Refutations and reasoning in undergraduate mathematics

This paper concerns undergraduate mathematics students’ understandings of refutation and their related performance in abstract conditional inference. It reports on 173 responses to a refutation instrument that asked participants to: 1) state ‘true’ or ‘false’ for three statements, providing counterexamples or reasons if they thought these false (all three were false); 2) evaluate possible counterexamples and reasons, where reasons were ‘corrected’ versions of the statements but not valid refutations; and 3) choose which of the counterexamples and the corrected statements were better answers, explaining why. The data show that students reliably understood the logic of counterexamples but did not respond normatively according to the broader logic of refutations. Many endorsed the corrected statements as valid and chose these as better responses; we analyse their explanations using Toulmin’s model of argumentation. The data further show that participants with better abstract conditional inference scores were more likely to respond normatively by giving, endorsing, and choosing counterexamples as refutations; conditional inference scores also and predicted performance in a proof-based course.