visual explorative approaches to learning mathematics - CiteSeerX

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VISUAL EXPLORATIVE APPROACHES TO LEARNING MATHEMATICS Zekeriya Karadag OISE/ University of Toronto [email protected]

Douglas McDougall OISE/ University of Toronto [email protected]

This discussion group focuses on visual explorative approaches to learning mathematics. We address several issues in the discussion such as technology use in mathematics education and its evolution from static to dynamic in conjunction with the visual characteristics of new learning form of mathematics. Among various representations used in mathematics and mathematics education, visual representations of mathematical concepts, the effects of the implementation of visual techniques, and more importantly the importance of visual exploration of mathematics and its effects to mathematics education will be discussed. Background The focus of this discussion group will be situated at the intersection of the technology use in mathematics education, representation systems in mathematics education, and developing a conceptual understanding in mathematics. An enormous corpus of literature has been accumulated on these topics, and many researchers discussed the topics in the various national and international meetings such as PME, PMENA, and ICMI (Arcavi, 1999; Duval, 1999; Hitt, 1999; Hoyles, 2008; Kaput, 1999; Kaput & Hegedus, 2000; Leatham, & McGehee, 2004; McDougall, 1999; Moreno-Armella, 1999; Presmeg, 1999; Radford, 1999; Santos-Trigo, 1999; Thompson, 1999). Yet, our main focus will be on the visual exploration of mathematics and on the visual versus algebraic understanding of mathematics, particularly in the technology supported learning environments. We will also explore the distinctions between visualization of mathematics, visual exploration of mathematics, and visual understanding of mathematics. We believe that the result of this conceptual exploration will lead us to improve our understanding of the theories on representational systems (Kaput, 1992) and distributed cognition between human and technology (Pea, 1993). Interestingly, the importance of exploration and visualization was emphasized in the past PMENA conferences. For example, in a paper presented in 1999 PMENA, the author points out the relationship between technology and representation: Technology, in all its forms, modifies, substantially, the process of knowledge production. Learning involves the construction of representations. It is through the construction of representations of an observed phenomena, (or of a mathematical concept) that we make sense of the (mathematical) world. Representations become mediational tools for understanding (Moreno-Armella, 1999, p. 99) Similarly, in another PMENA meeting, the authors explore what we know about technology use and its effects in teaching and learning mathematics: Much of what we know about the use of technology in the teaching and learning of mathematics is anectodal and might be referred to as “possibility” research…. What do we really know regarding teaching and learning mathematics with technology? What frameworks, methodologies and collaborations will support the research that will produce this knowledge?(Leatham & Peterson, 2005, p. 1) Some authors propose new perspectives for the use of technology rather than digital interpretations of traditional paper-and-pencil techniques. They suggest using dynamic, interactive, and collaborative features of technology: Swars, S. L., Stinson, D. W., & Lemons-Smith, S. (Eds.). (2009). Proceedings of the 31 st annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education. Atlanta, GA: Georgia State University.

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The advance of dynamic technological environments allows us to combine multiple individual cognitive acts of reference. This is possible since individuals can project their intentions and expressivity through the notations they create and share. They can also realize and generalize the structure of the mathematics through co-active collaboration with these environments. This can be made possible through the advances in representation infrastructures (dynamic mathematics software) and communication infrastructures (social and digital networks). (Moreno-Armella, Hegedus, & Kaput, 2008, p. 110) Various Types of Visualizations The literature documents at least three distinct meaning for the term visual mathematics such as (1) for studying advanced visual objects, (2) for visualizing algebraic rules, and (3) for exploring mathematics visually. The first one, studying advanced visual objects such as fractals and 3D functions, is a focus where mathematicians use technology to extend their imaginations and to make their ideas visible whereas the other two are related to mathematics education. Virtual manipulatives (NLVM, 1999; SAMAP, 2006), learning objects (Reis & Karadag, 2004; Sorkin, Tupper, and Harmeyer, 2004), and animations (Tchoshanov, n.d.) are the examples to visualize mathematical concepts, rules, and relationships. The idea behind creating this group of mathematical objects can be described as to make mathematical rules visible in order to improve students’ understanding. For example, the example from Tchoshanov’s (n.d.) collection visualizes a very well-known algebraic identity (figure 1). He visualizes the identity of a b2

a2

2ab b 2 by using animations.

