Demands and scaffolds for explaining the connection of multiple representations: Revisiting the bottle-filling task

Explaining the connections of multiple representations can enhance students’ conceptual understanding (e.g., in the bottle-filling task for functional graphs). But it poses high discursive demands that need to be further unpacked. The design research study qualitatively investigates the potentials and demands that fourteen second-language learners face when explaining the connection between functional graphs and filling glasses. The qualitative analysis of students’ pathways towards good explanations identifies (a) demands to construct a mental contextual representation of the filling process, (b) demands to unpack the holistic perspective into more refined concept elements of covariation and correspondence approaches, (c) highly intertwined demands to identify the relevant variables in view. For each of these underlying demands, we identify scaffolds to enable students – even recent second-language learners – to engage in mathematically and discursively demanding practices and to enable teachers to support them.