Toward a didactical phenomenology for the completeness axiom

The study is part of an instructional design project focused on introductory real analysis. The goal of the project is to develop a theoretically grounded and empirically supported instructional approach that builds on students’ experiences in the calculus sequence to engage them in the reinvention of the rigorous foundations of the calculus. An essential aspect of this foundation is the completeness of the real numbers. Drawing on the didactical phenomenology heuristic from the theory of Realistic Mathematics Education (RME), we conducted an iterative instructional design study focused on the completeness axiom. The work proceeded in two phases. First, we conducted an a priori phenomenological analysis to identify a context that would be experientially real to the students and that could be productively mathematized using the least upper bound concept and to develop a conjectured trajectory for the mathematization process. Second, we conducted a series of design experiments to develop a didactical phenomenology for completeness. The purpose of this empirical work was to test the viability of the design suggested by our a priori analysis and inform refinements to our design. The purpose of this paper is to share our design process, describe the resulting didactical phenomenology, and illustrate the corresponding reinvention process with examples from our empirical studies.