高中生建構向量概念的系統繪圖法之研發及學習成效研究

有鑑於向量概念對高中生是抽象的，一般考生在大學入考試中的表現也不盡理想，因此研究者希望發展一套系統繪圖法幫助高中生建立平面向量的概念。研究者以台北市某公立高中二年級為對象，APOS理論為引導，設計系統繪圖法教材並進行實驗教學。研發系統繪圖法的契機為，現高中課綱將向量歸類在幾何別，並之分為平面與空間然而教科書編排方式仍以代數表徵居多，研究者認為對於初學而言應給予足夠的實作繪圖機會掌握向量的概念，進而可以發展出將向量視為一個「物件」的心智模式。本研究目的是想探討這套系統繪圖法是否真的能夠幫助學生了解向量概念，以及心智結構的發展層次是否到達「物件」的階段，另一個目的是想比較使用系統繪圖法與講述式教學的生在認知結構向量概念上習成效有何差異。本研究採混合法，量化部分是採用準實驗研究法，並以研究者設計的前、後測試卷進行雙因子變異數分析；質性部以訪談九位學生詳細了解兩組學生分別在 APO認知結構的品質上有何區別。研究果分析發現，實驗組學生在接受系統繪圖法的教學後對於向量的大小，以及向量同時考慮方向與大小的表現比對照組優異，但是線性組合問題上則對照優於實驗。在認知結構「過程」的問題表現上實驗組優於對照，但是認知結構「動作」和「物件」兩組學生則是沒有統計上的顯著差異。透過質性分析，本研究發現實驗組學生在接受系統繪圖法後能形成較健全的「物件」結構，對照組學生則是傾向以計算的方式回答問題，解題能力較強但對於向量的「物件」結構比較薄弱。

關鍵字

APOS理論；動手操作；平面向量

並列摘要

The concept of vector is rather abstract for most senior high school students. Based on publicly released data, it was observed that many examinees performed unsatisfactorily on vector problems in the College Entrance Examination. This observation motivated the present researcher to develop a novel approach to help students learn the concept of plane vectors. The presentations of learning materials in all current textbooks are algebraic oriented. The present researcher believes that it is more important to take advantage of the geometric aspect of vector and allow the beginning students many opportunities to grasp the concept of vector in more concrete way. Thus the main idea of the new teaching method is to engage students in a systematic way of drawing vectors according to the APOS theory of Dubinsky. It is hope that under this approach students can subsequently perceive vectors as mathematical objects. The purpose of this study is to evaluate the effectiveness of this systematic method of drawing vectors by comparing it with the traditional teaching method. In particular, we would like to assess if students in both groups can progress to the level of perceiving vectors as mathematical objects. This study adopted the mixed methods approach as its research design. For the quantitative section of the study, a teaching experiment was implemented according to the quasi-experimental design. The data collected was analyzed by using a two-way analysis of covariance through SAS. As regards the qualitative section of the study, interviews were conducted on a total of nine students from both the experimental and the control group. The main focus of the interview is to clarify the differences in terms of the APO cognitive structure between the students in the two groups. It was found that students who had learned the systematic method of drawing vectors performed better than the control group in items related to finding the length of vectors and also in items that involved both the direction and length of vectors. However, students in the control group performed better in items related to linear combinations of vectors. As regards the APO cognitive structure, the experimental group performed significantly better than the control group in items related to “process,” but were no different in items related to “action” and “object.” However, analysis of the interview data revealed that students in the experimental group portrayed a more solid perception of vectors as mathematical objects, while students in the control group were relatively weaker in this aspect.