The pedagogy of proving in middle school mathematics

Although justifying and proving have gained a more prominent presence in discussions about school mathematics in all grade levels and across all content areas over the past decade, little research has examined how tasks designed to develop students' competencies in justifying and proving are enacted in the classroom. The literature shows that students, at all grade levels, struggle in developing mathematical arguments that are sufficiently general in nature and that teachers' conceptions and beliefs about proving in school mathematics call into question whether all students, regardless of age or ability, are expected to justify and prove as a part of their everyday mathematical practice. To date, however, there are few, if any, studies that have focused on how tasks designed to elicit justifications are enacted in the classroom. Thus, the central question of this dissertation research is to understand the outcomes of implementing proof-related tasks in the classroom.;To answer the central question of this dissertation, a study was conducted in seven 6th, 7th and 8th grade classrooms that utilized the Connected Mathematics Project (CMP) curriculum. Classroom observations took place during implementation of proof-related tasks---tasks that were identified as providing opportunities for students to produce generalizations or conjectures. Findings from this work suggest that middle school students' classroom experiences with justification and proof are insufficient for developing desired conceptions of mathematical proof as well as the skills necessary to produce appropriate justifications in subsequent mathematics courses. In addition to presenting findings from the classroom study, an integrated summary of research on students' and teachers' conceptions of justification and proof in school mathematics, along with issues related to enacting curriculum and implementing tasks of high cognitive demand, is presented. Finally, a secondary presentation of the results in light of how manipulatives and other concrete representations influenced the enactment of proof-related tasks is provided.