Assuming guided inquiry as a pedagogical ideal in mathematics education implies that teaching must connect students' thinking about a subject with curricular agendas and instructional goals. Because students are not typically inclined to consider their active inquiry as a route to acquiring the knowledge that is valued in school, such teaching must simultaneously elicit students' engagement in inquiry and legitimate inquiry as a route to learning. Within this conceptual framework, the empirical research reported in this paper describes several strategies used by secondary school geometry teachers as they attempted to practice a pedagogy of guided inquiry using the "Geometric Supposers." The teachers' strategies are discussed in terms of sociolinguistic theories about the teacher's role in defining the meaning of mathematical knowledge in the classroom. (Author)