高一學生視覺化轉化為幾何推理之過程及其特徵

推理是數學研究與學習中的重要工作，演繹推理更是國中幾何學習的重點，卻也是學生不易掌握的內容之一。在幾何推理的過程中，圖像與概念密切地相互作用，且從圖像中獲取的資訊會與個體心智中的想法作連結，從而進行幾何推理的視覺化過程，將能提供學習者在推理過程中所展現特徵的重要參考。隨著科技快速發展，科技工具提供有效的構圖、視覺化及推理支助。然而，幾何推理對學習者來說仍然存在相當的困難，這樣的工具也沒有在學習幾何推理時被廣泛使用。本研究目的在探討已習相關幾何內容的高一學生，透過質性訪談分析他們從視覺化轉化為幾何推理的過程及特徵。訪談內容包含三個幾何推理任務，學生以口述方式在紙筆或動態幾何環境下說明對這些推理任務的想法，並整合Toulmin論證模型和在動態幾何環境推論的特色，由此分析訪談逐字稿及錄影影像進行學生之推理過程。研究結果顯示不同知識程度的學生在論述策略的選取、尋找不變量以及直觀條件的使用對其推理歷程有較大的影響，使用動態幾何軟體則有助於他們發展一般化的推理結果。由學生從視覺化到幾何推理的過程中，藉由幾何知識與幾何物件之間的連結，可以分為以物件外觀為主導、以物件元素為主導、以幾何知識為主導和以邏輯關係為主導四個階層描述，其中依據子圖的層次關係與圖形的結構，以幾何知識為主導和以邏輯關係為主導的階層又各細分為兩個層次來描述。未來教學及研究可考慮兼顧推理過程中各個階段以及培養不同視覺化轉化幾何推理階層的任務設計，並探討學生在上述任務設計的表現及主要困難，以進一步幫助學生發展適當的視覺化以達到不同階層的幾何推理。

關鍵字

動態幾何；推理過程；幾何推理；視覺化

並列摘要

Reasoning and argumentation are essential for mathematical studies and learning. Deductive reasoning is the main learning objective of geometry in junior secondary school; however, it is one of the hardest skills to master for students. In the process of geometrical reasoning, figures and concepts interact simultaneously. The process from visualization to geometrical reasoning is mainly supported by the connection between the information from the figures and the ideas from individuals’ mind. An important reference can be provided by revealing students’ characteristics during their reasoning process. With the rapid development of science and technology, technological tools can provide effective support for construction, visualization, and reasoning. However, geometrical reasoning is still difficult for students and the tools are also not yet widely accepted in the learning of geometrical reasoning. This study aims at discussing the reasoning process and characteristics for the shifts from visualization to geometrical reasoning by students in senior one who have already learnt geometry. Using a semi-structured interview, three reasoning tasks are used. Students are asked to orally present their ideas about the tasks with a paper-and-pencil or dynamic geometry environment. Their reasoning process is analyzed by their transcripts and captions using the combination of Toulmin’s argument model and the reasoning features of dynamic geometry environment. Results showed that selecting argumentative strategies, finding the invariances, and using intuitive judgment are the main elements that can influence the reasoning process for students with different levels of knowledge. The geometrical imagination mainly supports an effective conjecture and then the dynamic geometry environment encourages the generalization of reasoning results. Hence, the shifts from visualization to geometrical reasoning can be described as the connection between geometrical knowledge and geometric objects by four levels: appearance-oriented, element-oriented, knowledge-oriented, and logic-oriented. By the structure of the figure and the relation of the sub-figures, the knowledge-oriented and logic-oriented levels are also described by two sub-levels respectively. Future studies may consider designing geometrical tasks that include various stages of reasoning process and different visualization levels for geometrical reasoning instruction. Students’ performance and difficulties with these geometrical tasks are required to analyze and discuss for developing suitable visualizations with different levels of geometrical reasoning.