  While proof is a central component of postsecondary mathematical study, proof
construction has historically posed significant difficulties for students who
intend to earn mathematics degrees at the undergraduate level. This work is
comprised of detailed literature reviews, an argument for a methodological
approach, and a descriptive research plan to study how university students use
their mathematical discourse to interpret proof tasks and ultimately engage in
constructing proofs in response to these tasks. The theoretical outcomes of
using the approach outlined therein are: (a) theorized models of undergraduate
students' patterned mathematical activity in response to proof tasks; (b)
identified words, phrases, or problem features that evoke this patterned
activity; and (c) identified similarities and differences in students'
descriptions of their uses for mathematical concepts vs. how instructors expect
students to think about and use mathematical concepts.
