台南地區高中學生對空間中直線方程式的錯誤類型分析

A Study of Error Patterns on Equation of Lines in Space for Senior High School Students in Tainan Area

本研究主要探討高中學生在「空間中的直線」單元的錯誤類型與錯誤可能形成之原因，藉此了解學生在此單元的學習困難點，提供教師教學參考。本研究採量的分析為主軸，輔以質性分析晤談，研究對象為台南市某私立中學高中學生47人。每年到高二課程時，幾乎會發現的現象便是第四冊第二單元「空間中的直線」的學習狀況有落差，學生雖然在第一單元「空間向量」已建立空間的基本概念，但進入第二單元之後，似乎對空間的認識仍然停留在空間平面的處理。對此單元所碰到的問題，不僅不知道該如何擬定解題，甚至會有錯誤的思考概念，所以本研究希望透過此次的自編測驗了解，一般學生在此單元內容學習上的盲點，進而找出協助教學的方式或是協助學生學習改進方式。研究並輔以晤談，瞭解學生在單元學習的真正錯誤情形並進一步修正學習上的錯誤概念。研究結果如下：一、台南地區學生在空間中的直線單元之錯誤情形為何？1.類型一：「直線的參數式」定義不清楚。2.類型二：「直線的對稱比例式」定義不清楚、混淆或計算錯誤。3.類型三：「直線和平面的關係」。不清楚直線和平面交點的意義、兩向量的外積概念與平面的法向量關係。4.類型四：「直線與直線的關係」。應用問題讀題、角平分線的特徵以及直線投影點的參數假設意義。二、台南地區高中學生在空間中的直線單元上之錯誤類型有：1.定義觀念不熟悉。2.參數式的應用不熟悉。3.直線的方程式轉換錯誤。4.外積運算錯誤。5.立體空間的觀念。三、空間中的直線的概念思考與運用之錯誤原因為何？1.先備知識錯誤或不足。2.條件忽略、忘記或誤加。3.粗心計算或者筆誤。4.敷衍應付或解題概念缺乏。5.可能有不適當的類化或錯誤的推廣、延伸。

This study focused on types of errors high school students may make when learning Equation of Lines in Space, and on what may contribute to these errors, hoping to learn more about students' difficulties in this unit and to provide teachers with some teaching strategies for reference in the future. The study is mainly conducted with quantitative analysis, supplemented by quantitative analysis of interviews, whose interviewees, all second graders, come from a private high school in Tainan city. When in the second year of high school, many students are found falling behind when learning the second unit in book Ⅳ- Equation of Lines in Space. Though students may have the basic concepts of space learned in the previous unit, Vector Space, their knowledge of space remains the same as that of space plane. What's more, not only do students not know how to solve the math problem, but they also figure out the problem in wrong ways. As a results, this study aims to find out the blind spots of students when they encounter this unit, and thus help teachers have the know-how to help students learn this unit as well. Accompanied by some interviews, the study reveals the scenarios in which students have difficulty leaning this unit; furthermore, it creates the possibility to fix students' errors and misconceptions. The results are as follows: What are the common types of errors in Equation of Lines in Space made by the high school students of Tainan? (1) They have vague ideas of the definition of parameter form of linear. (2) The definition of symmetric form of line is not quite clear to them, so confusions and miscalculations can often be seen. (3) They don't quite understand the relationship between line and plane: the meaning of line and cross plane and the relationship between the cross product of two vectors and normal vector of plane. (4) They have no clear conception of the relationship between lines; they are lack of understanding the questions, the characteristics of angle bisector, and the supposed parameter meaning of the linear projection point. 1. The three errors of the unit of linear in space are often made by the high school students of Tainan: (1) They are insufficient in understanding the definition. (2) They make errors transforming the equation of linear. (3) The applications of parameter form are not familiar to them. (4) They often make errors in the calculation of the cross product. (5) They are lack of the idea of three-dimensional space. 2. What are the reasons of making errors in thinking and using the lines in space? (1) Some students have or lack previous cognition. (2) They ignore the given conditions. (3) Forgetfulness, careless calculation and slip of pen are common sights. (4) They don't have prudent attitude and correct conception in answering the questions. (5) Inappropriate analogies and false inference and extend can be found, too.