Representing and defining irrational numbers: Exposing the missing link

Abstract

This article reports on a study of prospective secondary teach- ers' understanding of the irrationality of numbers. Specifically, we focused on how different representations influenced participants' responses with respect to irrationality. As a theoretical perspective we used the distinction between transparent and opaque representations, that is, representations that show some features of numbers and representations that hide some features. The results suggest that often participants did not rely on a given transparent representation (e.g., 53/83) in determining whether a number is rational or irrational. Further, the results indicate participants' tendencies to rely on a calculator, preference towards decimal over common fraction representation, and confusion between irrationality and infinite decimal representation regard- less of the structure of this representation. As a general recommendation for teaching practice, we suggest a tighter focus on representations and conclusions that can be derived from considering them. Definitions of irrational numbers provided at a school level are strongly linked to representations. This report gives insight into the extent to which represen- tational features of numbers are attended to when rationality or irrationality is considered, and into the ways in which different representations relate to one an- other, as perceived by preservice teachers. This report is part of ongoing research on understanding of irrational numbers by prospective secondary school teachers. Specifically, we focus here on how irrational numbers can be (or cannot be) rep- resented and how different representations influenced participants' responses with respect to irrationality.