Proof and reasoning are fundamental aspects of mathematics. Yet, how to help students develop the skills they need to engage in this type of higher-order thinking remains elusive. In order to contribute to the dialogue on this subject, we share results from a classroom-based interpretive study of teaching and learning proof in geometry. The goal of this research was to identify factors that may be related to the development of proof understanding. In this paper, we identify and interpret students' actions, teacher's actions, and social aspects that are evident in a classroom in which students discuss mathematical conjectures, justification processes and student-generated proofs. We conclude that pedagogical choices made by the teacher, as manifested in the teacher's actions, are key to the type of classroom environment that is established and, hence, to students' opportunities to hone their proof and reasoning skills. More specifically, the teacher's choice to pose open-ended tasks (tasks which are not limited to one specific solution or solution strategy), engage in dialogue that places responsibility for reasoning on the students, analyze student arguments, and coach students as they reason, creates an environment in which participating students make conjectures, provide justifications, and build chains of reasoning. In this environment, students who actively participate in the classroom discourse are supported as they engage in proof development activities. By examining connections between teacher and student actions within a social context, we offer a first step in linking teachers' practice to students' understanding of proof.KEY WORDS: classroom interaction, geometry, link between teaching and learning, pedagogical choices, proof and reasoning, secondary school mathematics In order to make sense of mathematics and to communicate mathematical ideas, it is essential to be able to assess and produce mathematical arguments, including formal proofs. Although student difficulty with proof has been well established in the literature (Chazan, 1993;Hart, 1994;Martin and Harel, 1989;Senk, 1985), research connecting pedagogical factors to the learning of geometric proof is limited (Herbst, 2002). Our work addresses this gap in the literature (Martin and McCrone, 2003;McCrone and Martin, 2004). We focus on reasoning and formal proofs in Euclidean geometry because, in the U.S.A., students typically are first required to write formal proofs in the context of Euclidean geometry in the secondary school. In fact, the geometry content is often taught by building a formal system of axioms, definitions, and theorems.Educational Studies in Mathematics (2005) 60: 95-124

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This study assessed the ability of university students enrolled in an introductory calculus course to solve related-rates problems set in geometric contexts. Students completed a problem-solving test and a test of performance on the individual steps involved in solving such problems. Each step was characterised as primarily relying on procedural knowledge or conceptual understanding. Results indicated that overall performance on the geometric related-rates problems was poor. The poorest performance was on steps linked to conceptual understanding, specifically steps involving the translation of prose to geometric and symbolic representations. Overall performance was most strongly related to performance on the procedural steps.

This analysis of the five NSF–funded secondary mathematics curricula describes their alignment with the Process Standards and Content Standards in Principles and Standards for School Mathematics. Distinctive features and examples are included.

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