Author(s):
Prior research has documented several persistent challenges students face in introductory proofs courses. These challenges include a tendency to conflate the truth of an implication with the truth of its converse and difficulties transforming logical implications in general. Students experience additional difficulties in quantifying logical implications and their negations and in interpreting multiply quantified statements. Within our Proofs Project, we have framed such challenges as epistemological obstacles. Research from the Proofs Project has identified students’ constructions of logical implications, as actions or as objects, as a key predictor for the kinds of epistemological obstacles they experience during instructional interactions. A student might treat a logical implication as an action by taking “If p, then q” as a command to act in the following way: determine whether p is true, and if so, conclude that q is true. As such, the implication has three components: two statements/properties, p and q, and an action of inference between them. In contrast, a student who treats a logical implication as an object has encapsulated this way of operating as a single entity. Many of the epistemological obstacles identified in prior research concern transforming and quantifying logical implications, so it stands to reason that constructing logical implications as objects might address them. After all, to transform or quantify something implies there is some thing to transform or quantify—a mathematical object. We share Proofs Project resources (tasks and instructional modules) that can support students’ constructions of logical implications as objects and address related epistemological obstacles.
Coauthors
Rachel Arnold, rlongley@vt.edu
