國中生在動態幾何軟體輔助下臆測幾何性質之研究

Junior High School Students Conjecture Geometric Properties in a Dynamic Geometry Software Environment

本研究探討國中生如何在動態幾何環境下臆測幾何性質。研究以程序性反駁模式為中介理論架構設計幾何臆測學習單，目的是瞭解國中生如何在動態幾何環境中建構圖形案例，並依據案例來臆測正確的幾何性質。其中，本研究特別強調將動態幾何軟體定位為「例子產生器」，結合幾何圖形案例測量值的紀錄表格，鷹架學生進行臆測活動。研究樣本為15位七年級國中生，以質性分析方法為主、量化資料輔助說明下，研究發現（1）動態幾何環境下，幾何性質本身涉及測量值關係的複雜程度及幾何性質是否容易在圖形上視覺觀察，影響學生造例與臆測表現。同時這兩個因素影響學生在動態幾何環境下的認知行為和學習困難；（2）具備良好的幾何物件分類系統是在動態幾何環境中成功臆測的重要關鍵；（3）學生對圖形進行分解與重組操作有助於在動態幾何環境中察覺圖形中蘊含的特徵或關係；（4）學生能拖曳不同圖形案例並不等同他們能察覺符合命題結果的正反例，進而影響臆測結果；（5）學生仍缺乏動態幾何環境知識以建構原本意圖產生的圖形案例。另，本研究也依據學生在結合動態幾何與案例記錄表格的表現，區辨出不同臆測認知策略：分別為有限隨機離散案例歸納、系統性調整案例臆測以及動態性調整案例臆測。

This study investigated how junior high school students conjecture geometric properties in a dynamic geometry software (DGS) environment. Using the proceduralized refutation model as an intermediate theoretical framework, we particularly examined the process and the difficulties that students may have when conjecturing. Specifically, we referred to DGS as an “example generator” and combined it with spreadsheets to support students in conjecturing geometric properties. A total of 15 seventh grade students participated in this study. Based on the qualitative analysis and quantitative data, we demonstrated that (1) the complexity of the relationship among measurements involved in a geometric property and the possibility of visualizing that property play important roles in determining students’ performance when conjecturing in a DGS environment; (2) being able to effectively classify geometric objects was the key to successfully perceiving geometric properties and relationships embedded in geometric diagrams; (3) decomposing and recomposing diagrams aided students in recognizing embedded geometric properties; (4) the ability to drag a geometric diagram into different shapes in a DGS environment did not guarantee the ability to discern supportive and counter examples or the ability to use those examples to correct false conditional statements; and (5) a lack of knowledge specific to DGS environments, particularly those related to dragging, hindered students’ effective construction of diagram examples. Additionally, we identified three types of conjecture approach: induction by randomly generating a finite number of discrete examples, conjecture by systematically making examples, and conjecture by dynamically altering examples.