We describe a four-level spectrum of pedagogical awareness (SPA) that emerged from the analysis of six undergraduate mathematics tutors' (1) conceptualizations of their first-year students' difficulties; (2) descriptive accounts of their strategies for facilitating the overcoming of these difficulties; and (3) self-reflective accounts regarding their teaching practices. These were recorded in 45 semistructured interviews conducted during the 8-week Oxford University term and following minimally participant observation of their tutorials. We exemplify the four levels of SPA, across these three strands, in 12 characteristic examples where tutors discuss episodes from the observed tutorials. The design and analysis of the study reflect three underlying theoretical perspectives: constructivist, sociocultural, and enactivist. Finally, we draw on participants' comments to suggest that our methodology encouraged reflection, raised awareness, and demonstrated the pedagogical potential of collaboration between mathematicians and mathematics educators. We see such awareness and collaborative goodwill as prerequisites to conceptualizing and implementing reform of pedagogical practice. (Contains 5 tables and 5 footnotes.)