高中生對於任意兩向量 可否作內積之概念心像

Abstract

本研究探討高中生對於內積概念中的基礎概念，「相同維度的任意兩向量皆可作內積」之概念心像。研究採問卷調查法，收集質與量的資料。研究抽樣採立意取樣，包括高數學程度、中數學程度學校之理組與文組四個班紋，共149位高中三年級學生。研究結果發現高達四成的學生對於起點不同的兩向量不具備皆可作內積的概念心像，僅有約四成的學生其任意兩向量皆可作內積的心像穩固，不會隨著題目所給的刺激而有所改變;同時學生的概念心像受到向量、角度、內積定義等相關概念心像的影響。

This study investigates the concept images of "any two vectors with same dimensions have an inner product" for senior high school students. A questionnaire survey was utilized to collect qualitative and quantitative data. A purposive sampling was used to sample 149 12th graders in four classes from a high or a middle level school with either science or liberal-arts oriented approach. The findings included that more than 40% students did not possess the concept images of having an inner product for vectors with different starting points; only about 40% students' concept images were stable that did not switch over different vector relations; and students' concept images in question were influenced by the related concept images of vectors, angles, and definition of inner products