Algebra Mal‐Rules and Cognitive Accounts of Error

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.