In this study we examines identity narratives of elementary teachers about 'good' mathematics teaching and about themselves as teachers. The Exploration Pedagogical Discourse (EPD), which is widely adopted by the mathematics education community, is demonstrated through a formal document of the Israeli Ministry of Education, and contrasted with the Acquisition Pedagogical Discourse (APD, akin to 'traditional' or 'teacher-centered' instruction). Our aim was to examine how teachers' identities and pedagogical discourse align with the EPD, contrasted with the APD, and how these alignments relate to their own forms of participation in the mathematical discourse. Twelve teachers were interviewed on the basis of teaching vignettes that were prototypes of different types of instruction. Almost all the teachers had some element of heteroglossity in their talk, drawing on the EPD, while voicing some narratives (often implicitly) that were more aligned with the APD. We focus on one teacher, showing that explicit identity narratives were aligned with the EPD while interpretations of a teaching vignette aligned with the APD. We contrast this case with another teacher who was more consistent in her adoption of the EPD. Finally, we show some connections found between teachers' adoption of APD as opposed to EPD and their own participation in a problem solving episode.