One of the main goals of this study was to listen to the voices of the teachers and students about their perceptions of two approaches to high school mathematics: inquiry-based instruction and conventional instruction. This study described the similarities and differences in the way teachers and students viewed upper-grade mathematics and mathematics instruction and how that affects their experiences in the classroom.As a multiple case qualitative study, various forms of data were collected including two perception continuums created by the author, teacher and student responses to prompts about what they did in mathematics class, teacher statements about their beliefs about mathematics, observations of each classrooms, and in-depth interviews with 15 students. This study found that students in the four classrooms viewed: (1) the conceptions of mathematics and mathematics instruction differently (2) adaptive reasoning and thinking in the mathematics classrooms differently and (3) the roles of teachers and students differently. There was greater distinction between the students' views in inquiry classrooms and students' views in the conventional classrooms. The author posited that inquiry has the possibility of having students think about mathematics in new and broader ways, which did not happen in the two conventional classrooms . This study also found that the views of certain sets of students in each classroom differed from their teachers. These differences were greater in the "purest" conventional classroom and the "purest" inquiry classroom.
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