學生的角概念及教師對角的教學相關知識之探討

Study of Students’Conceptions of Angles and Teachers’Knowledge about Teaching and Learning Angles

本研究在於探討國小四年級學生（N = 171）從生活中的傾斜、路徑交會、旋轉情境辨識角，以及角量測的解題表現，同時了解有經驗教師（N = 7）對角的教與學知識。研究結果發現：（一）學生的解題表現因角情境不同而呈差異，其在角量測與路徑交會情境的表現接近，而且皆優於其他情境；學生在傾斜與旋轉情境的表現相當但也最低落。（二）學生辨識角的理由說明包含五種類型：能明確以角定義說明（16%）、定義不全（18%）、其它（語義含糊，20%）、使用角形（19%），以及不知道如何說明（27%）。五類組能成功辨識角的百分比，以定義組最高（13%）；不知道組最低（6%）。（三）從分析教師對角的教學內容知識之結果發現：教師了解學生辨識角的表現受角形是否可見及情境類型影響；教師對學生理解圖形角的難度評估，符合學生的實際解題表現，但對旋轉角的概念理解則呈高估現象；教師的角教學活動重視觀察、操作與討論。本研究依據分析結果，針對角概念理解的教與學及課程與教學，提出討論與建議。

This study aimed to explore fourth-grade students’(N = 171) performance of identifying angles embedded in daily life situations, including slopes, path intersections, turns, and angle measurement. Experienced teachers’(N = 7) knowledge of teaching and learning angles was also examined. The findings showed a significant effect of angle situation on students’problem-solving performance. The students performed equally well on solving problems of angle measurement and path intersection situations. They also performed equally well on slopes and turns, but their performance on these two situations was poorer than that on the former two situations. The mathematical thinking that the students used for recognizing angles included five categories: using the mathematical definition of angles with accuracy and completeness (16%), using definitions with incompleteness (18%), vague explanations (20%), angle shape (19%), and don’t know (27%). For the overall percentage of accurate recognition of angles, the results showed that the accuracy and completeness group was the highest (13%), whereas the don’t know group was the lowest (6%). Furthermore, the teachers perceived that angle situations and whether the shape of angles was visible impacted students’performance. The teachers’evaluations of student understanding of angles with two arms accurately reflected the students’actual performance. However, they over-estimated students’understanding of angle rotations. For teaching the concepts of angles, the teachers highlighted the importance of exploring angles embedded in daily objects and operations with discussion. Teaching and learning about conceptual understanding of angles are discussed based on the findings. Suggestions for curriculum and instruction are also made.