本研究最主要有兩個目的,第一個目的是要瞭解國中生在解不同幾何問題中,應用等量公理的表現為何?其次是要瞭解學生的場地獨立或場地依賴性,以及他們有沒有正式且有系統地接受過幾何訓練,是否會與他們應用等量公理的表現有關?本研究的研究對象,選自台北市某兩所國中,每一所國中分別從一、二、三年中,在該校的導師之建議下,選擇了8位表達能力較佳的學生,其中國一和國二各選取1位學生,國三則各選取的2位學生,合計有4位國三學生,2位國二學生,2位國一學生參與本次研究。 本研究採取質性研究中的個案研究法,配合深度訪談,要求受試學生在解題的過程中,盡可能的將其想法說明或寫下來,並全程錄影與錄音,以資佐證。所採用的幾何問題包含五個不同的類型,但都隱含可以應用等量公理來解題的幾何問題。為了深入瞭解學生在等量公理應用上的表現,本研究在資料分析的部分,一來採取描述性統計的方式進行分析,此外在質的分析上,則以深入的個案研究方式來進行,並且採用資料來源三角校正法,以期得到更完整且真實的看法。藉此瞭解學生在不同類型的幾何問題中,應用等量公理之表現情形。 整體來說,研究結果顯示:(一)學生在不同幾何問題中,應用等量公理的表現上,有所差異。(二)學生場地獨立的程度,以及有沒有正式且有系統接受過幾何的訓練,與其在應用等量公理的表現上也有所差異。(三)學生在辨識圖形中相等的部分,容易受圖形的類型、有沒有塗色等因素所影響。(四)有學生藉由在圖形中描繪、畫圖,或者從不同向度觀察圖形之方式,來重新掌握圖形中元素與元素之間的關係,進而辨識出圖形中相等的部分(五)有些學生在辨識出圖形中相等的部分後,想到應用等量公理的方式有二,其一是藉由代入具體數字計算,其二則是在嘗試扣去圖形中不相等部分的情形下,才想到可以應用等量公理來解題。 根據上述的研究結果與發現,本研究建議教師應該多在適合的情境下,強調等量公理的概念。另外可以訓練學生畫圖的習慣與能力,以及從不同的角度、方向、距離來觀察圖形,幫助學生掌握圖形中元素與元素之間的關係,降低圖形影響視覺觀察的可能性。此外,本研究建議教材設計者,宜仔細考量教材中圖形著色的呈現方式。
There are two main purposes for this study. The first one is to understand junior high school students’ performance of using the Equality Axiom in solving different geometry problems. The second is to understand whether students’ performance of using the Equality Axiom in solving different geometry problems are related with field independent/field dependent or formal/informal in learning geometry. There were eight students involved in this research. They were chosen form two junior high schools in Taipei city. In each school, we chose 2 grade three students, 1 grade two student and 1 grade one student. So there are 4 grade three students, 2 grade two students and 2 grade one students. This study were an qualitative research and designed by the way of case study, matched with interview, and requested students try to speak out or write down their viewpoints during they solved the geometry problems. We also took all the process with a video recorder. In the part of data analysis, we adopted the way of descriptive statistic in quantitative analysis and the protocol analysis in qualitative analysis and used the data triangulation to get more objective viewpoints. In general the manifestation of the result in the research are: (a)Students’ performance of using Equality Axiom in solving different geometry problems is different. (b)Students’ performance between field independent or field dependent and formal or informal in learning geometry is different. (c)Students were easy influenced by the presentation of the color of the sketch. (d)Some recognized the equal parts in the sketch by drawing or observing the sketch from different angle of views to control the relation between the element and element in the sketch. (e) After students recognized out the equal parts in the sketch, they thought of using Equality Axiom by two ways. The first is arithmetic, the second is trying to deducts the not related part in the sketch. According to the result of the research and the detection of the above, we suggest teachers should emphasize the Equality Axiom under all kind of suitable situations, and educated students to cultivate ability and habits of drawing a sketch. And educated students to observe the sketch from different angle, direction, or distance, to help them control the relation between elements and elements in the sketch and reduce the possibility of the influence of the sketch. And suggest the designer of teaching material should thinking of the presentation of the colors in sketch carefully.