高中生建構平面向量線性組合概念之個案研究

本研究探討學生學習平面向量過程的特徵與困難，以及平面向量基本概念之認知結構，研究中選取會考成績達基礎以上乃至精熟之學生為樣研究對象，針對六位未學過的高一學生進行學習活動、訪談及後測。筆者依APOS理論發展平面向量基本概念之起源分解圖，包含平面向量基本意義、平面向量基本運算（加法、減法、係數積）以及平面向量線性組合；接著再依起源分解圖發展研究工具，設計對應之學習活動單與後測評量卷。資料收集分為兩階段，筆者此時同時扮演教學者與訪談者的角色，第一階段為平面向量基本概念之學習活動，過程中筆者不斷與個案學生互動，必要時會輔以提示、引導介入；第二階段為後測，測驗學生學習後具備之概念，在學生完成測驗後再訪談學生。研究結果顯示部分學生因幾何圖形性質不熟悉影響其將向量幾何表徵轉為坐標表徵；幾乎所有學生都無法自行由代數符號關係轉換坐標或幾何表徵求出係數積向量；學生傾向以物理情境或平移向量幫助自己建構與內化向量加法過程，而有些學生幾何加法過程有些反覆；學生在減法幾何中，難以反轉平行四邊形法加法，傾向以「加法與反向量過程合成」處理幾何減法；給定任意幾何圖形，學生很難將圖中向量去膠囊化，改寫成其它兩不平行向量的線性組合。筆者建議教學者設計活動讓學生主動連結向量坐標與幾何表徵；利用物理上位移、合力的類比幫助學生思考，加速學生內化向量幾何加減法的過程與意義；並幫助學生建立穩健的基本運算概念，建立各個基本運算間基模的連結；也幫學生統整複習其它幾何圖形的關係與性質；除了幾何直觀證明，也可利用二元一次聯立方程式說明兩不平行向量線性組合的存在性。

關鍵字

平面向量； APOS理論；認知建構

並列摘要

This research was trying to understand the cognitive construction of senior high school students in linear combinations of plane vectors. The author researched about the characteristics and obstacles of students while learning, and the cognitive constructions about elementary concepts of plane vectors after learning. Six freshmen participated the learning activities, joined the tests, and were interviewed. The author developed genetic decomposition diagrams (GD) of the elementary concepts in plane vectors according to APOS Theory, including basic meaning, elementary operations (vectors addition, subtraction and scalar multiplication were included) and linear combination. After then, research tools were developed. Learning activities sheet and post-test sheet were designed according to GD. The data were collected by 2 phases: the first phase were learning activities, in which phase the author interacted with the interviewed student at times; and the second phase were post-test, in which phase student was interviewed after finishing the test. The research result showed that some students had difficulties in transferring geometric representation into coordinate one. Almost all students couldn’t transferred coordinate and algebraic representation into geometric one and figured out scalar vectors by themselves. Students used physical situation and translated vectors to help interiorize the Process of addition. Some students had difficulties in reversing the Process of addition such that they intended to coordinate the Process of addition and inverse vectors while dealing with the subtraction problems. Students had difficulties in de-encapsulating a vector Object into the linear combination of other non-parallel vectors. The author suggested that teacher design activities to let students have opportunities in connecting coordinate and geometric representations. The analogy of “displacement” and “force” may help students’ learning, accelerate the interiorization of addition and subtraction Process. With intact concepts in elementary operation can help students develop the connection between the schema of those concepts. Teacher should also help student review the relation and properties of other geometric graphs. Except geometric intuition discussion, teacher can help students realize the existence of linear combinations between two non-parallel vectors with the help of linear equation with two unknowns.