Descriptions of expertise have focused on knowledge and characteristics of mathematics teacher educator (MTE) practice. To contribute to discussion of MTE expertise we share findings from a study of MTE discussion practice with prospective teachers (PTs) concerning pedagogy of teaching mathematics. Research on MTE discussion practices have shown there are key differences between discussions of mathematics and mathematics teaching in professional development, making clear MTEs' discussion practices in mathematics methods courses focused on mathematics and mathematics teaching must differ. In this paper we focus on MTEs' discussion practice in initial teacher education settings, building from empirical findings that include descriptions of MTEs' discussion practice focused on mathematics teaching. Specifically, we address the question: What is our mathematics teacher educators' pedagogy of discussion practice? Using self-study methodology and data from two primary sources: (1) transcripts of 8 critical friend conversations, and (2) evidentiary maps created using course artifacts, we use vignettes to describe three characteristics: anticipating preservice teachers' thinking, layering instructional activities, and preservice teachers' move toward independence in teaching practice. Findings from this study and mathematics education literature are used to support the claim that MTEs use anticipating, layering, and handover to independence as a system to promote PTs' discussion of mathematics teaching in methods courses. Findings are used to draw conclusions about MTE expertise of discussion practice and the importance of self-based methodologies in research on MTE expertise.