1994．第5卷第1期（Vol. 5 No. 1）．pp. 3341
Can Mathematical Rules and Procedures be Taught without Conceptual Understanding?
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.