| 摘要: | 數學幾何的學習，強調讓學生去靈活運用數學的多重表徵(Multiple Representations)，從虛擬教具(Virtual Manipulative)的操弄、數字與符號的書寫到言語的溝通和討論，使學生在實作、實測與直覺中，從視覺上去建構幾何物件的概念與知識。社會互動(Social Interaction)在幾何學習過程中扮演關鍵的角色，包含學習者與教師、學習者與同儕、學習者與教材內容、及學習者與學習工具之間的互動，透過互相觀摩、辯證、分享與合作，學生便能從群體的多元觀點思考數學解題的推理歷程並發展豐富的解題策略。文獻指出，學生對三維幾何的探索尤其重要，是個人身處世界中感知空間關係與理解各式表徵的基礎能力。在台灣九年一貫國民教育的數學指標中，三維幾何知識的建立著重在立體物件之體積與表面積的計算上，藉由單位立方體的堆疊活動來達成。因此，本研究以近年來被廣泛應用於教育學習的虛擬實境(Virtual Reality)為平台，發展「Interactive Virtual Mathematics Classroom (IVMC)」的多人互動虛擬學習環境(Multi-user Virtual Reality Interactive Learning Environments)，整合虛擬教具(Virtual Manipulative)與白板，設計一系列三維幾何的體積與表面積問題解決的活動，透過社會互動與探索虛擬世界裡的豐富表徵，增進學生三維幾何的學習成效，並檢視虛擬教具與白板在虛擬世界的問題解決中所發揮的效益，進一步分析學生對於問題解決所產生的社會互動行為與類別，對幾何學習中問題解決造成的影響。 本研究之實驗以兩個國小五年級班級為研究對象，分別為實驗組和控制組，以問卷調查方式與系統數據進行統計量的分析，並輔以錄音錄影與訪談進行質的分析。結果發現，使用IVMC進行幾何問題解決活動，實驗組的幾何學習成效優於控制組；實驗組在兩階段活動表現中，透過與同儕的社會互動，從虛擬教具的具體操作、白板的評論註記與言語的討論等面向思考解題過程與答案，發展出不同組合的幾何問題解決策略。 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. |
