Investigation of secondary school students' geometrical reasoning on the polygons

As Kant calls it, geometry, which is the field of synthetic a priori judgments and which Kant argues is synthesized on the basis of a priori intuitions about space, is an important sub-branch of mathematics and a field that gives individuals important abilities. Geometry gives individuals cognitive abilities such as generalizing, criticizing, summarizing what they have learned in diagrams and expressing them clearly. In order to achieve these abilities, it is important to reveal the types of perception used to approach the given figure in geometric tasks. Because, perception in geometry is closely related to how individuals see geometric shapes and how they interpret them in the mind. According to Duval, the subject called vision in geometry is also related to perception and, as an extension, visual images. For this, the way of looking at a geometric object is a very important cognitive factor in problem solving, reasoning and presenting evidence in geometry, and examining this way of looking is very necessary to reveal the nature of geometric reasoning. In order to understand the nature of geometric reasoning, it is important to reveal the cognitive processes that students activate during the solution of geometric tasks, as well as to what extent they manage shape-concept harmony in geometric tasks. In this study, it has been revealed to examine the mental thinking of middle school students about polygons and the nature of reasoning in their minds with the help of cognitive approaches to geometric reasoning. For this, Fischbein's the theory of figural concepts and Duval's Cognitive model were chosen as the theoretical framework of the research. The study group of the research consisted of 100 students continuing their education at the eighth grade in Karaman Province. The data of the study were collected with a semi-structured interview form and the three forms containing 28 open-ended questions, prepared for the purposes of the research, and analyzed by using the content analysis technique. As a result of the research, it was found that students had various misconceptions about the concepts of triangle, quadrilateral, pentagon and hexagon, that some semantic dilemmas, especially those arising from the linguistic meanings of quadrilaterals such as trapezoid and parallelogram, had negative effects on students' understanding of quadrilaterals, It has been revealed that students' drawings and definitions of polygons influenced by the prototype images in their personal figural concepts, and that their knowledge of figure precedes the conceptual knowledge while reasoning on the geometrical tasks. In addition, it was revealed that some individuals acted according to the perceptual apprehensions defined by Duval and could not form an appropriate reasoning by not activating their operative and discursive apprehensions appropriately. It has been revealed that some individuals could not manage their sequential apprehension, and have difficulties in forming the shape and developing reasoning, especially in geometric tasks given as shapeless. In addition to these, it has been revealed that individuals have some misconceptions and erroneous understandings about the concept of dimension. The results of the geometric reasoning of polygons were discussed with the support of the relevant literature and suggestions were made.