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題名：傳統中算家論證的個案研究
書刊名：科學教育學刊
作者：洪萬生
作者(外文)：Horng, Wann-Sheng
出版日期：2007
卷期：15:4
頁次：頁357-385
主題關鍵詞：中算家；程序性知識；論證；概念性知識；辭圖並用；Conceptual knowledge；Procedural knowledge；Reasoning；Synthesis of proposition and diagram；Traditional Chinese mathematician
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本文首先簡要評論《華人如何學習數學》，並結合有關數學史與數學教學的研究成果，從中析出可供對比傳統中算家論證的概念，然後，再據以分析劉徽、徐光啟、梅文鼎與李善蘭的幾個論證個案。至於研究方法，則採用歷史文獻分析法與比較史學方法，在一方面，對比劉徽和歐幾里得、劉徽和阿基米德，與劉徽和海龍，以掌握劉徽所代表的中算「固有的」論證特色。另一方面，考察徐光啟、梅文鼎與李善蘭如何呈現他們各自會通中西的論證特色。梅文鼎與李善蘭都為海龍公式提供了證明，尤其是我們進行中西對比的極佳切入點。綜合本文的論述，我們發現：劉徽、徐光啟與梅文鼎的論證，都包括了「程序性（算則）-程序性（算則）」、「程序性（算則）-概念性（含命題）」，以及「概念性（含命題）概念性（含命題）」等知識連結。不過，劉徽的「連結」方式多元，概念結構層次分明，而徐光啟與梅文鼎的論證，在作圖題上表現的「概念性-概念性」連結的邏輯缺陷，暴露了中算「辭圖並用」之限制。至於李善蘭的證明海龍公式，則企圖從「特定的」圖形解放，他針對正五邊形作圖及其證明時，則完全符合歐幾里得的證明規範，亦即：完全以「概念性（含命題）- 概念性（含命題）」之連結為主。
This article briefly reviewed How Chinese Learn Mathematics and thereby synthesized author's own studies concerning HPM. The aim was to abstract some relevant concepts upon which this study based to analyze mathematical reasoning of traditional Chinese mathematicians Liu Hui, Xu Guangqi, Mei Wending and Li Shanlan. As for the methodology, this study adopted the analysis of historical literature as well as comparative historiography. On the one hand, this study contrasted Liu Hui and Euclid, Liu Hui and Archimedes as well as Liu Hui and Heron in order to understand how Liu Hui makes reasoning on his own terms. On the other hand, this study investigated how Xu Guangqi, Mei Wending and Li Shanlan adapted the Western mathematics and integrated it with the Chinese mathe-matics under the influence from Western mathematics. The basic tool for comparative study was Mei Wending and Li Shanlan's proof on Heron's formula. As concluding remarks, the author comes to suggest three different connections, namely those between two forms of conceptual knowledge, between one form of conceptual and one procedural, as well as between two forms of procedural know-ledge, can be used to characterize some aspects of traditional Chinese mathematical argumentation in which the four mathematicians had due role to play. Since these terms are due to mathematics edu-cation, the author hopes this article can serve as a demonstration for integration of researches in mathematics education and those in history of mathematics.
