| dc.description.abstract | Learning geometry emphasizes the importance of exploring different representations such as virtual manipulative, written math formulas and verbal explanation, which help students build their math concepts and develop their critical thinking abilities for geometric problems solving. Besides for helping individuals construct math knowledge, social interaction of learning also plays one of crucial roles for contributing the development of mathematics understanding and geometry problem learning and solving. Regarding to social constructivism for geometry learning, peer learning behaviors of collaboration in social contexts with negotiating and sharing ideas are keys to deepen students’ understanding of geometric properties from multiple viewpoints. In this research, a Collaborative Virtual Reality Learning Environment (CVRLE) called interactive virtual mathematics classroom (IVMC) was proposed and developed for 3-dimensional (3-D) geometry problem solving. In IVMC, each student try to solve geometry problems related volume and surface area on their own tables with two representational tools, virtual manipulative and the whiteboard in CVRLE. The virtual manipulative allows multiple users to build, move and stack various shapes of 3-D objects for calculating volume and surface area of geometric problems. Students could validate their ideas by examining their manipulation and learn geometric concepts by observing peers manipulation in CVRLE. Regarding the whiteboard mechanism, it not only allows users to share ideas by writing mathematic formulas and drawing figures on their own whiteboards, but also to gives comments to help peers on peers’ whiteboards. Therefore, IVMC can provide a highly motivational and engaging learning environment for students to manipulate 3-D virtual objects and solve geometric problems individually or collaboratively. Various kinds of peers learning behaviors such as actively assisting peers in demonstrating, manipulating and criticizing or passively taking observation to reflect and revise solutions are explored and their effects on learning were further investigated.
One eight-week experiment was conducted with two classes of fifth-grade primary school students, one class as experimental group and the other as control group. Subjects engaged in two geometry problems solving: the calculation of volume and surface area of 3-D objects. Results showed that there was a statistically significant difference in learning achievement between the experimental group and control group. Further analysis showed that peers learning behaviors, virtual manipulative, mutual observation and comments on the whiteboard, significantly influenced learning achievement and their problem solving strategies as well. Interestingly, peer learning behaviors in the two kinds of geometric problems were different due to their different difficulty level of problems. In the easy geometric problem, volume calculation, subjects first worked out their solutions, afterwards actively helped peers to solve problems by manipulating the virtual objects or commenting on the whiteboard. The number of helping peers with the virtual manipulative was significantly related to the outcome of volume calculation. However, in the difficult problem, calculation of surface area, most subjects were found to observe peers’ solutions on the whiteboard, and then went back to try to solve their own. The subjects having good learning performance in math were observed frequently and also received many comments on their whiteboards from peers. It was found that the number of observation and comments from peers was significantly related to the outcome of surface area calculation.
Therefore, peers behaviors of social interaction of learning in IVMC were found useful to facilitate geometry problem solving with sharing ideas and exploring multiple representations in CVRLE. The difficulty level of the geometry problems would lead to various peers learning behaviors, which in turn cause impacts on learning achievement.
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