The Conceptualization Of Proof In The Context Of Elementary Mathematics

The Conceptualization of Proof in the Context of Elementary Mathematics



The Conceptualization Of Proof In The Context Of Elementary Mathematics

Introduction

There are currently increased efforts to make proof central to school mathematics throughout the grades. Yet, realizing this goal is challenging because it requires that students master several abilities. In this report we focus on one such ability, namely, the ability for deductive reasoning, and we review psychological research to enhance what is currently known in mathematics education research about this ability in the context of proof and to identify important directions for future research. We first offer a conceptualization of proof, which we use to delineate our focus on deductive reasoning. We then review psychological research on the development of students' ability for deductive reasoning to see what can be said about the ages at which students become able to engage in certain forms of deductive reasoning. Finally, we review two psychological theories of deductive reasoning to offer insights into cognitively guided ways to enhance students' ability for deductive reasoning in the context of proof.

The Conceptualization Of Proof In The Context Of Elementary Mathematics

Researchers and curriculum frameworks, especially in the United States, have recommended that proof become central to all students' mathematical experiences throughout the grades (e.g., Ball, Hoyles, Jahnke, & Movshovitz-Hadar, 2002; Hanna, 2000; Hanna & Jahnke, 1996; National Council of Teachers of Mathematics [NCTM], 2000; Schoenfeld, 1994; Yackel & Hanna, 2003). In the research literature, one can find three main reasons for the growing emphasis on proof, even in the early grades. First, proof is fundamental to doing mathematics--it is the basis of mathematical understanding and is essential for developing, establishing, and communicating mathematical knowledge (e.g., Ball & Bass, 2003; Carpenter, Franke, & Levi, 2003; Hanna & Jahnke, 1996). Second, students' proficiency in proof can improve their mathematical proficiency more broadly because proof is "involved in all situations where conclusions are to be reached and decisions to be made" (Fawcett, 1938, p. 120). Third, several researchers (e.g., Ball et al., 2002; Harel & Sowder, 1998; Marrades & Gutierrez, 2000; Moore, 1994; Usiskin, 1987) have identified students' abrupt introduction to proof in high school as a possible explanation for the many difficulties that secondary school (see, e.g., Chazan, 1993; Coe & Ruthven, 1994; Healy & Hoyles, 2000; Knuth, Choppin, Slaughter, & Sutherland, 2002) and university students (see, e.g., Epp, 1987; Harel & Sowder, 1998; Martin & Harel, 1989; Selden & Selden, 2003; Simon & Blume, 1996; Stylianides, Stylianides, & Philippou, 2004, 2007; Weber, 2001) face with proof, thereby proposing that students engage with proof in a coherent and systematic way throughout their schooling. Successful engagement with proof requires several abilities by students. One such ability is to recognize that a proof guarantees the truth of a statement for all the elements in the domain covered by the proof (Fischbein, 1982) but does not imply anything about the truth of the statement for elements outside that domain (Stylianides et ...