Figure 1. Screenshots from Tchoshanov's animations.

Similarly, virtual manipulatives developed by Utah State University team serve for the same purpose. For example, they use virtual algebra tiles to illustrate distributive law of algebra (figure 2). The figure illustrates a geometric representation to explain the rule for the example x (y + z) = xy + 2x.

Figure 2. Algebra tiles as virtual manipulatives.

Swars, S. L., Stinson, D. W., & Lemons-Smith, S. (Eds.). (2009). Proceedings of the 31 st annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education. Atlanta, GA: Georgia State University.

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However, these virtual manipulatives could also be used to explore mathematics visually. For example, student may develop an insight for patterning while using Towers of Hanoi example (figure 3). This example provides a scenario to encourage students explore the problems illustrated. Students, with or without guidance, are expected to realize the pattern while performing the task.

Figure 3. The Towers of Hanoi as virtual manipulatives.

Another exploratory environment to encourage students to improve their patterning skills is the Math Towers. The Math Towers provides scenarios such as billiard boards to engage students to develop some patterning relationships (figure 4). The scenarios illustrated in the Math Towers and the Towers of Hanoi aim to engage students to be part of a virtual environment and to explore mathematics visually.

Figure 4. The Math Towers.

Moreover, we have dynamic learning environments to create dynamic worksheets. For example, Geogebra, a free online dynamic software, allows us to create mathematical objects and to explore these objects visually and dynamically. In a study, we created a dynamic worksheet illustrating the relationship between unit circle and trigonometric functions (figure 5). We asked students to manipulate these mathematical objects and explore the relationship between them (they were not told that the functions were trigonometric).


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Figure 5. Visual exploration in Geogebra.

As seen in the examples described here, there are different types of visualizations. The cognitive collaboration between human and tools seem to be quite distinct in these examples although we call them all as visualization. Thus, it is important to identify this distinction and to develop theories explaining various visualization types. More importantly, some scholars argue that mathematics education evolves through integration of technology and needs a groundbreaking change to complete its evolution (Galbraith, 2006; Moreno-Armella, Hegedus, & Kaput, 2008). Moreno-Armella, Hegedus, and Kaput (2008) describe symbolic structures as “an environment that enable us to think deeper” (p. 100) and argue that “the nature of mathematical symbols have evolved in recent years from static, inert inscriptions to dynamic objects or diagrams that are constructible, manipulable and interactive” (p. 103). According to their perspective, students may think, reflect on their thoughts, and construct new mathematical knowledge in the dynamic learning environments. The dynamic learning mathematical environments enable students act mathematically, such as seeking visual patterns, define these patterns, and explore the properties of these patterns as they usually do symbolically in paper-and-pencil environments. The Rationale and Goals of Discussion Group The goals of this discussion group are to explore and discuss the dynamic and visual features of technology in mathematics education, to develop awareness on the potential effects of visual exploration in conceptual understanding of mathematics, and to set up a research agenda on the study of visual exploration in mathematics education. The discussion group will review the past and current use of technology as visual and dynamic cognitive tools and extend the theory of distributed cognition over the participatory mathematical activities. This discussion will deepen our understanding of the effects of visual explorative activities in learning mathematics and seek possible strategies to integrate web 2.0 technologies with dynamic learning tools. Objectives To perform a comprehensive review of various representational systems used in mathematics in junction with a linkage among them To explore various perspectives of visual learning in mathematics To explore the features of dynamic learning systems in conceptual understanding of mathematics To identify possible effects of visual and dynamic learning environments in mathematics education Swars, S. L., Stinson, D. W., & Lemons-Smith, S. (Eds.). (2009). Proceedings of the 31 st annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education. Atlanta, GA: Georgia State University.

