國中生線對稱概念學習研究

Abstract

摘要 對稱不僅是自然界中優美的造形，更是重要的數學概念。最近實施的九年一貫數學領域能力指標中更明列對稱概念的能力發展。本文第一部份以研究國中生對於線對稱概念所形成之概念心像為主，進而分析其迷思概念與其成因，並同時採取個案訪談的質性分析與問卷測驗的統計調查，依據線對稱概念心像的內涵︰典範現象、部份─全體推理及概念屬性的了解，設計面談診斷工具及問卷測驗工具。依平時數學表現，選取高、中、低三層次各兩位國二學生作為訪談對象分析學生之線對稱概念結構，並在大台北地區選取國一，國二，國三各兩班，約兩百位學生，實施開放性的線對稱概念測驗，分析中學生對於線對稱概念的了解、運用情形及迷思概念與推理策略。第二部分以Van Hiele(1984)發展層次為結構，設計線對稱概念試題，在北部選取國一、國二、國三各三班，約三百位學生作為問卷測驗對象，藉此了解中學生之線對稱概念發展情形。第三部分是以之前研究成果為參考，配合數學學習設計相關理論，設計實驗教材，以兩班國中二年級為對象，一班為引進GSP動態多重表徵教學環境的實驗組，另一班為傳統教學環境的對照組，實施線對稱概念實徵教學研究。 研究結果顯示（一）國中生對於線對稱概念呈現垂直或水平對稱軸的典範現象，且解題時多以典範例的概念心像而非採取概念定義處理問題。（二）國中生之線對稱概念大多在第一、二思維層次上，有中等程度的獲得，而二、三年級的思維層次沒有太大的差異（三）在GSP動態多重表徵環境教學環境中，學生對於概念心像的操作較為活躍，且典範現象之排他性也比傳統環境為低，在概念發展方面，實驗組在三、四層次的思維上有較高的提昇。但是，實驗組的低層次學生在各層次概念發展上，並沒有明顯進步。由此，我們認為，盡管是動態多重表徵環境，教師仍應注意學生學習情況，並且在適當時機作概念上的統整，以使學生的學習更為順暢與完整。

Abstract Symmetry is not only one of the most beautiful modeling in the natural world, but also the important concept in Mathematics. Recently, the ‘6-15 Curriculum Syllabus’ for elementary and junior high schools in Taiwan. Especially in the domain of Mathematics, was clearly defined the development of ability of symmetry concept. The major work of the first part of this paper is researching junior high school students’ concept image which carried from their line-symmetry concept, and proceed to the next step is analyzing the lost concept and the reasons it occur. In the meantime, we select qualitative research of case studies and adopt statistic investigate of questionnaire. In accordance with the intension of the concept image ： Prototypicality 、Part-Whole Reasoning and Attribute Understanding , we designed tools for interviewing and testing . According to their Mathematics grade , we took apart second grade students form junior high school to three levels : higher、middle and lower , and selected two students in each level for case studies to analyze their contexture of line-symmetry concept. Two classes a grade in Taipei, about two hundred students, we used an open questionnaire to analyze the understanding、operating、enginery using and lost of their line-symmetry concept. The second part of this paper is structured by Geometry development level ( Van Hiele, 1984)，depending on this idea, the questionnaire for line-symmetry concept was designed . Three classes a grade, we selected nine classes about three hundred students from three grade of junior high schools in Taipei area to be the object of the questionnaire. From this research we understood the development of line-symmetry concept of students who were in junior high school. The third part of this paper is based on previous results. Collocated theories of mathematics learning and previous results, we develop teaching materials for experiment. We selected two classes in second grade of junior high school for experiment of teaching and learning of line-symmetry concept. The one recommend GSP (The Geometer’s Sketchpad) which is dynamic-link multiple characteristics teaching and learning environment, and the other in traditional teaching environment is to comparison. The results of our research reveal three points : (1) The line-symmetry concept of students of junior high school has Prototypicality of vertical or horizontal symmetry axle. Instead of using definition of concept, they always use prototype model of concept image to solve problems . (2) On first and second thinking levels of line-symmetry concept of junior high school’s students almost reach intermediate acquisition. (3) In GSP learning environment of dynamic-link multiple characteristics, the operation of students’ concept is actively, and the exclusiveness of prototypical is lower than traditional environment. In aspect of development of concept, the class of experiment has promoted to higher grade in third and fourth thinking level. But the lower level students of this class have not clear advancement in each thinking level. So we conclusion that for the purpose of learning more smooth and more complete, the teacher should pay attention to the situation of students’ learning and summarize important concept in suitable time although in dynamic-link multiple learning environment.