Enhancing In-service Mathematics Teachers' Professional Knowledge with an HPM Approach as Observed in Teachers' Reflections

由教師反思中所見之在職數學教師專業知識增強的HPM進路

10.6278/tjme.20140307.003

蘇俊鴻(Jim-Hong Su)；英家銘(Jia-Ming Ying)

海龍公式 ； 數學史與數學教育 ； 學科教學知識 ； Heron's Formula ； history and pedagogy of mathematics (HPM) ； pedagogical content knowledge (PCK)

臺灣數學教育期刊

1卷1期（2014 / 04 / 01）

79 - 97

英文

本研究討論一個由在職教師為主所組成的數學史討論班課程中之教學活動，這個課程採用數學史與數學教育（HPM）的進路，希望能同時提升在職數學教師的數學知識與學科教學知識（PCK）。在這個教學活動中，四位在職高中數學教師閱讀古代數學文本，理解後撰寫讀書心得與反思。古代數學文本的素材包含四位古代數學家對海龍公式證明的古文原文或現代英文翻譯。研究者使用海龍公式證明作為閱讀素材的理由有三點。第一，海龍公式通常被用來計算三角形面積，而現代教材運用餘弦定律的證明方法，展現符號代數的威力，但古代證明通常使用歐式幾何的證明策略，所以，兩種證明策略之間的張力，隱含了幾何與代數雙重表徵的連結問題，使古代證明的閱讀與反思成為我們用以檢驗學習者是否掌握，以及教師能否反思表徵形式之相關能力的極佳範例。第二，不同證明策略，會導致不同的理解困難，而教師本身在遇到這些困難之後，是否能意會到學生學習也會遇到類似的困難，也是可檢驗教師教學內容知識的視點。最後，海龍公式的古代證明會使用到各類不同的先備知識，使得閱讀者有可能利用海龍公式進行數學知識的縱深統整（vertical integration），進而強化教師的教學內容知識。教師們的反思在撰寫完畢之後由研究者分析，可以看到透過數學文本的閱讀，教師們對於海龍公式數學內容知識的理解，補強現行教科書之不足，能夠延伸至面積課程教學的內容。此外，透過幾何表徵的映照，突顯代數方法的簡便性，使得教師對於整個教材的結構脈絡有全面性的觀照。研究者利用Veal與MaKinster的學科教學知識模型來解釋教師教學知識的增強，而文本的閱讀正是由此模型的下層的內容知識，往中層對學生的知識與上層教學策略知識連結，透過數學內容知識的增加，進而強化PCK的其他特質，教師整體的PCK就能提升。研究結果顯示，這四位在職高中數學教師，他們的內容知識、對學生的知識，以及教學策略知識都有增長。同時，從內容知識到上層教學策略中各個屬性間的縱向連結，特別是與脈絡、評量、教學法、課程與社會文化等五個屬性的縱向聯結，以及那些屬性之間的橫向連結皆有增強。

In this paper we discuss an HPM approach that may enhance in-service teachers' mathematics content knowledge and PCK at the same time. The method we used was to let four high school mathematics teachers read historical texts in mathematics, and ask them to write reports on their reflections. The historical texts consisted of four proofs of Heron's Formula provided by various scholars in history. The reports were then collected and analysed. We used the PCK model of Veal and MaKinster to explain the enhancement of the teachers' PCK. Their reflections indicated that their content knowledge, knowledge of students, and knowledge of instructional strategies improved. In addition, the vertical links from content knowledge to several attributes of PCK, as well as the horizontal links among those attributes were strengthened.

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1. Ying, Jia‐Ming,Su, Yi‐Wen,Huang, Jyun‐Wei(2015).An Exploratory Study on Influences of a Mathematical Culture Course on University Students' Mathematics Beliefs - the Case in a Medical University.臺灣數學教育期刊,2(2),1-24.