Session Information
24 SES 03, Creative Learning Systems in Mathematics Education
Paper Session
Contribution
This paper presents a case study for the learning mathematics in dynamic learning environments by revisiting Polya’s (1945) problem solving strategies and Kaput’s (1992) multiple and dynamic representation vision. We, then, focus deeply on a specific theme, the lack of understanding, emerged from the study.
The purpose of this research was to understand students’ problem solving activities in the dynamic learning systems, particularly GeoGebra, and to access their mathematical thinking processes guiding these activities. Therefore, we explored two main research problems: (1) How do students explore problem space in DLS and working on the problem? (2) How do they use the features of GeoGebra? Until this stage of the study, we have documented that the understanding stage of problem heuristics described by Polya (1945) seems a challenging stage leading students to a dead end while solving their mathematics problems.
Polya (1945) defines a heuristics about problem solving steps that has been widely accepted: (1) understanding the problem, (2) creating a plan to solve unknown, (3) carrying out the plan, and (4) looking back. He claims that people start working on the problems by attempting to understand the problem if the problem itself is unfamiliar enough to be a problem.
We assume that understanding a problem and developing conceptual understanding stand on similar cognitive skills. Regarding understanding the problem and developing a conceptual understanding, Kaput (1992) envisions the use of multiple representations and a linkage among these representations that constitutes a theoretical framework for dynamic learning environments. Students in technologically rich classrooms are more familiar to develop multirepresentational views of mathematics. Some technologies provide them to develop almost a kinematic understanding of functional relationships (Heid, 2005, p. 357-358). Dynamic learning environments provide users an explorative environment allowing multiple representations, which users can create mathematical objects, and connections among these objects, manipulate these objects and connections with various levels of constraints, and observe change in other representations while manipulating one representation.
GeoGebra as an example of these systems provide visual, algebraic, written, and numeric representations depending on users’ preference (Hohenwarter, 2002, para. 2). Students can create mathematical objects linked to each other. It allows students observing the change in one of the objects when they manipulate the other. Kaput’s (1992) and our assumption are that students can also mentally visualize their interaction when they manipulate and observe their interaction in this environment. Of course, mental interaction is completely different than what is seen on the screen, because the mental interaction involves conceptual definitions, a definition of a function and the derivative function of that function, and connection between those definitions, such as derivative function and the function itself. Moreover, students make use of algebraic representations provided by GeoGebra to explore algebraic relations between objects. For example, in our research, students could see the change in measures of angles and the lengths of sides while manipulating the corners of the figure given in the problem.
Method
Expected Outcomes
References
Hohenwarter, M. (2002). What is GeoGebra? Retrieved from http://www.geogebra.org on January 13, 2010. Heid, M. K. (2005). Technology in Mathematics Education: Tapping into Visions of the Future. In W. J. Masalski and P. C. Elliot (Eds). Technology-Supported Mathematics Learning Environment. Reston, VA: The National Council of Teachers of Mathematics. Kaput, J. (1992). Technology and mathematics education. In D. Grouws (Ed.), Handbook of research on mathematics teaching and learning, pp. 515–556. New York: Macmillan. Karadag, Z. (2004). Hatalardan ogrenme yontemi: Koordinat duzlemi ve simetri ornegi. [Learning from mistakes: An example on coordinate plane and symmetry]. 4th International Educational Technology Symposium. Sakarya, Turkey. Karadag, Z. (2009). Analyzing students’ mathematical thinking processes in technology supported environments. Toronto: Unpublished PhD dissertation. Kunichika, H. Hirashima, T. and Takeuchi, A. (2006). Visualizing errors and its evaluation. In M. Ikeda, K. Ashley, and T.-W. Chan (Eds.). ITS 2006, LNCS 4053, pp. 744-746, 2006. Polya, G. (1945). How to solve it. Princeton: Princeton University Press.
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