An Investigation of Pedagogical Factors Influencing Student Understanding of Geometric Proof

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The purpose of the three-year study is to develop an empirically grounded theoretical model that relates pedagogy to student understanding of geometric proofs. By engaging in the logical reasoning that is associated with mathematical proof, students develop vitally important critical thinking skills. In addition, proof is fundamental to the discipline of mathematics because it is the convention that mathematicians use to establish the validity of mathematical statements. This study focuses on geometric proof because geometry is traditionally the course in which students are first required to construct proofs. Despite the fact that student difficulty with proof has been well established in the literature, existing empirical research on pedagogical methods associated with the teaching and learning of geometric proof is insufficient. This research study, focuses on the effects of curriculum and pedagogy on student understanding of geometric proof.In order to assess the effectiveness of the pedagogical methods used by participating teachers, the project focuses on two components of student understanding of proof, namely, students' beliefs about what constitutes a proof and students' proof-construction ability. Specifically, the project has four objectives:1. To document student understanding of proof in order to update and expand existing research in this area;2. To characterize evolving student beliefs about what constitutes a proof (using Dreyfus and Hadas' (1987) six principles) in proof-based geometry classes and to link these characterizations to aspects of the pedagogy including sociomathematical norms, nature of the activities, and teacher beliefs;3. To characterize students' evolving proof-construction ability in proof-based geometry classes and to link these characterizations to aspects of the pedagogy including sociomathematical norms, nature of the activities, and the teacher's instructional philosophy; and4. To describe the effects of participation in a research study focused on their classrooms on teachers' practice and perspectives that guide their practice. In order to achieve these objectives, the principal investigators will spend two years in the classrooms of several teachers recording the beliefs and proof construction ability of the students as well as the beliefs held and the pedagogical methods used by the teachers. Teachers will read and reflect on the existing research on geometric proof, will work collaboratively with the researchers to develop instructional materials, and will implement these materials in their classrooms. In addition, the study will focus on the sociomathematical norms established in each of the classrooms, as these norms relate to students' understanding of proof. Data will be analyzed as it is collected as well as in the third year of the study in order to develop models that describe how pedagogical interventions influence student understanding.Findings from this project will be disseminated to the mathematics education research and teaching communities. Research findings will have implications for curriculum, pedagogy, and teacher enhancement