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研究生: 蘇意雯
論文名稱: 數學教師專業發展的一個面向:數學史融入數學教學之實作與研究
指導教授: 洪萬生
學位類別: 博士
Doctor
系所名稱: 數學系
Department of Mathematics
論文出版年: 2005
畢業學年度: 93
語文別: 中文
論文頁數: 294
中文關鍵詞: 數學史HPM 教師專業發展模型反思詮釋學
英文關鍵詞: History of Mathematics, Teacher’s Model for Professional Development, Reflection, Hermeneutics
論文種類: 學術論文
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  • 摘要
    在數學教學上輔以歷史取向,自從1970 年代初創立的數學史與數學教學的
    關聯之國際研究群(International Study Group on the Relations between the History
    and Pedagogy of Mathematics, HPM),就是以此為首要目標。如何讓數學史可以
    在數學的「教學」和「學習」中,扮演更有效的角色,是有心從事HPM 教學的
    教師相當關心的課題。本研究是以學校為中心,由同校教師組成實踐社群,讓學
    校不只是學生們的學習環境,同時也成為教師們在職教育的場所,並透過協同行
    動研究的方式,觀察參與教師在學習HPM 教學中,如何調融數學史與數學知識,
    經由自我詮釋進行教學的轉變歷程。具體而言,我們企圖解答如下兩個問題:
    1. 以學校為中心,甚麼是HPM 為進路的教師專業發展之策略?
    2. 以學校為中心,HPM 為進路的專業成長策略下,參與之數學教師有了什
    麼轉變?
    在本研究進行過程中,參與教師(連同研究者共有四位) 經歷了直觀期、面向擴
    張期、適性期三個階段。這其中所發展出的策略有:
    ‧ 廣泛閱讀數學史及數學教學相關書籍
    ‧ 利用認知三面向的學習工作單之設計,引動教師融入數學史於數學教學
    ‧ 藉由教學後實作心得促成反思
    ‧ 多方面參與和數學教育/ HPM 有關之座談與研討
    ‧ 定期專家諮詢
    ‧ 以學校為中心之實踐社群方式帶動共同成長
    其中涵蓋邏輯、歷史、學生認知三面向的HPM 學習工作單,連結了參與教師對
    於HPM 理論的了解與實作。根據參與教師學習HPM 的教學過程,研究者仿造
    詮釋學理論,提出「HPM 教師專業發展模型」,從中顯示參與教師如何在HPM
    學習工作單的設計過程中,體會C1 (由教科書編者、課程標準與教科書內容所組
    成之循環),以及C2 (由古代數學家、數學物元、數學理論所組成之循環) 之精神,
    經過自我詮釋之後進行教學。此外,我們也利用外顯的教學行為觀察,將參與教
    師在數學課堂上對於HPM 教學的概念化後的實作表現,刻劃成為「分離」、「外
    加」、「引介」、「執行」、「和諧整合」、「決策」幾種狀態。最後,並提出「優選」
    狀態,作為教師專業發展未來所努力的理想境界之一。
    最後,本論文闡明上述所發展的策略可以促成參與教師如下的轉變:
    ‧ HPM 教學者身分的轉變
    ‧ 科普寫作的參與
    ‧ 反思、批判性能力增強
    ‧ 引動數學知識的統整
    ‧ 導向以學生為主體的教學
    有關本研究之結論如何推廣至在職教師訓練課程,以及對於有心想要自我充實或
    學習HPM 教學的教師,研究者也提出具體建議,從而證成數學史融入數學教學
    的發展與實踐,可以成為數學教師專業發展的進路之一。

    Abstract
    There has been a growing interest in adopting the historical approach in
    mathematics teaching since the 1970s. How can history play an effective role in
    improving the teaching and learning of mathematics? Teachers who are concerned
    about HPM would have regarded this as a primary goal. If we extend the
    pedagogical concern to initiating more mathematics teachers in applying the history of
    mathematics into their teaching, we believe that this would be beneficial not only to
    students but to teachers themselves as well. Teachers’ education is very important.
    So we must know well what the impact of the history of mathematics is on the
    development of mathematics teachers. In order to deal with the above questions, we
    undertook one school-based research during a two-year period, from August of 2002
    to July of 2004. A community of teaching practices in terms of HPM was developed
    in one of the Taipei municipal senior high school. By way of collaborative action
    research, we observed participating teachers’ process of transformation in which they
    adjusted and melt the history of mathematics and mathematics knowledge by means
    of interpretation and teaching. Therefore, in this thesis, I attempt to answer the
    following two questions:
    1. What are the strategies for teacher's professional development on HPM
    approach on the school-centered base?
    2. And what are the changes of these participating mathematics teachers
    under this HPM approach?
    The research was conducted in a partnership among three teachers, T1, T2, T3 and
    the researcher herself (hereafter abbreviated participants). In the light of the HPM,
    the participants went through three phases of professional development. They
    learned to search for primary sources, to read related articles and to engage in critical
    discussions, which includes practices from both Western and Eastern methods of
    teaching in order to design and create HPM worksheets. They were encouraged to
    write down their reflections to make public their private ideas. Apparently their
    reflective narration could fortify knowledge, make their innovative works accessible
    to others, and go on to enhance their professional knowledge. We believe that,
    through this kind of professional practice, the participants can increase their personal
    and professional knowledge, which in turn contribute to their teaching.
    The strategies the participants adopted are: reading a lot of articles about
    mathematics teaching, designing HPM worksheets including the logical aspect of
    mathematical knowledge, the historical aspect of mathematical knowledge and theaspect of student’s cognition. Finally, the researcher suggests a Teacher’s Model for
    Professional Development in terms of HPM, which can explain the practices of these
    teachers through the process. In this model, teachers enter the hermeneutic circle,
    say C1, to look into the ideas of the editors of textbooks, the mathematics knowledge
    and the contents of textbooks. Then they enter another hermeneutic circle, say C2, to
    learn the ancient mathematicians’ ideas, mathematical objects, and mathematical
    theories. After the teachers interpret the essence of C1 and C2 by themselves they
    then start to teach. In practice, we can characterize in six different manners, the
    teachers’ use of the history of mathematics in the classroom: isolation, addition,
    introduction, execution, integration, and decision-making. In the end, the researcher
    suggests that “optimization” to be the goal for future development of the teachers.
    By the end of the two-year project, it is obviously that the participants have
    enhanced their professional expertise in terms of the HPM in following ways, namely,
    1) they can begin to write popular mathematics articles; 2) they are more reflective
    into their teaching than ever; 3) they are able to integrate their mathematics
    knowledge into a broad picture; and 4) they start to care about the students' thinking.
    As a conclusion, this thesis suggests that an HPM approach can do to help
    mathematics teacher’s professional development in an efficient way.

