Teaching for conceptual change: The case of infinite sets.

In this chapter we show that the instructional design principles deriving from the conceptual change approach offer a valuable framework for analyzing and reflecting on instructional interventions in mathematics. More specifically, we focus on the mathematical notion of equivalency of infinite sets, using the instructional design recommendations of the conceptual change approach to analyze and reflect on related learning environments. We evaluated the impact of traditional courses with little or no emphasis on students' intuitive tendencies to overgeneralize from finite to infinite sets, and of courses that were developed in line with the principles listed above, on high-school students and on prospective mathematics teachers' intuitive and formal knowledge of Cantorian Set Theory. Our findings in the different studies indicate that instruction that implemented these principles led to the promotion of the reconstruction of knowledge structures. These interventions promoted learners' awareness of the differences between finite and infinite systems, and of the contradictions that result from interchangeably applying different criteria when comparing infinite sets. Looking at these instructional interventions through different lenses could provide additional insights into their pros and cons. (PsycINFO Database Record (c) 2016 APA, all rights reserved)