Proof is accepted as a fundamental aspect of mathematics and mathematics education by many researchers (Heinze and Reiss, 2003; Jones, 2000; Mariotti, 2006; Schoenfeld, 1994). Moreover, reasoning and proof are one of the process standards determined by National Council of Teachers of Mathematics (NCTM, 2000). Proof definition of Bell (1976) was “a directed tree of statements, connected by implications, whose end point is the conclusion and whose starting points are either in the data or are generally agreed facts or principles" (p. 26). Hanna, de Bruyn, Sidoli, and Lomas (2004) defined mathematical proof as a set of explicit givens such as axioms, principles and logic rules which are utilized in constructing deductive arguments. Despite the fact that proof is an important component of mathematics, students usually have difficulty in the construction of proof (Harel and Sowder, 1998; Knapp, 2005; Weber, 2001). Proof is one of the main subjects in the university level mathematics courses. However, some students may graduate with a poor grasp of proof concept (Jones, 2000). In the study of Gibson (1998), it was found that students have difficulty in the comprehension of the rules and the nature of the proof, conceptual understanding, proof techniques and cognitive load.

Not only students but also mathematics teachers have difficulty in mathematical proof. Moreover, mathematics teachers’ perceptions about proof have a critical effect on mathematics education (Harel and Sowder, 1998). Considering the case that pre-service elementary mathematics teachers are future teachers, their conceptions of proof were investigated in the present study. In more details, the study of Jones (2000) and the study of Kinchin and Hay (2000) were taken into account. Jones (2000) investigated the undergraduate students’ conception of proof through concept map in terms of the number of key terms and relationship between the key terms they used. The study conducted by Jones (2000) revealed that most of the undergraduate students have poor understanding of proof. Kinchin and Hay (2000) identified three main concept map structures as chain, spoke, and net. The structure of chain involves a linear sequence of key terms in which each key term is only linked to the previous key term. In the structure of spoke, key terms were directly linked to the core concept but not linked to each other. The structure of net has an improved and extensive form and shows a deep understanding of the concept.

The purpose of the study is to examine pre-service elementary mathematics teachers’ conceptions of mathematical proof. In addition, their conceptions of mathematical proof were investigated in terms of year levels. In the line of these purposes, the research questions of the study are identified as:

1. What are the pre-service elementary mathematics teachers’ conceptions of mathematical proof?

2. How do pre-service elementary mathematics teachers’ conceptions of mathematical proof differ in terms of year levels?

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