Role of Conceptual Knowledge in Mathematical Procedural Learning

Two experiments were conducted to explore the relation between conceptual and procedural knowledge in the domain of mathematics. The simultaneous activation view, which argues that computational errors arise from impoverished concepts and that errors can be eliminated by giving concrete referents to symbols, was compared with the dynamic interaction view, which argues for distinct systems that interact diachronically and for a progressive independence of procedural knowledge with expertise. Experiment 1 revealed that many 4th- and 6th-grade children possess significant conceptual knowledge but made computational errors nevertheless. In Experiment 2, a Longitudinal Guttman Simplex analysis revealed that 5th graders mastered conceptual knowledge before they mastered procedural knowledge. Results across studies support the dynamic interaction view. Recently, a number of cognitive theorists have argued for a distinction between conceptual and procedural knowledge. This distinction has been applied broadly to various aspects of cognition, such as memory (Anderson, 1983; Schacter, 1989), language (Anderson, 1983), propositional reasoning (Byrnes, 1988; Keating, 1988), causality (Inhelder & Piaget, 1980), and mathematics (Hiebert, 1987). Conceptual knowledge, which consists of the core concepts for a domain and their interrelations (i.e., "knowing that"), has been characterized using several different constructs, including semantic nets, hierarchies, and mental models. Procedural knowledge, on the other hand, is "knowing how" or the knowledge of the steps required to attain various goals. Procedures have been characterized using such constructs as skills, strategies, productions, and interiorized actions.

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