An alternative framework to evaluate proof productions: A reply to Alcock and Inglis

Alcock and Inglis (2009) propose a framework for categorizing individuals' proof productions that depends on considering whether or not these individuals used examples, diagrams, or other objects outside the representation system of proof. They note that their framework is able to account for "all existing case studies of undergraduate proof productions," which makes it "useful for analyzing large-scale datasets." We agree that the framework does have this virtue, but argue that this is a consequence of its simplicity. The cost of such simplicity is that when the framework is applied, a great deal of interesting information is lost. As a result, Alcock and Inglis' framework may be useful to answer some interesting questions in large-scale studies, but it is inadequate to address questions regarding the ways in which different types of syntactic and semantic reasoning contribute and complement each other in students' proof constructions. In this paper, we propose an alternative framework that may be useful for addressing some of these questions. Our framework is surely messier than the one proposed by Alcock and Inglis. As it is interpretive, it will be more difficult and time-consuming to implement and will almost certainly achieve lower inter-rater reliability. However, we believe it is also more useful for studying interesting theoretical questions regarding students' reasoning in proof production tasks.