This commentary addresses the role of theoretical frameworks in building models of students’ mathematics. Specifically, it compares ways that the Learning Through Activity framework (LTA) and scheme theory explain and predict students’ mathematical activity. Both frameworks rely on Piagetian constructs – especially reflective abstraction – to build explanatory models for teaching and learning. LTA attempts to provide the teacher-researcher with a greater degree of determination in student learning trajectories, but then the teacher-researcher must address constraints in the students’ available ways of operating. These issues are exemplified in the case of teaching students about multiplying fractions. Additional theoretical issues arise in explaining logical necessity in students’ ways of operating and the role of reflective abstraction in organizing new ways of operating.
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