國一學生由算術領域轉入代數領域呈現的學習現象與特徵

戴文寶 ； 邱守榕

《科學教育》 10期 (2000/08) Pp. 148-175

https://doi.org/10.6767/JSE.200008.0148

摘要

本研究主要探查國一學生初學代數的困難以及克服困難所需的輔導。本研究採用質的研究法，先做課堂觀察，再對十位班後段學生進行建構式教學實驗，在晤談中其解代數問題時建構數學知識的行為。根據教科書、習作、校內考卷，且與合作教師討論，編成四層次的晤談問題：(1)代數式求值(2)代數式的化簡(3)解方程式(4)應用問題，之後以二位學生之試驗性晤談檢驗其內容效度。正式晤談共三次，第二次藉前三層次的問題探查八位元學生初學文字數的現象，第二、三次僅選其中三位元，藉後二層次的探查學生在文字數概念上的發展情況；第三次再深入探究學生對文字數的代數表徵與圖形表徵之間的對照是否理解。晤談實況全程錄音，再轉錄成原案腳本據以分析。先製作學生初始錯答行為分佈表、學生對各類題的作答行為分佈表以及輔導策略分佈表，並且以師生對話為證據，殿現國一鵼姝共同困難與邊思概念以及輔導策略的得失。經三角校正比較分析後發現：
（一）學生解題行為中的共同性：制約性、包容性與自發性；雖然學生最初看到問題的的直接反應是：不會（制約性），但並未拒絕學習，相對地欣然接受研究者的繼續輔導（包容性），並且開始嘗試解題（自發性）。
（二）學生的迷思概念如下：
1.有關文字數與代數式的意義：在代數式求值問題中，學生誤把3x作為3＋x。
2.有關「同類項」的意義與全併規則：(1)只處理含有x的「同類項」，不含x的項（常數項）不算作同類項，因而未予合拼，例如3x＋4＋5x＋3＝8x＋4＋3；(2)不接受化簡至含加號的代數式，如3x＋4為答案，以致繼續誤併不同類項，例如將3x＋4併為7x或併為7；(3)不確知x＝1x，而把x看作“0x”計算，如3x+x=3x。
3.含括弧的化簡問題：(1)括弧外的數字只和括弧內的第一項相乘，忽略第二項；(2)括弧外的數字若是負數，未隨著變號；(3)不曉得括弧內的算式要和括弧外哪一項進行運算。
4.無法同時對照文字數的代數表徵與圖形表徵：亦即學生徒具程式性的知識，會畫二元一次方程式的圖形，知道二元一次聯立方程式的解可以以丙直線的交點表示，但缺乏概念性的理解，不知直線上的點代表方程式的解，即單一線性方程式圖形的意義極難以理解。
（三）以數線為「鷹架」表徵代數式求值問題，有且于學生理解題意。問題中的算式圖像化後，學生皆能線數線正確的算出答案。
本研究足以反駁社會上對中學生不肯學數學的刻板印象，以實證性資料顯示：大部分國一学生要得到解題與輔導機會，都願意且有能力進入代數領域。

並列摘要

This research explores the learning difficulties of beginning algebra encountered by first year secondary school students with low academic performance, as well as the effectiveness of remedial instructions to help overcome the difficulties. Qualitative methodology is used to investigate students’mathematics behaviors with constructivist teaching experiment in which ten students were involved, and the questionnaires were designed to use as the instructional material to elicit constructive learning. In order to comply with school curriculum, textbook, homework problems and officially administered tests were analyzed and teachers’opinions were solicited. Four levels of test items are: (1)substituting the symbolic letters with given values (2)to simplify algebraic expressions (3)to solve simple algebraic equations (4)word problems with one unknown.
In order to examine the content validity, two students were interviewed in the pilot study. In the formal stage, the whole procedure was undertaken three times. 8 students were interviewed at the first time, and 3 of them proceeded with the second and the third time. The first interview is to explore learning phenomena and characteristics when the students learned algebra the first time; The second one, which adds in word problems, is to explore the students’concept development about algebra after they have studied for a period of time; The third one is to explore whether the students can understand the mapping from the algebra representation to graphic representation.
The research information was collected by observation and deep interview, including the videotapes on classroom observation and on the interviews. The analysis concentrated in talk record and was supported with teaching videotapes to analyze the protocols. To show the behavior of junior high students who study algebra for the first time and to analyze how junior students understand basic algebra concepts, the researcher set up the distribution chart about students’wrong answers first time, their behavior when answering each classified questions and the strategies to help them. The conversation between teachers and students was used as the evidence to show the common behavior students have and the classified problems were used to describe students incorrect judgement; All of these were showing the trouble and misconception students have in common. After comparing these charts, we found that
1. the common properties of students’ problem solving behavior stimulatingreflectingness, wideness and activeness; though the direct response of students seeing the questions was “I quit “ (stimulating-reflectingness) ,but they didn’t refuse to learn; on the contrary, they were willing to accept the assistance from the researcher (wideness) and started to learn to solve the problem (activeness)
2. The incorrect judgments made by students toward literal term were as follows
(1)Students thought that only the terms including“x”were like terms and were needed to be combined, and that the constant was not like terms so that no further simplification was executed, for example,3x+4+5x+3=8x+4+3… Besides, students usually combined the unlike terms in a wrong way or had no idea what lx was That resulted in the following situation: 3x+4=7x,5x+ 1 =6x,3x+x=3x…
(2)The misconception about parentheses questions were as follows: (a)The coefficient always multiplicated the first term inside the parentheses, and failed with the second term. (b)If coefficient was negative, students always fail to change the sign of the product. (c)Students didn’t know which term was needed to be calculated with the terms inside the parentheses.
3. Students couldn’t map the algebraic representation and graphic representation at the same time. They could draw the diagram of a linear equation, but they didn’t know the line represented the solution of the equation. It meant that students only have procedure knowledge and lacked conceptual understanding. Compared with signal linear equation, the meaning of two-linear equations was more understandable.
4.The researcher used the number line representing the algebraic expressing. Visualizing the problems make students understand more the meaning of problems. With the number line as a scaffolding, they could move from total incapability to figuring out the correct answer.