期刊論文
1.洪萬生(2000)。《算數書》初探。師大學報：科學教育類，45(2)，77-91。  延伸查詢new window
2.Haapasalo, L.、Kadijevichvich, D.(2000)。Two Types of Mathematical Knowledge and Their Relation。Journal für Mathematk Didaktik，21(2)，139-157。  new window
3.蘇俊鴻(2006)。海龍公式的各樣證法之特色。HPM通訊，9(4)，35-40。  延伸查詢new window
4.Berggren, J. L.(1990)。Proof, pedagogy and the practice of mathematics in medieval Islam。Interchange，27(1)，36-48。  new window
5.Sfard, Anna(1991)。On the dual nature of Mathematical conceptions: reflections on processes and objects as different sides of the same coin。Educational Studies in Mathematics，22(1)，1-36。  new window
6.Gray, E.、Tall, D.(1994)。Duality, ambiguity and flexibility: A proceptual view of simple arithmetic。Journal for Research in Mathematics Education，25(2)，115-141。  new window
7.洪萬生(2000)。《算數書》的幾則論證。臺灣歷史學會會訊，11，44-52。  延伸查詢new window
8.陳春廷(2006)。李善蘭如何證明海龍公式？。HPM通訊，9(4)，16-22。  延伸查詢new window
9.李建勳(2006)。海龍公式的流變－由徐光啟到梅瑴成。HPM通訊，9(4)，9-15。  延伸查詢new window
10.洪萬生(2005)。從程序性知識看《算數書》。師大學報：人文與社會科學類，50(1)，75-89。new window  延伸查詢new window
11.洪萬生(2004)。教改爭議聲中，證明所為何事？。師大學報：科學教育類，49(1)，1-14。  延伸查詢new window
12.洪萬生(2006)。當梅文鼎遇上《幾何原本》。科學月刊，37(7)，504-508。  延伸查詢new window
13.程和欽(2006)。對有關李善蘭證明海龍公式的一點心得。HPM通訊，9(4)，52-52。  延伸查詢new window
14.洪萬生(1994)。數學史上三個公式積圓面。科學月刊，25(7)，539-544。  延伸查詢new window
15.Fauvel, J.(1988)。Cartesian and Euclidean Rhetoric。For the Learning of Mathematics，8(1)，25-29。  new window
16.Lam, L.-Y.、Ang, T.-S.(1986)。Circle Measurement in Ancient China。Historia Mathematica，12，325-340。  new window
17.Chemla, K.(2005)。Geometrical Figures and Generality in Ancient China and Beyond: Liu Hui and Zhao Shuang, Plato and Thabit Ibn Qurra。Science in Context，18(1)，123-166。  new window
18.Siu, M. K.、Volkov, A.(1999)。Official Curriculum in Traditional Chinese Mathematics: How Did Candidates Pass the Examinations?。Historia Scientiarum，9(1)，87-99。  new window
19.Siu, M. K.(1993)。Proof and Pedagogy in Ancient China: Examples from Liu Hui's Commentary on Jiu Zhang Suan Shu。Educational Studies in Mathematics，24，345-357。  new window
會議論文
1.大學入學考試中心(2001)。指定科目考試參考手冊。0。  延伸查詢new window
2.Horng, W.-S.(2001)。Intrinsic Cognitive Dimension of the HPM: Text versus Context。0。  new window
3.Tall, David(2002)。Differing Modes of Proof Belief in Mathematics。0。  new window
4.Duval, Raymond(2002)。Proof Understanding in Mathematics: What Ways for Students?。0。  new window
5.Haapasalo, L.、Kadijevichvich, D.(2003)。Simultaneous Action of Conceptual and Procedural Mathematical Knowledge by Means of ClassPad。0。  new window
6.Cullen, C.(2002)。Learn from Liu Hui? A Different Way to Do Mathematics。0。783-790。  new window
7.Haapasalo, L.(2003)。Linking Procedural and Conceptual Mathematical Knowledge in Technology-based Learning。0。98-101。  new window
8.Volkov, A.(2004)。History of Ideas or History of Textbooks: Mathematics and Mathematics Education in Traditional China and Vietnam。臺中市。57-80。  new window
圖書
1.洪萬生(1991)。同文館算學教習李善蘭。近代中國科技史論集。臺北市：中央研究院近代史研究所。  延伸查詢new window
2.洪萬生、林倉億、蘇惠玉、蘇俊鴻(2006)。數之起源：中國數學史開章《筭數書》。臺北市：臺灣商務印書館。  延伸查詢new window
3.Fan, L. H., Wong, N. Y., Cai, J. F.,、Li, S. Q. Eds..(2004)。How Chinese Learn Mathematics: Perspectives from Insiders。Singapore：World Scientific。  new window
