從程序性知識看《算數書》

Abstract

在本文中，我打算運用『程序性知識』的面向來考察《算數書》的內容。過去數學史家曾運用『算則』，來刻畫中國古代數學的特色。現在，『程序性知識』則出自數學教育，我們因而可以援引數學教育的研究成果，來豐富我們對於中國古代數學特徵的理解。其實，『數學史研究』與『數學研究』固然可能彼此互惠，同理，『數學史』與『數學教育』當然也可以互相發明。基於此，我將根據數學教育專家的論述，舉例說明程序性知識 vs. 概念性知識，以及此一『對比』如何關聯到數學論證上。然後，我們針對《算數書》中的幾個問題及其解法，來檢視它們如何與程序性知識相關連。特別地，我也將試著運用Eddie Gray & David Tall所謂的『程序成概念』，以說明某些『術曰』所呈現的知識類型。最後，我們再從此一角度，考察這些『術曰』中涉及的數學論證之類型。In this article the author will explore the Suan Shu Shu in terms of the notion of procedural knowledge. In the past three decades, historians of Chinese mathematics have successfully adopted the term "algorithm" - a concept from computer science - to characterize some aspects of ancient Chinese mathematics. Thus we can now refer to research in mathematics education in order to enrich our historical understanding of ancient Chinese mathematics. In some sense, the study of mathematics and the study of the history of mathematics can benefit each other, and the same goes for studies of the history of mathematics and studies in mathematics education. Therefore, here I will first contrast procedural knowledge with conceptual knowledge and show the relevance of this contrast to studies in mathematical reasoning. By looking carefully into the mode of presenting and solving problems in the Suan Shu Shu, I will then investigate how these problems can be explained in terms of procedural knowledge and/or Gray & Tall's notion of the "procept". In conclusion, I will try to show the connection between and among some aspects of the ancient text's methodology in order to clarify the types of mathematical reasoning involved here.