"This cannot be"-refutation feedback and its potential affordances for proof comprehension

Professors in proof-based mathematics courses often intend that the feedback they provide on students' flawed proofs will promote proof comprehension. In this theoretical article, we investigate how such feedback can be formulated. Drawing on Lakatos's process of proof and refutation, we propose the notion of heuristic refutation feedback for feedback on a flawed proof that contains a mathematical argument, possibly incomplete, that logically implies that the student's proof is invalid. Such feedback is heuristic in the sense that interpreting and utilizing it invites mathematical reasoning that can contribute to development of proof comprehension. We build on Toulmin's model of argumentation to analyze possible variations in the formulation of feedback. Based on data from a Real-Analysis course, we highlight two key decisions entailed in formulating refutation feedback: deciding what flawed (possibly implicit) claim in the student's proof to refute, and deciding how explicitly to present the various elements of the refutation argument. We exemplify these decisions in two particular types of refutation feedback that we have identified: refutation by counter-example and refutation by false implication. We show how different formulations of refutation feedback may afford different opportunities for student-engagement with particular facets of proof comprehension. Our findings suggest how professors can purposefully tailor feedback in order to achieve particular didactic goals.