4.洪萬生(1999)。孔子與數學。孔子與數學。臺北市：明文。  延伸查詢new window
5.郭書春(1998)。九章算術。遼寧教育出版社。  延伸查詢new window
6.高大倫(1995)。張家山漢簡《引書》研究。成都：巴蜀書社。  延伸查詢new window
7.Dauben, Joseph W.(1979)。Georg Cantor: His Mathematics and Philosophy of the Infinite。Georg Cantor: His Mathematics and Philosophy of the Infinite。Cambridge。  new window
8.Van der Waerden, B. L.(1983)。Geometry and Algebra in Ancient Civilizations。Geometry and Algebra in Ancient Civilizations。Berlin, Germany/ Heidelberg, Germany/ New York, NY。  new window
9.Clavius, C.(1591)。Euclidis Elementorum Libri XV。Euclidis Elementorum Libri XV。Coloniae。  new window
10.Bulmer-Thomas, I.(1941)。Selections Illustrating the History of Greek Mathematics, Vol. II。Selections Illustrating the History of Greek Mathematics, Vol. II。Cambridge, MA。  new window
11.Heath, T. L.(1956)。Euclid: The Thirteen Books of the Elements, Vol. 3。Euclid: The Thirteen Books of the Elements, Vol. 3。New York, NY。  new window
12.(1987)。The History of Mathematics: A Reader。The History of Mathematics: A Reader。London, UK。  new window
13.Bunt, L. N. H.、Jones, P. S.、Bedient, J. D.(1988)。The Historical Roots of Elementary Mathematics。The Historical Roots of Elementary Mathematics。New York, NY。  new window
14.李繼閔(1992)。九章算術及其劉徽注研究。九章算術及其劉徽注研究。臺北市。  延伸查詢new window
15.中村元(1995)。中國人的思維方法。中國人的思維方法。臺北市。  延伸查詢new window
16.郭書春(1995)。古代世界數學泰斗－劉徽。古代世界數學泰斗－劉徽。臺北市。  延伸查詢new window
17.Grattan-Guinness, I.(1997)。The Fontana History of the Mathematical Sciences: The Rainbow of Mathematics。The Fontana History of the Mathematical Sciences: The Rainbow of Mathematics。London, UK。  new window
18.Engelfriet, P.(1998)。Euclid in China: The Genesis of the First Chinese Translation of Euclid's Elements Books I-VI。Euclid in China: The Genesis of the First Chinese Translation of Euclid's Elements Books I-VI。Boston, MA/ Köln, Germany/ Leiden, Netherlands。  new window
19.(2000)。History in Mathematics Education。History in Mathematics Education。Boston, MA/ Dordrecht, Netherlands/ London, UK。  new window
20.(2005)。華人如何學習數學。華人如何學習數學。南京。  延伸查詢new window
21.洪萬生(2006)。此零非彼0：數學、文化、歷史與教育文集。此零非彼0：數學、文化、歷史與教育文集。臺北市。  延伸查詢new window
22.Davis, R. B.(1986)。Conceptual and Procedural Knowledge in Mathematics: A Summary Analysis。Conceptual and Procedural Knowledge: The Case of Mathematics。Hillsdale, NJ/ London, UK。  new window
23.Volkov, Alexei(1992)。Analogical Reasoning in Ancient China: Some Examples。Regards Obliques sur L'argumentation en Chine。Paris, France。  new window
24.梅文鼎(1993)。勾股舉隅。中國科學技術典籍通匯－數學卷，四。鄭州。  延伸查詢new window
25.劉鈍(1993)。勾股舉隅、幾何通解提要。中國科學技術典籍通匯－數學卷，四。鄭州。  延伸查詢new window
26.梅文鼎(1993)。平三角舉要。中國科學技術典籍通匯－數學卷，四。鄭州。  延伸查詢new window
27.歐幾里得(1993)。幾何原本，十五卷。中國科學技術典籍通匯－數學卷，五。鄭州。  延伸查詢new window
28.Horng, W.-S.(2000)。The Pythagorean Theorem in Different Cultures。History in Mathematics Education: The ICMI Study。Dordrecht, Netherlands。  new window
29.Horng, W.-S.(2000)。Euclid versus Liu Hui: A Pedagogical Reflection。Using History to Teach Mathematics。Washington, DC。  new window
30.Horng, W.-S.(2001)。The Influence of Euclid's Elements on Xu Guangqi and His Successors。Statecraft and Intellectual Renewal in Late Ming China: The Cross-cultural Synthesis of Xu Guangqi (1562-1633)。Boston, MA/ Köln, Germany/ Leiden, Netherlands。  new window
