Revisiting Popular Frameworks of Geometric Thinking: The Case of Mariah’s Thinking About Hierarchical Relationships and Diagrams

Understanding hierarchical geometric relationships represents an important step in the development of geometric thinking. However, researchers have reported challenges in the teaching of this topic to both K–12 students and preservice teachers. Prominent frameworks regarding geometric thinking may influence researchers to focus on certain aspects while ignoring others. Most notably, the roles of hierarchical diagrams and explicitly stated inclusive definitions in the process of learning hierarchical geometric relationships have received insufficient attention. We conducted a teaching experiment with a preservice teacher whom we call Mariah. Over the course of the sessions, she made substantial progress in her thinking about hierarchical geometric relationships. We present the results in the form of two arguments, which we summarize as follows: (a) it is possible for a learner to entertain definitions and their consequences without necessarily changing their conceptions of the shapes involved; and (b) communication with diagrams is nontrivial and can be intertwined with learners’ conceptualizations of relationships among shapes; furthermore, learners’ conceptualizations of hierarchical relationships do not necessarily follow trivially from their conceptions of the shapes involved. We offer implications for the teaching of hierarchical relationships and for further research into the learning of hierarchical relationships.