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研究生:吳碧智
研究生(外文):Bi-Tsz Wu
論文名稱:圓形複合圖形面積計算-以「假設性學習軌道」理論設計教學實驗
論文名稱(外文):Calculations of Circle Complex Graph Area: Use “Hypothetical Learning Trajectory” Theory to Design Teaching Experiments
指導教授:陳嘉皇陳嘉皇引用關係
指導教授(外文):Jia-Huang Chen
口試委員:袁媛李心儀陳嘉皇
口試委員(外文):Yuan YuanSin-Yi LiJia-Huang Chen
口試日期:2014-06-06
學位類別:碩士
校院名稱:國立臺中教育大學
系所名稱:數學教育學系在職進修教學碩士學位班
學門:教育學門
學類:普通科目教育學類
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:161
中文關鍵詞:教科書複合圖形假設性學習軌道迷思概念
外文關鍵詞:textbookcomplex graph areahypothetical learning trajectorymisconception
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本研究針對現行的南一版數學教科書,關於圓形複合圖形面積計算的單元內容進行教學探討,以瞭解課程內容的編排模式,並以Simon(1995)的「假設性學習軌道(Hypothetical Learning Trajectory)」理論自行設計教學課程內容及編製課堂上所需的學習單來進行教學實驗,並於事後找學生進行訪談,以瞭解國小六年級學童在學習軌道下的數學表現,並針對不同層次的學生所產生的迷思概念及學習困難之處做教學省思,檢討自身教學不足的部分及思索如何改進原本的教學內容,以幫助學生在計算圓形複合圖形面積試題時,能具備更完整的幾何思維及解題策略,進而擁有正確解題之能力。研究結果顯示如下:
一、依據「假設性學習軌道」理論可設計圓形複合圖形面積教材,包含「找圓心、認識圓形」、「圓心角度與幾分之幾圓」、「圓面積公式的由來」、「『分割』與『移補』」、「『填補』與『其它策略』」。
二、認知層次不同的學生,在「圓形複合圖形面積學習軌道作業」下的數學表現亦不同。
三、經由學習軌道教學實驗回溯省思後,教學者對於學生概念的理解、教學方法的改進及課程設計的修正等方面產生具體且正向的改變,以增進自身的教學專業能力。

最後,研究者針對教學實驗歷程提出三點建議,以做為未來圓形複合圖形面積教學的參考內容。建議內容如下:
一、檢視學生的數學先備知識,以利學習軌道的設計。
二、重視圓面積公式的演變由來,使學生能將公式形成長久且穩固的記憶。
三、培養計算過程工整的觀念,以提高其事後檢查的意願及作答的正確率。
The study aims at current textbooks of Nan-Yi to explore the teaching of the content of calculations of circle complex graph area, in order to understand the arrangement mode of course content, and use “hypothetical learning trajectory” theory of Simon(1995) to design the course content of teaching and prepare the learning sheets for teaching experiments by oneself. After the teaching experiments, the researcher interviews some students, in order to understand sixth grade students’ mathematics performances under the learning trajectory, and aims at the misconceptions and the difficulties in learning of students of different levels to implement the reflection on teaching. The researcher makes self-examinations about the insufficient parts of teaching and considers how to improve original teaching content for helping students to have complete geometry thinking and problem-solving strategy while they are solving the questions of circle complex graph area. And then, they can have problem-solving ability. The findings are as follows:
(1) According to “hypothetical learning trajectory” theory, we can design teaching material of circle complex graph area, including “Finding the center of a circle and understanding circles”, “Degrees of a central angle and a few fractions of a circle”, “The origin of circle area formula”, “Division and moving up for filling”, “Filling up and other strategies”.
(2) Students of different cognitive levels also perform differently under the operation of the learning trajectory of circle complex graph area.
(3) The teacher produces concrete and positive changings in aspects of understanding students’ concepts, improvement in teaching methods, and correcting of course design via retrospective reflection of teaching experiments of learning trajectory, for improving own professional competence in teaching.

Finally, the researcher aims at the progress of teaching experiments to provide three suggestions for the reference content of the teaching of circle complex graph area in the future. The suggestions are as follows:
(1) Viewing students’ prior mathematic knowledge, it’s beneficial to design learning trajectories.
(2) Putting stock in understanding the origin of circle area formula, it can help students to form the long-time and stable memory of the formula.
(3) Cultivating the concept of neat calculation process, it can increase students’ willingness of checking answers and raise the rate of correct answers.
摘要................................................................................................................................I
目次...............................................................................................................................V
表次............................................................................................................................VII
圖次...........................................................................................................................VIII
第一章 緒論
第一節 研究動機............................................................................................1
第二節 研究目的............................................................................................4
第三節 名詞釋義............................................................................................5
第四節 研究限制............................................................................................6
第二章 文獻探討
第一節 「假設性學習軌道」理論之相關研究.......................................... 7
第二節 兒童幾何認知的發展理論..............................................................14
第三節 兒童圓形概念之相關研究..............................................................19
第四節 九年一貫國小數學課程幾何教材分析..........................................23
第五節 南一版本數學圓形複合圖形面積教材分析..................................27
第三章 研究方法
第一節 研究架構..........................................................................................31
第二節 研究對象..........................................................................................34
第三節 課程內容設計..................................................................................35
第四節 資料蒐集與處理..............................................................................46
第五節 研究的信度與效度..........................................................................49
第四章 研究結果與討論
第一節 找圓心、認識圓形..........................................................................51
第二節 圓心角度與幾分之幾圓..................................................................65
第三節 圓面積公式的由來..........................................................................80
第四節 「分割」與「移補」......................................................................97
第五節 「填補」與「其它策略」............................................................112
第五章 結論與建議
第一節 研究結論........................................................................................137
第二節 研究建議........................................................................................144
參考文獻
中文部分............................................................................................................147
外文部分............................................................................................................149
附錄
附錄一................................................................................................................151
附錄二................................................................................................................154
附錄三................................................................................................................156
附錄四................................................................................................................159
附錄五................................................................................................................161
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