Analyzing errors in elementary mathematics
William M. Bricken · 1987
An accurate model of the process by which novices make errors in elementary mathematics would contribute to our understanding of learning and to our style of teaching. Research in Intelligent Tutoring Systems has focused on the automated description and classification of errors in order to enhance the tutor's student model. An error is defined by an error schema, a transformation pattern whose left-hand-side matches the problem before transformation, and whose right-hand-side matches the incorrect result. This thesis characterizes errors as unique. Most error schema occur only once in a sample of classroom responses. Errors that are stable over time for one student are rare, and confined to specific problems rather than to classes of problems. Common errors that occur for many students are common only to the extent that they describe features of the subject domain. On close inspection, students do not share common error processes, even when they get the same wrong answer. The implication is that errors can be modelled by formal computational systems only by eliminating reference to the cognitive processes that create them. In a study of a minimal formal system which permitted accurate tracking of errors, each subject exhibited an idiosyncratic set of errors. Statistical, protocol, and clinical analysis of longitudinal data from an eighth grade pre-algebra class provided no support for the existence of algorithmic mal-rules. Students did exhibit demand-driven errors, goal-driven errors, compensatory errors, extra-domain errors, multiple solution techniques, and abrupt correction of misinterpretations. These observations are consistent with those error studies that incorporate detailed individual process data. The conclusion is that error end-products do not adequately describe error processes, and are highly sensitive to the degree of abstraction of the data they do model. The future of automated tutoring may be to model correct procedures and to identify syntactic pattern errors common to each domain, freeing teachers for remediation of individual semantic misunderstandings.