Peer Instruction (PI) is an evidence based pedagogy commonly used in undergraduate physics instruction. When asked questions designed to test conceptual understanding, it has been observed that the proportion of students choosing the correct answer increases following peer discussion; however, relatively little is known about what takes place during these discussions or how they are beneficial to the processes of learning physics [M.?C. James and S. Willoughby, "Am. J. Phys." 79, 123 (2011)]. In this paper a framework for analyzing PI discussions developed through the lens of the "resources model" [D. Hammer, "Am. J. Phys." 64, 1316 (1996); D. Hammer "et al.", "Information Age Publishing" (2005)] is proposed. A central hypothesis for this framework is that the dialogue with peers plays a crucial role in activating appropriate cognitive resources, enabling the students to see the problem differently, and therefore to answer the questions correctly. This framework is used to gain greater insights into the PI discussions of first year undergraduate physics students at the University of Edinburgh, UK, which were recorded using Livescribe Smartpens. Analysis of the dialogues revealed three different types of resource activation corresponding to increasing cognitive grain size. These were activation of knowledge elements, activation of linkages between knowledge elements, and activation of control structures (epistemic games and epistemological frames). Three case studies are examined to illustrate the role that peer dialogue plays in the activation of these cognitive resources in a PI session. The implications for pedagogical practice are discussed.