Investigating students' geometrical proofs through the lens of students' definitions

The current study is part of a comprehensive research on linking visualization, students' construction of geometrical concepts and their definitions, and students' ability to prove. The aim of the current study is to investigate the effect of learners' understanding of definitions of geometrical concepts on their understanding of the essence of geometrical proofs and their ways of proving. By "understanding" (geometrical definitions and/or proofs), we mean knowing the definition's and/or proof's meaning and their role within the logic structure of geometry and also the ability to define and/or prove in "geometrically correct" ways. Ninety grade 11 students from an Arab high school in Israel participated in the comprehensive study in geometry of which the current study forms a part. Research tasks for the investigation of the current study's aim were designed, constructed, and used in a questionnaire to the whole research population and in interviews with about 10% of the population. The findings point clearly to effects of elements within the students' understanding of definitions on their understanding of proofs and on their ability to prove. These elements are as follows: (a) The difficulty to internalize that an incomplete definition is an incorrect definition and may lead to an incomplete proof. (b) The difficulty to deal with non-economical definitions forms the basis to non-economical proofs. (c) Students have difficulties to accept equivalent definitions to the same geometrical concept. (d) The students' lack of understanding the origin of a constructive definition of geometrical concept.