Teacher Knowledge Vol.2-891 WAYS OF REASONING AND TYPES OF PROOFS THAT MATHEMATICS TEACHERS SHOW IN TECHNOLOGY-ENHANCED INSTRUCTION

In this report we document and analyze the ways of reasoning and types of proofs employed by high school mathematics teachers to validate conjectures and to justify procedures in order to solve problems, which emerge when they work in a dynamic geometry environment. Proof is a fundamental activity in mathematical practice (Hanna, 2000; Weber, 2001), also is a key element in school mathematics (National Council of Teachers of Mathematics, 2000). However, research in mathematics education evidences that students have serious difficulties in understanding and presenting deductive proofs (Harel and Sowder, 1998; Schoenfeld, 1985). Why do students experience difficulties in constructing mathematical proofs? One reason may be related to the way used to introduce aspects of proofs in mathematical instruction. Balacheff, 2000 argues that instructional methods rely on asking students to imitate their teachers behaviors. In this context, we consider that in order for students to identify the proving activity as a central in their mathematical experiences, teachers need to have solid understanding of what the concept of proof entails (Stylianides, 2005), not only deductive proof, but also the use of arguments and justifications in general. In this context, it is necessary to carry out researc