What do mathematicians wish to teach teachers about the discipline of mathematics?

This study investigates what mathematicians, who teach academic mathematics courses to secondary school mathematics teachers, want to teach them about the discipline of mathematics. Data sources included interviews with five research mathematicians. The data analysis revealed that expanding teachers’ knowledge about the discipline of mathematics was one of the interviewees’ main objectives. The mathematicians referred to nine characteristics that can be grouped into three key aspects: (1) essence of mathematics, (2) doing mathematics, and (3) worth of mathematics. The characteristics of the discipline of mathematics generated in our study are in line with ideas and practices widely advocated in the last decades for the teaching and learning of mathematics, and with contemporary philosophical approaches regarding mathematics. The potential of the characteristics to serve as a conceptual framework in studies that focus on teacher knowledge about the discipline of mathematics is discussed.