31.Engelfriet, P.、Siu, M.-K.(2001)。Xu Guangqi's Attempts to Integrate Western and Chinese Mathematics。Statecraft & Intellectual Renewal in Late Ming China: The Cross-cultural Synthesis of Xu Guangqi (1562-1633)。Boston, MA/ Köln, Germany/ Leiden, Netherlands。  new window
32.馬若安(2002)。17、18世紀中國天文學與數理天文學著作中的時空觀。法國漢學，第六輯（科技史專號）。北京市。  延伸查詢new window
33.洪萬生(2003)。數學文化的交流與程序性知識。理性、學術和道德的知識傳統。臺北。  延伸查詢new window
34.Li, J.(2004)。A Chinese Cultural Model of Learning。How Chinese Learn Mathematics: Perspectives from Insiders。Singapore。  new window
35.An, S.-H.(2004)。Capturing the Chinese Way of Teaching: The Learning-questioning and Learning-reviewing Instruction Model。How Chinese Learn Mathematics: Perspectives from Insiders。Singapore。  new window
36.Huang, R. J.、Leung, Frederick K. S.(2004)。Cracking the Paradox of Chinese Learners: Looking into the Mathematics Classrooms in Hong Kong and Shanghai。How Chinese Learn Mathematics: Perspectives from Insiders。Singapore。  new window
37.Siu, M. K.(2004)。Official Curriculum in Mathematics in Ancient China: How Did Candidates Study for the Examination?。How Chinese Learn Mathematics: Perspectives from Insiders。Singapore。  new window
38.Gu, L. Y.、Huang, R. J.、Marton, F.(2004)。Teaching with Variation: A Chinese Way of Promoting Effective Mathematics Learning。How Chinese Learn Mathematics: Perspectives from Insiders。Singapore。  new window
39.李瑾(2005)。中國文化的學習模式。華人如何學習數學。南京。  延伸查詢new window
40.蕭文強(2005)。中國古代官學數學課程：考生是怎樣學習和準備考試的？。華人如何學習數學。南京。  延伸查詢new window
41.蔡金法、維克多．西弗賴利(2005)。中國學習者的數學思維特徵－一個跨國比較研究的視角。華人如何學習數學。南京。  延伸查詢new window
42.黃榮金、梁貫成(2005)。中國學習者悖論的質疑：透視香港與上海數學課堂。華人如何學習數學。南京。  延伸查詢new window
43.徐斌豔(2005)。外在表徵對中國學生數學學習的作用。華人如何學習數學。南京。  延伸查詢new window
44.安淑華(2005)。追逐中國的教學方式：學－問和學－思的教學模式。華人如何學習數學。南京。  延伸查詢new window
45.洪萬生(2005)。從古今翻譯看數學文化交流。迎接新世紀：重新檢視香港數學教育－蕭文強教授榮休文集。香港。  延伸查詢new window
46.Wong, N. Y.(2006)。From 'Entering the Way' to 'Exiting the Way': In Search of a Bridge to Span 'Basic Skills' and 'Process Abilities'。Mathematics Education in Different Cultural Traditions: A Comparative Study of East Asia and the West。New York, NY。  new window
47.李善蘭(1867)。天算或問。  延伸查詢new window
其他
1.王鼎勳(2006)。有關李善蘭證明海龍公式的心得與討論，0。  延伸查詢new window
2.（明）徐光啟(1983)。勾股義，0。  延伸查詢new window
3.(1983)。幾何原本，0。  延伸查詢new window
4.Duval, R.(1999)。Questioning Argumentation，0。  new window
圖書論文
1.Wong, N. Y.(2004)。The CHC learner's phenomenon: Its implications on mathematics education。How Chinese learn mathematics: Perspectives from insiders。Singapore：World Scientific。  new window
2.黃毅英(2005)。儒家文化圈(CHC)學習者的現象--對數學教育的影響。華人如何學習數學。南京：江蘇教育出版社。  延伸查詢new window
3.Hiebert, J.、Lefevre, P.(1986)。Conceptual and procedural knowledge in mathematics: An introductory analysis。Conceptual and procedural knowledge: The case of mathematics。Lawrence Erlbaum Associates。  new window
4.Cai, J. F.、Cifarelli, V.(2004)。Thinking Mathematically by Chinese Learners。How Chinese learn mathematics: Perspectives from insiders。World Scientific。  new window
5.顧泠沅、黃榮金、費蘭倫斯．馬頓(2005)。變式教學：促進有效的數學學習的中國方式。華人如何學習數學。南京：江蘇教育出版社。  延伸查詢new window
 
 
 
 
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