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研究生:林千鑫
論文名稱:國中二年級學生一元二次方程式解題歷程分析之個案研究
指導教授:蕭龍生蕭龍生引用關係
學位類別:碩士
校院名稱:國立高雄師範大學
系所名稱:數學系
學門:數學及統計學門
學類:數學學類
論文種類:學術論文
論文出版年:2014
畢業學年度:102
語文別:中文
論文頁數:113
中文關鍵詞:解題歷程解題策略一元二次方程式
外文關鍵詞:Problem-solving processesProblem-solving strategiesQuadratic equation with one unknown
相關次數:
  • 被引用被引用:2
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摘要

本研究旨在探討國中二年級學生在一元二次方程式相關概念的發展,以例題中的解題歷程、解題策略、解題成敗因素來進行分析。本研究採用質性研究中的個案研究法,藉由對個案所搜集的資料進行分析,並探討個案學習成就低落的因素。綜合本研究結果提出下列幾項結論,最後根據研究中所發現的結果提出建議,以做為未來「數學課程」在教學上的參考。
一. 解題歷程方面
(一) 閱讀題目的關鍵在於是否能了解題目的敘述,抓住題目本身的關鍵字句,了解該如何使用何種因式分解的方法,或是該如何對未知的部分加以假設未知數,進而依循線索逐步解題。過於輕浮的態度面對問題,往常常會產生漏掉一些細微的條件。
(二) 實施策略進行解題時,解題者將需要一定程度的解方程式的能力,了解該怎麼假設未知數並列式之後,只剩一些解方程式因式分解的演算。運算的能力不足或是過於自信輕忽,則容易產生計算錯誤的情形。
二. 解題策略方面
(一) 學生在解一元二次方程式的應用問題時,所使用的策略太過於依賴記憶與背誦,只記得什麼樣的題型就該用怎麼的假設未知數,將解題過程流於形式,只是單純一昧的模仿不加思考,而非了解為什麼該這樣做,因此在應用上產生很大的障礙。
(二) 面對一元二次方程式的應用問題,當又是計算題時,學生對於如何將解題系統化的表達感到恐懼,因此若能勇敢的踏出第一步,嘗試著將想法寫下,假設未知的部分,在想辦法從中找出等式的關係來列式,就算結果錯誤,只要找到問題的癥結點,那也能為下一次成功累積舊經驗,強化自己的實力。
三. 解題成敗因素方面
(一) 數學知識:基模的建立影響學生解題成敗的關係甚大,在面對問題時須從既有的舊經驗中去連結。當所學的經驗較豐富時,能夠提取的線索也越多了解題意分析後,掌握關鍵字句,再運用相關的基模有系統的整合,才能完整的解題成功。
(二) 後設認知:是否了解自己的解題進度、是否了解自己的解題情況、是否察覺自己問題的所在,這在解題的過程中扮演著自己監督自己的角色,來完成問題的解決。
(三) 情意態度:若能主動提升面對問題時的態度與解題時的勇氣及信心,對解題的成敗會有一定的影響,即使這次失敗,卻也能累積經驗為下一次做準備。

