本研究以放聲思考法及半結構事後晤談，探討現階段國一學生在一元一次方程式的解題表徵運用、解題歷程與解題策略。研究工具為一份自編的一元一次方程式測驗；測驗題型包含多元性的圖形與符號文字，並經三位資深教師修訂刪裁而定稿。研究對象以分層立意取樣，在研究者任教的高雄市立某國中三個國一常態班級中，依九十四學年上學期第三次段考成績，將各班區分為低分（後27％）、中分（46％）與高分（前27％）三組，各班自高分組與中分組各選出一位口語表達能力較佳之學生做為施測對象。六位受試者先接受三個星期的放聲思考訓練，再個別接受測驗及晤談；測驗及晤談全程錄音錄影，並轉譯成逐字稿，以分析其解題表徵、解題歷程表現與解題策略的應用。
主要研究結果依解題表徵、解題歷程與解題策略三個面向分述如下：
一、受試者解題表徵：在一元一次方程式的解題表現，受試者大多運用文字、代數及數字的多元解題表徵，較少利用圖形的解題表徵。
二、解題歷程的分析：
1.有效地利用圖形表徵輔助解題者，其解題速度較快；
2.對於文字敘述較長的題目，受試者重覆讀題與分析題目的次數會相對地增加；
3.在解題失敗的題目中，一半以上的受試者其解題歷程呈現探索、執行與計畫三階段同時進行；
4.解題歷程常呈現驗證階段的受試者，其解題成功的比率亦較高；
5.對於解題計算較繁複或對結果有疑慮的測驗題，受試者較常出現驗證的解題歷程；
6.題目內容若較切合受試者的生活情境，其解題成功率會相對地提高；
7.高分組所運用的解題時間較中分組短，其解題成功率亦較高。
三、解題策略的應用性：
1.高分組受試者所應用的解題策略一致性較高，中分組受試者使用的解題策略差異較大、較具豐富性。
2.受試者解一元一次方程式試題時較常運用的解題策略有：列方程式、同類項合併、移項法則與等量公理。

This study employed thinking aloud and semi-structured interviews to explore problem-solving representations, problem-solving processes, and problem-solving strategies of six grade seven students on word problems of linear equation in one variable. The instrument of the study was a researcher-designed test with literal, graphics and/or symbolic descriptions and was examined and revised by three senior secondary mathematics teachers. According to their mathematics scores of 3rd midterm exam last semester, students were divided into three achievement groups－－low achievement group (the lowest 27%)，middle achievement group (46%) and high achievement group (the highest 27%). One subject was selected from each of middle and high achievement groups of three grade seven classes. Six subjects, in total, had taken thinking aloud training for three weeks, and then they took the paper and pencil test individually with a follow-up interview. All the processes of individual tests and interviews were video recorded. The videotapes were transcribed and provided the major evidence of the analyses of participants’ performances of problem-solving processes, their problem-solving representations, and their problem-solving strategies.
The results of problem-solving representation, problem-solving process, and problem-solving strategy were reported separately as follows:
(1)Problem-solving representation. Participants applied literal, algebraic and numeral representations to solve one-variable leaner equation problems more often than used graphic one.
(2)Problem-solving process.
(a)When graphic representation was applied in this test, the time of problem solving could be shortened effectively.
(b)The times that Participants repeat to read and analyze the topic increased relatively in the topics with more writing narration.
(c)In more than one half of the fault problem-solving cases, the three stages of exploration, implementation, and planning were administered simultaneously.
(d)The more verification was applied during participant’s problem-solving process, his/her opportunity of success was higher.
(e)Verification was often administered in problems with complex computations or questionable topics.
(f)The relevance was higher between problem content and daily life, the opportunity of success was higher.
(g)The time that the high achievement group used to solve problems was shorter than the middle achievement group used, and the opportunity of success was also higher than the middle achievement group.
(3)Problem-solving strategy.
(a)The problem-solving strategies applied by participants of high achievement group were more consistent, and the problem-solving strategies among participants of middle achievement group were more diverse.
(b)The problem-solving strategies that participants often used to solve word problems of linear equations in one variable were translating the word problem into an equation, simplification of equation by collecting terms, using inverse operations, and properties of equality.

第一章 緒論 1
第一節 研究動機 1
第二節 研究目的與問題 3
第三節 名詞釋義 4
第二章 文獻探討 6
第一節 數學解題的表徵形式探討 6
第二節 數學解題歷程模式的探討 9
第三節 代數方程解題相關因素的探討 18
第四節 一元一次方程在國中代數課程的定位 26
第三章 研究方法 42
第一節 研究設計流程 42
第二節 研究對象 43
第三節 研究工具 44
第四節 實施程序 51
第五節 資料分析 53
第四章 研究結果 66
第一節 受試者解題歷程分析 66
第二節 解題歷程的綜合分析 92
第三節 解題策略的運用 98
第四節 解題表徵的呈現 131
第五章 結論與建議 139
第一節 結論 139
第二節 建議 142
參考文獻 145
附錄一：一元一次方程式測驗 151
附錄二：各受試者解題歷程分析 154
附錄三：受試者解題原案分析 166

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