Networking theories of quantitative reasoning and mathematical reasoning to explore students’ understanding of functions

This study examines the reasoning processes of secondary school students while they solve tasks that model functional relationships. It networks theories of quantitative reasoning (QR) and mathematical reasoning (MR) by comparing and contrasting while also coordinating and combining covariational reasoning (CR) with MR to investigate students’ understanding of functions within a quantitatively rich problem-solving process. The analysis draws on data from a small-scale teaching experiment involving eight participants. Students operating at Mental Actions 1–2 primarily relied on memorized strategies and surface-level properties, leading to rigid, procedural responses. In particular, students often invoked graphical forms and symbolic conventions—such as the visual shape of a graph or algebraic templates—as fixed cues for function type. This use of shape thinking and conventions reinforced imitative reasoning, as students applied familiar patterns without analyzing underlying quantitative relationships. Conversely, students demonstrating Mental Actions 3–5 exhibited creative reasoning, engaging deeply with covarying relationships to construct well-supported mathematical arguments. This study underscores the bidirectional relationship between CR and imitative reasoning, suggesting that reliance on procedural strategies both arises from and perpetuates limited conceptual understanding.