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To explore the possible scenarios for the future of the mathematics education supported with contemporary technologies such as Web 2.0, dynamic learning environments, and visual representation Questions to frame our discussion Which representation types are used in mathematics education? What types of visualizations do we use and how could they be identified? Which representation is more compatible with the nature of human learning? Could algebra be a barrier to learn mathematics? How can we observe or track the effects of visual versus symbolic representations in learning mathematics and their effects on concept development? How could the Web 2.0 technologies and dynamic learning environments be integrated to engage students for visual exploration of mathematical concepts? Format for the Discussion Group Meeting Day 1: Identifying the Current Situation and Contemporary Perspectives The organizers of the group will begin the discussion with a brief introduction of the topic and some quotes from literature. Then, we will ask the participants to reflect on these quotes in small groups and share their reflections with whole group. The second hour of the discussion will focus on the technology, representation, and visualization. We will engage the participants to brainstorm on the expectations of technology in math education, on the various forms of representations used in mathematics, and on the meanings of visualization in small groups and share their reflections with whole group. Introducing group members Introducing discussion group topics and goals Reviewing selected quotes form previous work and engaging participants to reflect on these quotes Brainstorming on the expectation of technology in mathematics education Brainstorming on the various forms of representations used in mathematics Brainstorming on the meanings of visualization The quotes will be the evolution of the mathematics education (Moreno-Armella, Hegedus, & Kaput, 2008) and the metaphor of the technology use (Galbraith, 2006). Day 2: Projecting on the Future We will start the meeting by briefing previous day’s discussion and by outlining themes emerged from the discussion. Then, we will demonstrate a couple of dynamic worksheets created by Geogebra and ask participants to develop scenarios in their small groups on how to use these dynamic worksheets in classrooms and to share their scenarios with the whole group. During the last half hour, we will encourage the participants to discuss the opportunities of integrating Web 2.0 technologies (i.e. wikis) with these dynamic worksheets. Day 3: Summarizing the Discussions and Setting Goals for the Future Collaboration Opportunities Summarizing previous discussions Reminding the metaphor by Galbraith (2006) and engaging to brainstorm on the mathematics education for the next fifty years Forming an international working group on the topic Setting an agenda for the future Swars, S. L., Stinson, D. W., & Lemons-Smith, S. (Eds.). (2009). Proceedings of the 31 st annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education. Atlanta, GA: Georgia State University.

