Enhancing student’s understanding of unit conversion by explicating underlying structures: design research on schematization in middle schools

Abstract Schematization from visual models can contribute to enhancing students’ understanding of mathematical procedures when the underlying mathematical structures are focused and explicitly articulated. While students’ learning with schematization approaches has been intensively studied for arithmetical procedures, less is known for a similar case of procedures of unit conversion. The current design research study, therefore, pursues the research question of how to enhance students’ understanding of unit conversion so that they can progressively learn to explain and justify the conversion procedure for mass units. We developed a schematization trajectory with tasks and scaffolds starting from manipulating visual models to focus on structures, progressively schematizing them into a formal procedure and justifying this procedure. In Design Experiment Cycle 1, we qualitatively analyzed 20 middle school students’ explanations and justifications to empirically specify the relevant structures involved in unit conversions (bundle structures, place-value structures, and refinement structures) and their relation to each other. In Design Experiment Cycle 2, we redesigned the tasks and scaffolds in the schematization trajectory so that students could progressively unpack relevant structures. The qualitative analysis traces two focus pairs’ learning pathways and substantiates the role of explicitly articulating relevant structures for schematization that were typical for the analyzed 18 students. The outcome of the paper is not only a revised learning environment but also an empirically grounded contribution to theorizing on focusing and articulating different mathematical structures in schematization processes.