## 高中生建構平面向量線性組合概念之個案研究

2017
##### Authors

Shen, Hsiang-Ping
##### Abstract

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.