How can ideas be connected afterwards? Decomposing teachers’ facilitation practices for conceptual learning in a case of formal volume calculation

Facilitating whole-class discussions based on tasks that support inquiry and provide conceptual learning opportunities has been identified as a central, challenging task of teaching. We drew upon practice-based and interactionist research traditions to decompose how eight experienced Grades 5–7 mathematics teachers facilitated whole-class discussions after their students had determined volumes for rectangular prisms. The discussions highlighted ways that students worked with wooden cubes, imposed multiplicative structures, and developed formal calculations. Qualitative analysis of the video-recorded lessons revealed five practices that the teachers used to facilitate whole-class discussions. Three of the practices supported extensive, explicit connections between higher-order and lower-order knowledge elements related to volumes of rectangular prisms. We characterized such instruction as providing deep conceptual learning opportunities. The remaining two practices jumped quickly to calculations and did not make as many explicit connections to the organization of cubes in rows, columns, and layers. We characterized such instruction as providing shallow learning opportunities. Identifying the five facilitation practices contributes to theorizing about relations between teaching and learning that support conceptual understanding and underscores the importance of examining such relations at a micro-level and in the context of specific mathematical topics. Moreover, the results can inform professional development programs that seek to help teachers to enact productive conceptual learning opportunities during whole-class discussions.