Algebra Mal-Rules and Cognitive Accounts of Error

We report on emplrlcol study of elementory algebra errors, conducted In three separate schools. The errors ore diagnosed uslng mal-rules, as proposed by Slee-man (1984, 1,985). Our anolysls uncavers the fallowlng properties of algebra mal-rules: The frequency of mat-rules Is severely skewed, there are many infrequent mol-rules and few frequent ones: mal-rules are very unstable, students typically use mal-rules very irregularly: different mal-rules hove explanatory power In dlf-ferent schools (many of our most powerful mal-rules are prevlausly unreported): mal-rule dlagnosls is mare successful with mare skilled students: students' can fldence ratings da 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. Controry to commonplace assumptions, we argue that It is lmpossible to make a clear dis-tlnctlon 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 argln.