Students’ difficulties with argumentation, proving, and the role of counterexamples in proving
are well documented. Students in this study experienced an intervention for improving their
argumentation and proving practices. The intervention included the eliminating counterexamples
(ECE) framework as a means of constructing and critiquing viable arguments for a general claim.
This framework involves constructing descriptions of all possible counterexamples to a conditional
claim and determining whether or not a direct argument eliminates the possibility of
counterexamples. This case study investigates U.S. eighth-grade (age 13) mathematics students’
conceptions about the validity of a direct argument after the students received instruction on the
ECE framework. We describe student activities in response to the intervention, and we identify
students’ conceptions that are inconsistent with canonical notions of mathematical proving and
appear to be barriers to using the ECE framework.