關鍵字:解題歷程、解題策略、一元二次方程式









Abstract

The main purpose of this study is exploring the eighth graders’ related concept development of quadratic equation with one unknown and analyzing the case by problem-solving processes, problem-solving strategies and the factors that determine success or failure of problem-solving. The study adopts the case research methodology of qualitative research methods to analyze the data that were collected from the case and explores the factor of low performance on mathematics. Based on the research result, we offered some conclusions and suggestions as below to be the reference on the further teaching of mathematical courses.
1. For problem-solving processes,
&;lt;1> The key point of reading questions is based on whether the case can understand the description of the subject clearly or not, by catching the key words of the subject itself to know how to use which Factorization method or hypothesize a variable for unknown part to solve the problem gradually by clues. Some slight conditions always are missed when the case uses the careless attitude to solve the problem.
&;lt;2>When practicing the strategy to solve the problem, the case has to have a certain degree of calculation ability on equation; after knowing how to hypothesize variable parallel formula is followed by the process of equation factorization calculation. The calculation mistake happens easily to the person with the poor calculation ability or over self-confidence and carelessness on solving the problem.
2. For problem-solving strategies,
&;lt;1> Students rely on recollection and reciting strategies excessively when solving the application problem of Quadratic equation with one unknown, they only remember which kind of question can go with which hypothesis variable method and then the process of solving the problems will be lead to become an formalization without thinking why. Therefore, it will cause a great obstacle to solve the problem.
&;lt;2>When facing the application and the calculation of the quadratic equation with one unknown at the same time, students fear for how to answer the questions systematically; therefore, if they can try to write down their thinking to suppose the unknown part and find the way to calculate from equality relationship, learn to accept trial and error and find out the core problems, it will strengthen the problem-solving ability due to the trial and error experiences.
3. For factors that determine success or failure of problem-solving,
&;lt;1> Knowledge about mathematics: The establishment of the right schemes affects the students to answer the questions successfully or not a lot. Students have to make a connection with the past experience when facing the problem. With more abundant experience, the more clues will be adopted. Understanding the meaning of the question ,and knowing well the key words and then applying the related schemes to integrate systematically, the problem can be solved completely.
&;lt;2> Meta-cognition: It is important if the case herself/himself understands the progress in solving the problem, knows the situation of solving the problem and is aware of her/his problem. The case must learn to superintend herself/himself to complete the answer to the question.
&;lt;3> Affective attitude: That the case screws up the courage and confidence to solve the problems will influence the results greatly. Failure this time can be a preparation for the next challenge.

Key Words: Problem-solving processes, Problem-solving strategies,
Quadratic equation with one unknown


目錄
第一章 緒論 1
第一節 研究背景與動機 1
第二節 研究目的與待答問題 4
第三節 名詞釋義 5
第四節 研究範圍與限制 7
第二章 文獻探討 9
第一節 建構主義與認知發展的探討 9
第二節 數學解題及心理學理論 15
第三節 數學解題歷程與策略 17
第四節 數學解題的成敗因素與相關研究 27
第五節 數學解題歷程之研究方法 36
第六節 一元二次方程式的相關研究 42
第三章 研究方法設計 45
第一節 研究問題 45
第二節 研究對象 46
第三節 研究方法與步驟 47
第四節 資料搜集與處理分析 50
第五節 研究流程 51
第四章 結果與分析 53
第一節 「一元二次方程式」課程 53
第二節 個案的學習歷程 55
第三節 個案在晤談、寫作業與答考卷呈現出之學習概念 64
第四節 綜合討論 94
第五章 結論與建議 97
第一節 結論 97
第二節 建議 100
參考文獻 103
一、中文部分 103
二、英文部分 105
附錄 111
附錄一:同意書 111
附錄二:教室觀察記錄表 112
附錄三:課程進度表 113

表次
表 2 1 AUSUBEL之學習分類表 14
表2 2 POLYA之解題歷程表 17
表2 3 SCHOENFELD的解題分析架構與相關問題 20
表2 4數學解題歷程比較表 22
表2 5 SCHOENFELD 常用之解題策略表 24
表2 6 數學解題思考步驟及程序表 25
表2 7 中外學者數學解題成敗因素比較表 27
表2 8 各學者對數學知識論點分析比較表 30
表2 9 各學者對後設認知的定義與觀點比較表 32
表 4 1 小美的歷年成績表 55
表 4 2 小美作業作答題數統計表 61
表 4 3 小美考試得分統計表 63

圖次
圖2 1記憶系統之訊息處理模式 11
圖 3 1研究流程圖 51
圖 4 1 小美全部作業選擇題作答情況統計圖 61
圖 4 2 小美全部作業填充題作答情況統計圖 62
圖 4 3 小美全部作業計算題作答情況統計圖 62


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