學生在動態、互連、多重表徵函數的電腦環境下學習平移及標尺變換之思考模式

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Title:	學生在動態、互連、多重表徵函數的電腦環境下學習平移及標尺變換之思考模式 Thinking Types of Students' Learning of Translation and Scaling in Dynamic, Linked, Manipulable Environments
Authors:	林保平
Date:	1995-06
Issue Date:	2011-12-07 13:25:22 (UTC+8)
Publisher:	臺北市立師範學院
Abstract:	中文摘要　　有關函數和方程式圖形之平移和標尺變換的教學，通常均只重視代數式表徵。本研究利用動態，互連，多重表徵函數的電腦環境，來探討學生如何經由直接移動和變換圖形來學習平移及標尺變換。在這個電腦環境下學生可自由移動及變換圖形和坐標軸。對照三種不同的平移和標尺變換是本研究的特色。我們發現學生似乎作三種不同層次的思考－表徵的、實體的、以及操作運思的。雖然學生都經常直接操作圖形來幫助他們思考（實體層次），但他們將操作當作思考對象的運思卻有不同，他們的操作運思似乎正好與他們的數學學習經驗相反，高三的學生似乎不如高二及高一的較有創造性，而直接明白地討論逆反操作似乎可以幫助學生發展操作運思的能力。三名學生中，有兩名使用皮亞傑的群關係。在平移和標尺變換的學習上這種群關係幫助學生解題和組織他們的知識。三名學生都使用了倒推替代的基模。但是物件移動及變化的處理方式似乎也帶來了一些視覺和直觀上的錯覺。我們發現學生在圖形和坐標軸的平移及標尺變換上都有反轉的誤謬，當他們純粹依賴直覺時，經常把點對點的變換與函數對函數的變換弄混。學生也十分依賴外在環境所提供的線索來得到認知衝突而沒有自行監控自己的思考過程。協助學生發展自動回饋並因而能肯定自己的答案似乎應為數學教育的重要目標。 Abstract 　　The teaching and learning of translation and scaling related to graphs of functions or equations often emphasize the algebraic aspect the representation. The teaching experiments described in the paper used the manipulable, dynamic, linked, multiple representation computer environments to help students understand the relationship of the movement or variation of the equation graph and its algebraic representation. In these environments, students could manipulate (translating or scaling) the graphic objects and the coordinate system. Contrasting three kinds of translation and scaling was emphasized-transforming the graph only, transforming the coordinate system only, and trans-forming both the graph and the coordinate system simultaneously. Three students with different levels of mathematical learning experience participated in this study. They were in algebra I , algebra. II and algebra III courses respectively. Students were found to think in three planes-representational, physical, and operational planes. Although all of the students often used the function provided in the environments to manipulate the graphic objects (thinking in the physical plane), there were differences in thinking operationally. The degree of operational thinking seems to correspond in reverse to their mathematical experiences. The student with the most mathematical experience appeared to think less creatively and less operationally than the other two students. The student's algebraic sophistication may have inhibited her operational thinking, while the explicit discussion of the inverse action of translation seems to have been helpful in developing this operational thinking. Two students were found to use Piaget's INRC group relationship. In the realm of translation and scaling, these relationships helped them solve problems and organize their knowledge. But the basic scheme all the students used was the backward-substitution scheme. The object movement or variation approach seemed aslo to provoke some visual and intuitive errors. The reversal errors were found in both translation and scaling of graph and axes. Students often confused the point-to-point transformation with the function-to-function transformation when they relied purely on their intuition. Students were also found to rely on external stimuli to provide cognitive conflicts and did not appear to monitor their thinking although they were all viewed as "good" in mathematics. Helping students to develop an automatic feedback system to monitor the thinking process and hence confirm the results automatically by themselves seems to be a very important aim to mathematics education.
Relation:	臺北市立師範學院學報 26期 393-422
Appears in Collections:	[Office of Academic Affairs] 臺北市立師範學院學報 (19-36卷1期)

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