Spatial Structuring and Units Coordination: Building and Illustrating a Coordinated Theoretical Framework

In this article, we present a coordination of two prominent theoretical perspectives in mathematics education—spatial structuring and units coordination—for building models of students’ spatial enumerations and measurement. The authors, each experienced in either theoretical perspective, hypothesized that a coordination of the two theories may be possible, given their common grounding in Piagetian theory. We argue that such a coordination is not only possible, but important for supporting collaboration across research communities and for building more robust explanatory models of students’ mathematics. Indeed, a central claim of this article is that by building a coordinated theoretical framework, researchers can explain students’ mathematics better than if they apply either theory alone. Methodologically, our coordination of the two theories was empirically driven. We conducted clinical interviews with pre-service elementary school teachers, analyzed interview data from each theoretical perspective separately, then engaged in a process of coordinating our spatial-structuring- and units-coordination-based models. In addition to our proposed coordination of these two theoretical perspectives, secondary contributions of this article include revised definitions of certain theoretical constructs, a case study of analyzing one pre-service elementary school teacher’s reasoning through our coordinated framework (and how our coordinated analysis contributes to insights into this student’s spatial-numerical reasoning), and an illustration of how two epistemologically-compatible theories focusing on students’ mathematical thinking can be coordinated.