Students' conceptions of the definitions of congruent and similar triangles

Abstract

This study addresses high school students’ conceptions of mathematical definitions of congruent and similar triangles. The findings indicate that many of the participants differentiated between definitions and theorems and did not always accept the congruent and similar triangles theorems as formal definitions of congruency and similarity. Based on the participants’ explanations of their responses and from the interviews performed, it appears that two issues prevented some participants from accepting or preferring these theorems as definitions. The first was a concern for uniformity: there is only one known and accepted definition of each concept. The second was a focus on the essence of the concepts: the essence of the concepts of similarity and congruency lies primarily in the lengths of the sides of a triangle. The students who accepted these theorems as formal definitions explained their reaction as arising from the equivalence and from the theorems including necessary and sufficient attributes. Students’ difficulties in understanding the characteristics and roles of mathematical definitions of geometric concepts affected their understandings of mathematical and geometric definitions. This behaviour indicates a tendency to interpret the content of theorems incorrectly and an inability to unpack the logical structure of the theorem.