RESEARCH ON VIRTUAL MANIPULATIVES This paper has focused on research related to teacher use of mathlets and the design of tools to support that research. As part of the enlvm project, an external evaluation was performed, case study research was conducted [4], and artifact analysis was performed [6]. A number of other sources report research on student learning from virtual manipulatives [7][8]. and to create web pages that contain the applets. Future work will focus on developing more complete virtual manipulativebased curriculum for targeted courses, developing manipulative-based online assessments, and the evaluating, improving, and demonstrating the effectiveness of manipulative-based online lessons. ACKNOWLEDGMENT The work described in this project was supported by National Science Foundation grants IMD and IMD REFERENCES [1] The Mathematics Education Center (2009). Abstracts of research on manipulatives. Retreived May 26, 2010 from [2] Drickey, N.A. (2001). A comparison of virtual and physical manipulatives in teaching visualization and spatial reasoning to middle school mathematics students. Unpublished doctoral dissertation, Utah State University. [3] Duffin, J. (2004). Theory for authoring tools that support teacher adaptation of mathlets. Unpublished doctoral dissertation, Utah State University. Available at: [4] Buffington, P. & Granofsky, B. (2008). The enlvm project: Promoting student achievement in mathematics through educator collaboration. Retrieved Mat, 26, 2010 from anofsky.pdf [5] Duffin, J. (2009). Research using materials from the NLVM. _Using_Materials_from_the_NLVM [6] Duffin, J. (September, 2006). When teachers reuse and remix interactive online resources. Proceedings of the Open Education 2006 conference. Logan, UT. [7] Center for Implementing Technology in Education (2006). Learning mathematics with virtual manipulatives. Retrieved May, 26, 2010 from [8] Deubel, P. (2010). Math manipulatives. Retrieved May, 26, 2010 from Computing Technology for Math Excellence Web site: VII. FUTURE DIRECTIONS The NLVM has been translated into Spanish, French, Simplified Chinese, Danish, and Arabic. Efforts are ongoing to translate the NLVM into additional languages including Portuguese. Desktop versions of the NLVM have been developed that provide additional functionality over the web version including the ability to access the tools without being connected to the internet, the ability to save and print applets,

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