    i 目次 第1 章緒論............................................................1 第1.1 節研究動機.....................................................................1 第1.2 節研究目的.....................................................................2 第1.3 節名詞界定.....................................................................4 第1.3.1 節HPM……………………………………………..4 第1.3.2 節數學教師專業發展…………..………………….4 第1.3.3 節學習型組織及實踐社群.………………………..6 第1.4 節待研究問題…………………………………………10 第1.5 節研究範圍與其限制…………………………………10 第2 章文獻評論.....................................................11 第2.1 節教師專業發展…………………………….………...11 第2.1.1 節教師專業發展的重要性及實施方式................11 第2.1.2 節反思實作.……………………...……………….15 第2.1.3 節實踐社群的學習理論………………………….17 第2.1.4.節實踐社群與教師學習………………………….18 第2.1.5 節國外數學教師專業發展實作討論…………….19 第2.1.6 節國內數學教師專業發展實作討論....................24 第2.2 節數學史與數學教學之關聯.......................................27 第2.2.1 節運用數學史於數學教學之理念........................27 第2.2.2 節HPM 教學素材的研發及使用…..…………….30 第2.2.3 節有關HPM 的教師專業發展文獻探討….….…34 第2.2.4 節有關數學史的實作研究…………………….…34 第2.3 節以學校為中心之教師在職進修...............................41 第2.3.1 節教師在職進修的歷程發展…………………….41 第2.3.2 節教師在職進修的制度及理念………………….42 第2.3.3 節以學校為中心的進修方案……………….…....44 第2.4 節總評…………………………………………………47 第3 章研究方法.......................................................49 第3.1 節研究場域及參與人員………………………………49 第3.1.1 節研究場域……………………………………….49 第3.1.2 節參與人員……………………………………….49 第3.2 節研究進行方式………………………………………51 第3.2.1 節HPM教學發展模式……………………………54 第3.2.2 節研究策略……………………………………….59 第3.2.3 節實踐社群進行之時程表……………………….60 第3.2.3.1 節實踐社群進行之活動……………………..63 第3.2.3.2 節為HPM 擬定目標…….…………………..64 第3.2.3.3 節打造HPM 之多元面向..….………………66 第3.2.3.4 節確立HPM 教學發展模式..……………….69 第3.2.3.5 節檢視成長狀態嘗試與國際接軌…………..69 第3.3 節資料收集……………………………………………69 第3.4 節資料分析……………………………………………71 第3.5 節研究時間安排……………...……………………….72 第4 章研究結果.......................................................73 第4.1 節參與教師的初始教學狀態………………………....73 第4.1.1 節日進老師的初始教學………………………….73 第4.1.2 節積極老師的初始教學………………………….76 第4.1.3 節海綿老師的初始教學………………………….81 第4.1.4 節研究者的初始教學…………………………….86 第4.2 節參與教師HPM 教學的發展與實踐………………..91 第4.2.1 節日進老師HPM 教學的發展與實踐.………….91 第4.2.1.1 節直觀期……………………………………..92 第4.2.1.2 節面向擴張期………..….…………………..95 第4.2.1.3 節適性期………………....….……………..106 第4.2.2 節積極老師HPM 教學的發展與實踐.…………108 第4.2.2.1 節直觀期……………………………………108 第4.2.2.2 節面向擴張期………..….………………….110 第4.2.2.3 節適性期………………....….……………...118 第4.2.3 節海綿老師HPM 教學的發展與實踐.…………120 第4.2.3.1 節直觀期……………………………………120 第4.2.3.2 節面向擴張期………..….………………….124 第4.2.3.3 節適性期………………....….……………...129 第4.2.4 節研究者與社群成員之互動歷程…...…………132 第4.2.4.1 節直觀期……………………………………132 第4.2.4.2 節面向擴張期………..….………………….133 第4.2.4.3 節適性期………………....….……………...138 第5 章結論與建議.................................................141 第5.1 節結論……………………..……..………….……….141 第5.1.1 節HPM為進路的教師發展策略……….……….141 第5.1.1.1 節HPM 教師專業發展模型……..…………142 第5.1.1.2 節造成參與教師改變之策略………………145 第5.1.2 節參與教師在HPM教學中之轉變……..……...146 第5.2 節建議……………………………….. ……..……….164

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