Acquisition of children's addition strategies: A model of impasse-free, knowledge-level learning

Abstract

When children learn to add, they count on their fingers, beginning with the simpleSum Strategy and gradually developing the more sophisticated and efficientMin strategy. The shift fromSum toMin provides an ideal domain for the study of naturally occurring discovery processes in cognitive skill acquisition. TheSum-to-Min transition poses a number of challenges for machine-learning systems that would model the phenomenon. First, in addition to theSum andMin strategies, Siegler and Jenkins (1989) found that children exhibit two transitional strategies, but not a strategy proposed by an earlier model. Second, they found that children do not invent theMin strategy in response to impasses, or gaps in their knowledge. Rather,Min develops spontaneously and gradually replaces earlier strategies. Third, intricate structural differences between theSum andMin strategies make it difficult, if not impossible, for standard, symbol-level machine-learning algorithms to model the transition. We present a computer model, calledGips, that meets these challenges.Gips combines a relatively simple algorithm for problem solving with a probabilistic learning algorithm that performs symbol-level and knowledge-level learning, both in the presence and absence of impasses. In addition,Gips makes psychologically plausible demands on local processing and memory. Most importantly, the system successfully models the shift fromSum toMin, as well as the two transitional strategies found by Siegler and Jenkins.