ABSTRACT
INTRODUCTION In 1973, Stephen Erlwanger published a groundbreaking article in the Journal of Children’s Mathematical Behavior reporting on a case study of one sixth grade student’s mathematical thinking. “Benny’s conceptions of rules and answers in IPI [Individually Prescribed Instruction] Mathematics,” or “Benny” as it came to be called, employed protocol analysis of clinical interviews to reveal the “rules” Benny had developed to operate on fractions and decimals. Benny was a successful student in IPI, a curriculum in which students individually progress through sequentially ordered behavioral objectives via continuous cycles of assessment and feedback. Erlwanger showed that while Benny demonstrated mastery on the IPI assessments using his rules, he had little conceptual understanding of fractions and decimals. Perhaps more importantly, he also argued that Benny’s “rules” could be seen as a sensible effort on the part of the learner to construct meaning out of instructional experiences that made little mathematical sense.
