Abstract

In 1973, Stephen Erlwanger published a groundbreaking 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, analyzed 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, 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 of the learner to construct meaning out of instructional experiences that made little mathematical sense.