國中七年級低成就學生學習 「二元一次聯立方程式-代入消去法」之個案研究

Case study of low-attaining seventh graders’learning of linear equations with two unknows

陳雅雯(Ya-Wen Chen)

指導教授 ： 許慧玉

國立清華大學/竹師教育學院/數理教育研究所碩士在職專班/碩士(2017年)

摘要

本研究旨在透過國中二元一次聯立方程式的「代入消去法」單元設計，結合國小怎樣解題的「雞兔同籠」，配合具體的表徵、視覺化的觀察、動手操作、表格設計、閱讀理解策略，設計本教學活動，讓學生能在進入正式課程時，從具體的表徵連結到抽象的代數思考，建構自己對於代入消去法的心向，每個活動都在奠定後面的學習基礎，營造學生的成功機會。
本研究採用個案研究，針對新竹縣某偏鄉國中六位低成就學生進行個案研究，從活動中觀察個案學生學習表現，由研究者記錄學生思考的學習歷程，各活動對應代入消去法達到的程度，將學生於活動中出現的情形整理如下:
1、學生能透過自由造例能觀察圖像表徵、造例及變化。
2、學生能透過有規律遊戲，從不同數字的布題及規則的操作、觀察及猜測規律。
3、學生能透過活動列出兩個二元一次方程式。
4、觀察表格中紀錄的數字能將同樣的數學算式找出規則，如:從x+y=10，得到x=10-y的逆運算。
5、能閱讀文字描述的二元一次聯立方程式解題過程進行符號化。
6、能閱讀理解二元一次聯立方程式-代入消去法的解題過程，但無法遷移到解題上。
研究中發現，本活動能加深低成就學生對於代數解題的認知及需求性，依據每個活動的目標，設計難易度由淺入深，低成就學生建立成就感後，都願意說出想法，表述出每個關係，進而能說出代入消去法及雞兔同籠的具體連結，非一般的抽象計算而已。

關鍵字

二元一次聯立方程式 ； 代入消去法 ； 代數 ； 低成就學生

並列摘要

Case study of low-attaining seventh graders’learning of
linear equations with two unknows
Abstract
This study aims to design a teaching activity that combines
the problem solving method of junior high schools (i.e., “elimination by substitution” of simultaneous linear equations using two variables) with that of elementary schools (i.e., “chickens and rabbits in a cage”), and incorporate concrete representation, visual observation, manual operation, form design and reading comprehension. This will help students develop their own mind-set using “elimination by substitution” and associate concrete representations to abstract algebraic thinking once they begin their formal education. Each activity aims to establish a future-focused learning that provides opportunities for students to succeed.
The case study method was used in the research with six low-achieving students from a remote village in Hsin Tsu County as subjects. The students were observed and assessed in each activity based on their learning performance and learning process with respect to the “elimination by substitution” method.
The following observations were made:
1. The students are capable of understanding image representations, giving examples and making changes through these examples.
2. The students are able to determine the answer to the problem based on different numbers and rules of the game.
3. Through the activity, the students are able to create two linear equations using two variables.
4. By observing the numbers recorded on the list, the students are able to learn the rules from the same mathematical formula, e.g. obtaining the inverse calculation of x=10-y from x+y=10.
5. The students can understand the problem-solving process expressed in words to describe linear equations using two variables in order to proceed to symbolization.
6. Despite understanding the problem-solving process involved in linear equations using two variables - elimination by substitution, the students are unable to solve the problem.
This study concludes that the activities can enhance learning and understanding of algebraic problem solving among low-achieving students. The degree of difficulty is integrated to achieve the objective of each activity, enabling all low-achieving students to express their thoughts and perceptions. Moreover, the students are able to associate the “elimination by substitution” method with the “chickens and rabbits in a cage” method aside from general abstract calculation.
Keywords: linear equations with two unknows、elimination by substitution 、algebra low-attaining student

並列關鍵字

linear equations with two unknows ； elimination by substitution ； algebra ； low-attaining student

參考文獻

林清山、張景媛 (1993)。國中生後設認知、動機信念與數學解題策略之關系研究。教育心理學報(26), 53-74。

林清山、張景媛(1994)。國中生代數應用題教學策略效果之評估。教育心理學報(27), 35-62。

張景媛 (1994)。數學文字題錯誤概念分析及學生建構數學概念的研究。教育心理學報(27), 175-200。

郭生玉 (1990)。心理與教育測驗。台北：精華。

黃怡婷 (2013)。數位遊戲式學習應用在二元一次方程式學習之研究。屏東科技大學，未出版，屏東縣。