Argumentation in a Geometry Class: Aligned with the Teacher’s Conception of Proof1 AnnaMarie Conner The University of Georgia, USA [email protected] Abstract: While a straightforward relationship between argumentation and proof is not universally accepted, it is reasonable that argumentation, as a tool for student learning, would be instrumental in the teaching of proof in high school geometry classes. This study examines the argumentation in one student teacher’s high school geometry classes and suggests a possible relationship between the observed argumentation and the student teacher’s conception of proof. Aspects of the student teacher’s conception of proof could be seen in how she supported argumentation in her classes. Theme: Research, focused on explanation, argumentation, and proof in geometry education The importance of the teaching and learning of proof and justification is undisputed within the mathematics education community, and the relevance of argumentation to student learning of mathematics has been demonstrated at various levels of school mathematics (Krummheuer, 2000; Yackel, Cobb, & Wood, 1999; Zack & Graves, 2001). Recent research has investigated the relationship between argumentation and proof, demonstrating the efficacy of Toulmin’s (1958/2003) scheme for the analysis of this relationship (Pedemonte, 2007) and the importance of attending to qualifiers and rebuttals in addition to claims, data, warrants, and backings in more advanced analyses (Inglis, Mejia-Ramos, & Simpson, 2007). This paper provides a description of argumentation within a geometry classroom and presents evidence of a possible relationship between a student teacher’s conceptions of proof and her facilitation of collective classroom argumentation in a high school geometry class. Framework The teaching and learning of proof is of major concern to secondary and tertiary mathematics education, both because of the centrality of proof to the practice of mathematicians and because of the status given to proof and justification in curriculum standards (e.g., National Council of Teachers of Mathematics, 2000). Research on the teaching and learning of proof has examined students’ proof schemes (Harel & Sowder, 1998; Housman & Porter, 2003), students’ difficulties with proof (Bell, 1976; Chazan, 1993; Moore, 1994; Weber, 2001), students’ ability to validate texts as proofs (Selden & Selden, 2003; Stylianides, Stylianides, & Philippou, 2004), the relevance of technology to the learning of proof in geometry (Laborde, 2000; Leung & Lopez-Real, 2002; Mariotti, 2000), and students’ perceptions of proof (Almeida, 1995; Recio & Godino, 2001). Building on these researchers’ work in the various aspects of proof and proving, I define a person’s conception of proof as the person’s ability to prove and analyze arguments, perception of the role and need for proof in mathematics, and affective perception of proof. 1 This paper is based on the author’s doctoral dissertation, completed at The Pennsylvania State University under the direction of Rose Mary Zbiek, supported in part by the National Science Foundation under Grant No. ESI0083429 to the University of Maryland with a major subcontract to The Pennsylvania State University and by a research initiation grant from The Pennsylvania State University College of Education Alumni Society. Any opinions, findings, and conclusions or recommendations expressed in this document are those of the author and do not necessarily reflect the views of the National Science Foundation or the College of Education Alumni Society.Toulmin’s (1958/2003) model of argumentation involves claims, data, warrants, backings, qualifiers, and rebuttals, related as shown in Figure 1. According to Toulmin, a claim is the statement whose truth is being established, data is evidence presented in support of the claim, a warrant is a bridge between the data and claim, giving reasons that the particular data presented is relevant to the claim, and backing, which is usually implicit, is support for the warrant’s validity in the particular field in which it is used. Qualifiers and rebuttals are less often used in analysis of argumentation in mathematics education, but were shown to be useful in the analysis of arguments presented by more advanced students of mathematics (Inglis, Mejia-Ramos, & Simpson, 2007). A qualifier is indicative of the strength of the warrant (usually a word such as “probably”), and a rebuttal is a description of circumstances under which the warrant would not be valid (Toulmin, 1958/2003). Figure 1: Diagram of Toulmin’s Model, adapted from Toulmin (1958/2003, p. 104) Krummheuer (1995) adapted Toulmin’s (1958/2003) model of argumentation to mathematics education research by defining as acceptable warrants and backings those that are accepted by the particular mathematics community that is engaging in the argumentation rather than what is accepted by the field of mathematics in general. Like Krummheuer, I define collective argumentation as “a social phenomenon, when cooperating individuals tried to adjust their intentions and interpretations by verbally presenting the rationale of their actions” (Krummheuer, 1995, p. 229). Collective argumentation differs from traditional Aristotelian argumentation because it is concerned with a group reaching consensus rather than an individual convincing a group2. I define a proof as logically correct deductive argument built up from given conditions, definitions, and theorems within an axiom system. This definition of proof suggests that a proof is a particular kind of argument, and thus presupposes a relationship between argumentation and proof. The nature of this relationship is not, however, straightforward. Balacheff (1999) suggests that argumentation may present an epistemological obstacle to a person’s learning to prove, while others (e.g., Pedemonte, 2007) propose that argumentation may lead to proof, although the structure of the proof may be different from the structure of the related argumentation. The relationship that I propose is linked to the collective nature of argumentation in classrooms and to beliefs about the role of the teacher in facilitating learning in classrooms. In brief, part of the teacher’s role in the classroom is to act as a representative of the mathematics community in negotiating sociomathematical norms, including normative standards of argumentation