The Position of the Student in Geometry Instruction: A Study from Three Perspectives.

Abstract

This study aims to understand the student’s position in instruction. I conceptualize instruction as interactions between the teacher, students, and mathematics, in educational environments (Cohen, Raudenbush & Ball, 2003; Lampert, 2001). In the three manuscripts contained in this dissertation, I look at the position (Harré & van Langenhove, 1999) of student from the perspective of the teacher, the student, and the mathematics. “Mathematical Arguments in a Virtual High School Geometry Classroom” looks at the position of the student from the perspective of mathematics. It examines the mathematical arguments that could be made by learners in response to a virtual classroom discussion by comparing arguments made by a learner who had taken a geometry class to arguments made by a learner who had not. It shows the virtue of the two-column proof in its affordance to support chains of implications in arguments. However it also shows the drawback of the two-column proof in its lack of flexibility to support backings and rebuttals in arguments. “Teachers’ Perceptions of Geometry Students” looks at the position of the student from the perspective of the teacher. It examines teachers’ perceptions of students that are instrumental in the work of teaching. It shows that while ‘making conjectures’ teachers perceive students in terms of engagement, ignoring the mathematical value of students’ work. While ‘doing proofs’ teachers perceive students in terms of the mathematical content at stake. These different perceptions of students crucially influence how students are supported in their mathematical work. “The Work of ‘Studenting’ in High School Geometry Classrooms” looks at the position of the student from the perspective of the student. It examines the work that students do in instruction and the tacit knowledge that could guide this work. A theoretical model that describes ‘studenting’ is developed as well as a model for the rationality that supports ‘studenting.’ Each group of participants involved in this study responded to the same scenario of geometry instruction, depicting a geometry class working on an open ended mathematical problem. These data sets provide three points of view on instruction. Together they serve to inform the instructional position of students.