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The goal of this paper is to increase our understanding of differentapproaches to proving in advanced mathematics. We present two case studiesfrom an interview-based investigation in which students were asked to completeproof-related tasks. The first student consistently took what we call areferential approach toward these tasks, examining examples of the objects towhich the mathematical statements referred, and using these to guide reasoning. The second consistently took what we call a syntactic approach towardthese tasks, working logically with definitions and proof structures withoutreference to examples. Both students made substantial progress on each of thetasks, but they exhibited different strengths and experienced different difficulties. In this paper we: demonstrate consistency in these students' approachesacross a range of tasks, examine the different strengths and difficulties associated with their approaches to proving, and consider the pedagogical issuesraised by these apparent student preferences for reasoning in certain ways.

Referential and syntactic approaches to proving: Case studies from a transition-to-proof course