Developing secondary students' understanding of the generality and purpose of proof
In spite of their fundamental role in mathematics, prior studies have found that high school students struggle to develop conceptual understanding of proof, and in particular, understand the generality requirement for proofs. Using design research methodology, I investigated an alternative method of introducing ten advanced 9th grade students to proof using problems and instructional strategies that emphasized the generality and purpose of proofs. The design study consisted of 14 sessions, each lasting approximately 30 minutes, that were held twice a week for seven weeks. Semistructured interviews were also conducted with students at the beginning and end of the study to track changes in students' understanding of proof. Findings from this dissertation study are presented as three stand-alone articles. In the first article, I present a framework for assessing students' understanding of different proof components based on Stylianides' (2007) definition of proof. I argue that the proposed framework provides a more nuanced perspective of students' understanding of proof and allows the researcher or teacher to identify aspects of proof that students understand as well as ways that they can improve their argument. In the second article, I analyze my design conjecture that using universal claims, or claims involving the universal quantifier "all" or "any", would afford opportunities for students to 1) engage in the reasoning-and-proving process, 2) talk about reasoning-and-proving, and 3) develop an intellectual necessity for proof. In the third article, I trace one student's successes and challenges in constructing proofs that adhered to the generality requirement over the course of the study. In this article, I demonstrate how her understanding of the purpose of proof supported her transition from empirical to deductive arguments on proof tasks. I conclude in chapter 5 with implications for research and teaching.
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