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

Abstract

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.