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We report an empirical study of elementary algebra errors, conducted in three separate schools. The errors are diagnosed using mal‐rules, as proposed by Sleeman (1984, 1,985). Our analysis uncovers the following properties of algebra mal‐rules: The frequency of mal‐rules is severely skewed, there are many infrequent mal‐rules and few frequent ones; mal‐rules are very unstable, students typically use mal‐rules very irregularly; different mal‐rules have explanatory power in different schools (many of our most powerful mal‐rules are previously unreported); mal‐rule diagnosis Is (...) more successful with more skilled students; students' confidence ratings do not partition the total set of mal‐rules, every mal‐rule we find is associated with high confidence ratings by at least one student. The Implications of our data for cognitive theories of error generation are discussed. Contrary to commonplace assumptions, we argue that It is impossible to make a clear distinction between slips and mistakes; most errors depend on properties of the knowledge base and the cognitive architecture. Errors In a procedural skill cannot be assumed to be purely syntactic In orgin. (shrink)

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We explore the extent to which Anderson's (1987) theory of knowledge compilation can account for the relationship between instructions and practice in learning to use a simple device. Bibby and Payne (1993) reported experimental support for knowledge compilation in this domain. This article replicates the finding of a performance cross‐over between instruction type and task type that disappears with practice on the tasks. The research is extended by using verbal protocols to model the strategies of novice and more experienced individuals. (...) Production system models of these strategies suggest that knowledge compilation only provides an adequate account of practice for one of two Instruction groups. To model the strategy shifts for the second group, we employ a “procedure modification heuristic” (after Neches, 1987), which relies on access to a declarative model of the configuration of the device. This suggests that instructionally derived declarative knowledge may have a subtle ongoing effect on the changes in procedural knowledge with practice. (shrink)

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