Students’ individual schematization pathways - empirical reconstructions for the case of part-of-part determination for fractions

Abstract

According to the design principle of progressive schematization, learning trajectories towards procedural rules can be organized as independent discoveries when the learning arrangement invites the students first to develop models for mathematical concepts and model-based informal strategies; then to explore the strategies and to discover pattern for progressively developing procedural rules. This article contributes to the theoretical and empirical foundation of the design principle of progressive schematization by empirically investigating students’ individual schematization pathways on the micro-level for the specific case of part-of-part determination of fractions. In design experiments series in laboratory settings, nine pairs of sixth graders explored the part-of-part determination and progressively schematized their graphical strategies before discovering the procedural rule. The qualitative in-depth analysis of 760 min of video shows that progressive schematization is a multi-facetted process that cannot be described by internalization of graphical procedures alone. Instead, the compaction of concepts- and theorems-in-action is crucial, especially for the goal of justifiable procedural rules.