傳統中算家論證的個案研究

Case Studies of Traditional Chinese Mathematical Reasoning: Liu Hui, Xu Guangqi, Mei Wending and Li Shanlan

洪萬生(Wann-Sheng Horng)

《科學教育學刊》 15卷4期 (2007/08) Pp. 357-385

https://doi.org/10.6173/CJSE.2007.1504.01

摘要

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。

關鍵字

中算家 ； 程序性知識 ； 論證 ； 概念性知識 ； 辭圖並用

並列摘要

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 mathematics 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 knowledge, 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 education, the author hopes this article can serve as a demonstration for integration of researches in mathematics education and those in history of mathematics.

並列關鍵字

Conceptual Knowledge ； Procedural Knowledge ； Reasoning ； Synthesis of Proposition and Diagram ； Traditional Chinese Mathematician

參考文獻

大學入學考試中心()。

中村元(1995)。中國人的思維方法。台北:學生書局。

有關李善蘭證明海龍公式的心得與討論（未出版）

安淑華、范良火、黃毅英、蔡金法、李士錡主編(2005)。華人如何學習數學。南京:江蘇教育出版社。

李建勳(2006)。海龍公式的流變－由徐光啟到梅玦成。HPM通訊。9(4)，9-15。