Learning the single-digit multiplication facts: The role of the commutative principle

Abstract

The role of the commutative principle of multiplication in learning the single-digit multiplication facts was explored in this study. Two theoretical positions are described that depict the manner in which these arithmetic facts are taught and learned. The associationist position states that the single-digit multiplication facts are learned through memorization and should be taught using drill and practice. The meaningful learning conception views learning this set of facts as developing the mathematical concepts underlying multiplication. Proponents of this approach advocate using a variety of experiences directly related to multiplication to establish a conceptual framework for multiplication. The influence of the commutative principle is absent from the associationist formulation, but is quite prominent in the meaningful learning notion of single-digit multiplication fact learning. The purpose of this investigation was to ascertain whether instruction on the commutative principle promotes the transfer of learning from practiced to unpracticed single-digit multiplication facts.

A conceptually-based, explicit instructional model was used to teach the single-digit multiplication facts. An array, skip counting, and repeated addition notion of multiplication was used to establish an initial concept of multiplication and to teach the facts from 2 $\times$ 2 to 5 $\times$ 9 to all subjects. Subjects were randomly assigned to two groups based on pretest fact recall performance. The conceptual plus commutativity group received instruction on the commutative principle, whereas the conceptual only group received extra drill.

The data were analyzed using an analysis of covariance for each dependent measure. The results from a restricted sample of the transfer subtest data (i.e., subjects who scored less than 2 on the transfer pretest measure) indicated that the commutative principle of multiplication does promote transfer of learning from practiced to unpracticed single-digit facts. These results are discussed in terms of the associationist and meaningful learning theories of single-digit multiplication fact learning. The implications of the results for rule versus concept learning are discussed. Also presented are the educational implications of this study and directions for further research.