To define mathematical proof is difficult since it may have different roles depending on the author, the community or the audience, and the methods of the proof (Cadwallader Olsker, 2011; Staples, Bartlo, & Thanheiser, 2012). One of the most simple and explicit definitions of mathematical proof was stated by Bruyn, Sidoli and Lomas (2004) as “a mathematical proof, by definition, can take a set of explicit givens (such as axioms, accepted principles or previously proven results), and use them, applying the principles of logic, to create a valid deductive argument” (p.82). Based on the related literature review, it can be seen that definitions of proof have different school of thoughts (VanSpronsen, 2008). For example, some definitions of proof focused on the characteristic of being a logical and deductive argument or the structure of proof (e.g., Griffiths, 2000; Hanna & Barbeau, 2002; Weber, 2005) and some of them mainly focused on the functions/roles of proof (e.g., Goetting, 1995; Hanna, 1989; Hersh, 1993).

Yopp (2011) implied that there is not a clear and strict distinction between the terms role, purpose and function of mathematical proof and these terms were used interchangeably in the studies. While the roles of proof were examined in some studies in detail by considering many applications, there are also studies which examine the roles of proof under more general titles. According to Hersh (1993), there are two roles of proof which are convincing and explaining. Similarly, Volmink (1990) accepted conviction as the most important role of proof. Schoenfeld (1994) expressed the roles of proof as communicating ideas with others, thinking, exploring and understanding mathematical arguments. However, proof might involve some additional roles depending on the educational setting (Nordby, 2013).

Students should have experience with activities involving justification, proof and reasoning processes in mathematics classrooms (Bieda, 2010; Bostic, 2016). To be able to integrate these processes into mathematics classrooms effectively, mathematics teachers should also have necessary knowledge and experience regarding proof and reasoning. In addition to the effects of mathematics teachers’ knowledge of proof on students’ understanding of proof, mathematics teachers’ proof definitions also affect students’ views, ideas, attitudes and perceptions of proof. In this respect, how mathematics teachers describe mathematical proof is an important issue. To state differently, since prospective middle school mathematics teachers are future teachers, their experiences, perceptions and ideas regarding proof might have effect on framing their future instructions (Bostic, 2016). Thus, one of the aims of this research is to investigate prospective middle school mathematics teachers’ definitions of mathematical proof. Moreover, students should be able to choose appropriate proof method in the proving process of the given statement (Rota, 1997) however they may have difficulties in naming the proof methods in some cases even though they know how to apply the method (Türker, Alkaş, Aylar, Güler, & İspir, 2010). Considering this issue, proof methods that prospective middle school teachers know while proving the given mathematical statements were also investigated in the present study. In this respect, this study was guided by two main research questions:

1. How do prospective middle school mathematics teachers define mathematical proof?

2. Which proof methods do prospective middle school mathematics teachers know while proving the given mathematical statement?

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