四、五、六年級學生在平行概念上的差異—從概念心像與概念定義的觀點探討

Abstract

本研究以四、五、六年級各249、246、237人進行四個平行概念的作業，結果尚未正式學習的四年級學生顯著較五、六年級差，剛學過的五年級和學過一陣子的六年級差異不大，而兩個語文作業的表現都較圖形作業差。繪圖作業因技巧的限制，方位與點線關係的效果雖顯著，但變化不明顯。平行圖形的辨識率，四年級明顯受方位、相對位置與相對長度的顯著，而五、六年級僅傾斜且不等長的辨識率稍低，顯示典型心像明顯影響四年級的表現，而對五、六年級影響較小。四年級較能寫出或判斷平行線等距離的敘述，等距離可能是自發性的概念定義；五、六年級在學過永不相交的定義後，皆能正確應用於不同情境；垂直於第三線則難度較高，學習後僅能記住數學定義，但無法轉換到日常情境。本研究從Vinner與Tall的理論指出，正式課程前的四年級學生是從所經驗的平行圖形自行抽離屬性，且概念心像的範例有限，而五、六年級則有較為多樣化的概念心像，並習得一些正式的定義，圖形作業在喚起豐富的概念心像後表現不錯，而語文作業的表現則視能否提取適當的概念定義而定。本研究顯示數學課程中提供多樣化的平行圖形是必要的，有待破除的等長迷思尤其應該增加範例，直觀性較高的等距離意義是可以善加利用的概念。

This is a study on 4 tasks of parallel concept on a group of elementary school pupils; 249 in grade 4, 246 in grade 5, and 237 in grade 6. The results indicated that grade 4 pupils, who have not yet formally learned the concept, show significant difference from grade 5 and 6. There are marginal difference between grade 5 who have just learned the concept and grade 6 who have learnt the concept for a while. Their performance in verbal task is not as good as in their figural task. Restrained by the techniques of drawing, the effect of coordinates and the relation with points and lines is significant but the variation is insignificant. The accuracy rate for grade 4 pupils in identifying parallelograms varied significantly. Direction, relative position, and length significantly affected performance. Performance of grade 5 and grade 6 pupils only the capacity in identifying oblique and unequal length was low, indicating typical image affected the performance of grade 4 pupils at significant level and not as significant for grade 5 and grade 6 pupils. Pupils in grade 4 could write, determine, and describe the equidistance of parallel lines. Equidistance may be a spontaneous concept definition. After learning the definition of non-intersection, pupils in grade 5 and grade 6 can correctly apply the concept in different situations. It is more difficult to learn the 3rd vertical line and pupils can remember only the definition in mathematics and can not apply the concept in real life situations. This study started from the theories of Vinner and Tall. Grade 4 pupils who have not yet received formal learning of the concept demonstrated their spontaneous deviation from the attributes of parallelogram and the scope of their concept image is limited. Grade 5 and grade 6 pupils demonstrated their diversities in their concept image and learn certain formal definitions. Their performance in figural task is good after abundant concept image have been aroused. Their performance in verbal task varied with proper retrieval of concept definition. This study indicated that it is necessary to provide a diversity of parallelograms in mathematics courses. Additional examples are also necessary for breaking the misconception of lines with the same length. The highly intuition notion of equidistance can be properly applied as well.