HKIER

Abstract

Many mathematics teachers share a common belief that a firm foundation of numerical manipulations must be instilled into students' minds first before they can appreciate the mathematical concepts and relationships involved. Thus teachers, particularly those teaching students of primary grades or of lower mathematics achievement, are under the strong temptation to teach algorithmic rules stripped of conceptual meaning. This paper presents a conceptual analysis to counteract this belief. By delineating the intricacy underlying students' learning of mathematical algorithms, it is shown that this taken-for-granted possibility of separating the two is highly problematic. It will be argued that to teach algorithmic manipulations and procedural rules devoid of substantial conceptual meaning is a self-defeating policy which could just lead to more problems and difficulties in fostering mathematical understanding in the long run. The issue of learning a mathematical algorithm is explored from two different approaches, namely, first from a psychological perspective by considering the cognitive process involved in executing or reproducing a procedure, and second from a philosophical perspective by considering the unique epistemological character of mathematical knowledge and its relation to the communication of meaning component related to the procedure to guide successful retrieval of procedural steps. On the other hand, based on the so-called epistemo-logical triangle of mathematical knowledge, it is demonstrated that conceptual/theoretical knowledge of mathematics can only be properly developed when the "concept" aspect can be appropriately related to "object" and "symbol" respectively in a teaching situation, thus keeping the necessary distinction between "object" and "symbol" in tension. Lastly, with reference to several different conceptualizations of mathematical understanding as a desirable goal of mathematics education, the above claims are further strengthened.

摘要

不少數學教師均認為學生必先有穩固的運算或處理數式的基礎才能進而掌握有關之數學概念及關係，因此他們傾向教授運算規則而不涉及概念理解；在小學或成績低下的班級中，尤其如此。針對這種想法，本文試圖以理論分析，說明學習數學公式或程序的複雜性與及將理解與學習運算分離所存有的問題，並指出這種抽離數學意意的教學方式，長期來說只會帶來更多學習問題和困難。本文從兩個不同進路展析，一方面從心理學角度考察進行或重覆運算程序時的智性過程及記憶容量的限制等；另一方面則從哲學角度，藉著「數學認知的三角關係」，說明數學知識的獨特性與及其意義溝通的關係。最後綜合幾種確認理解作為重要的數學教育目的的理論觀點，更加強以上的結論。