Teacher quality has become a national policy concern in the US, especially in mathematics. This study provides insights into the conceptualisation of high quality mathematics teaching from the perspective of approximately 750 students in grades nine through twelve. Results from Rasch analysis yield information about the quality of the "Mathematics Quality Survey" constructed for this study and the hierarchy of items representing varying levels of quality as perceived by the students. Group interviews with teachers and open-ended responses from students are used to support the findings. This study lays the groundwork for understanding the difference between being qualified and being a quality teacher. (Contains 2 tables and 1 figure.)