Co-emergence of reasoning and representing as social practice in small-group interactions

This study investigates students' co-emergence of covariational reasoning (CR) and representational fluency (RF) while solving quantitatively rich tasks. Specifically, it examines how students' reasoning about quantities and quantitative relationships developed through representational activity in small-group interactions. A teaching experiment was conducted in an after-school setting with eight secondary students at a community centre. Retrospective analysis revealed that graphical representations emerged as the primary medium for both basic and advanced CR, suggesting that graphs may offer an accessible context for reasoning about dynamic quantitative relationships. In addition, two distinct patterns of reasoning emerged. Students exhibiting Relational Connections and engaging in Mental Actions 3–4 created and connected multiple representations to model the underlying covarying quantitative relationships. In contrast, students demonstrating Multistructural Connections and Mental Actions 1–2 tended to treat quantities as fixed values or as simply increasing or decreasing together. The representations they produced were often fragmented and disconnected, providing little evidence of how each representation conveyed or deepened understanding of the quantitative relationship. The findings reveal that the co-emergence of reasoning and representing is socially mediated through dialogue, negotiation, and joint construction of representations, underscoring the need for classroom environments that support these intertwined processes.