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Possible Future Agenda Items Set up a working group for dynamic and visual learning Create possible research questions Seek possible strategies to disseminate the themes emerged through discussion References Arcavi, A. (1999). The role of visual representations in the learning of mathematics. In F. Hitt & M. Santos (Eds.), Proceedings of the Twenty First Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, 55-80. Duval, R. (1999). Representation, vision and visualization: Cognitive functions in mathematical thinking: Basic issues for learning. In F. Hitt & M. Santos (Eds.), Proceedings of the Twenty First Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, 3-26. Galbraith, P. (2006). Students, mathematics, and technology: assessing the present –challenging the future. International Journal of Mathematical Education in Science and Technology, 37(3), 277-290. Geogebra (2001). [Web site]. http://www.geogebra.org. Hitt, F. (1999). Representations and mathematics visualization. In F. Hitt & M. Santos (Eds.), Proceedings of the Twenty First Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, 137-138. Hoyles, C. (2008). Transforming the mathematical practices of learners and teachers through digital technology. 11th International Congress on Mathematical Education. Monterrey, Nuevo Leon, Mexico. Kaput, J. J. (1999). On the development of human representational competence from an evolutionary point of view: From episodic to virtual culture. In F. Hitt & M. Santos (Eds.), Proceedings of the Twenty First Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, 27-48. Kaput, J. J. (1992). Technology and mathematics education. In D. A. Grouws (Ed.), Handbook of Research on Mathematics Teaching and Learning. USA: The National Council of Teacher of Mathematics. Kaput, J. J. & Hegedus, S. J. (2000). An introduction to the profound potential of connected algebra activities: Issues of representation, engagement and pedagogy. Proceedings of the 28th International Conference of the International Group for the Psychology of Mathematics Education, 3, 129-136. Leatham, K. & McGehee, J. (2004). Geometry and technology. In McDougall, D.E & Ross, J. A. (Eds.), Proceedings of the twenty-sixth annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, 1, 65-66. Leatham, K. R. & Peterson, B. E. (2005). Research on teaching and learning mathematics with technology: Where do we go from here? In G. M. Lloyd, M. Wilson, J. L. M. Wilkins, & S. L. Behm (Eds.), Proceedings of the 27th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education. MathTowers (2002). A mathematics learning environment for Grades 7 & 8. [web site]. http://www.math-towers.ca. McDougall, D. (1999). Geometry and technology. In F. Hitt & M. Santos (Eds.), Proceedings of the Twenty First Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, 135-136. Swars, S. L., Stinson, D. W., & Lemons-Smith, S. (Eds.). (2009). Proceedings of the 31 st annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education. Atlanta, GA: Georgia State University.

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Moreno-Armella, L. (1999). On representations and situated tools. In F. Hitt & M. Santos (Eds.), Proceedings of the Twenty First Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, 99-104. Moreno-Armella, L., Hegedus, S. J., & Kaput, J. J. (2008). From static to dynamic mathematics: Historical and representational perspectives. Educational Studies in Mathematics, 68, 99111. NLVM (1999). National Library of Virtual Manipulatives. [Web site]. http://nlvm.usu.edu. Pea, R. D. (1993). Practices of distributed intelligence and designs for education. In G. Salomon (Ed.), Distributed cognitions: Psychological and educational considerations. USA: Cambridge University Press. Presmerg, N. C. (1999). On Visualization and Generalization in Mathematics. In F. Hitt & M. Santos (Eds.), Proceedings of the Twenty First Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, 1, 151155. Radford, L. (1999). Rethinking Representations. In F. Hitt & M. Santos (Eds.), Proceedings of the Twenty First Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, 147-150. Reis, Z. & Karadag, Z. (2004). 3D Method in Computer Based Instruction. In L. Cantoni & C. McLoughlin (Eds.), Proceedings of World Conference on Educational Multimedia, Hypermedia and Telecommunications 2004, 3259-3264. Chesapeake, VA: AACE. SAMAP (2006). Sanal Matematik Manipulatifleri. [Virtual Mathematics Manipulatives]. [Web site]. http://samap.ibu.edu.tr/. Santos-Trigo, M. S. (1999). The use of technology as a means to explore mathematics: Qualities in proposed problems. In F. Hitt & M. Santos (Eds.), Proceedings of the Twenty First Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, 139-146. Sorkin, S., Tupper, D., & Harmeyer, K. (2004). Teaching Mathematics, Science, and Technology Teachers How to Create Instructional Multimedia. In L. Cantoni & C. McLoughlin (Eds.), Proceedings of World Conference on Educational Multimedia, Hypermedia and Telecommunications 2004, 3432-3437. Chesapeake, VA: AACE. Tchoshanov, M. (n.d.). Visual mathematics. http://dmc.utep.edu/mouratt/ . The Math Towers (2002). A mathematical environment for Grades 7 & 8. [Web site]. http://www.math-towers.ca/ Thompson, P. (1999). Representation and evolution: A discussion of Duval’s and Kaput’s papers. In F. Hitt & M. Santos (Eds.), Proceedings of the Twenty First Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, 1, 49-54.
