Investigation of eight grade students' geometric reasoning processes in the context of instructional design based on learning trajectories

The aim of this study is to examine the geometric reasoning processes of eighth grade students after geometric construction instruction designed with learning trajectories. For this purpose, design-based research was preferred as a research method. Participants consist of 36 students in 8th grade. The geometric thinking levels of these students were categorized as low, medium and good. One student from each category was selected and interviews were conducted, and students' geometric reasoning was examined in these interviews. In this study, learning trajectories for eighth grade students' construction of basic geometric concepts were designed and implemented. These trajectories have been revised by making retrospective analyzes of the implementation process. With these learning trajectories, the students' developmental progress and difficulties in geometric construction studies are presented with an interpretive framework. The geometric reasoning processes of the students were examined in terms of cognitive and perceptual processes according to Duval's cognitive model, from a cognitive perspective, with the interviews made after the teaching. In addition, students' geometric thinking levels and attitudes were examined before and after the application. As a result of the research, it was observed that the geometric thinking levels of the students improved and their positive attitudes towards geometry increased after the geometric construction teaching. In addition, a statistically significant difference was observed in favor of the post-test only at the 2nd and 3rd levels when the geometric thinking levels were detailed. According to this result, it was seen that geometric construction studies were effective in the transition of students from the 2nd level to the 3rd level, which is a critical transition. It has been seen that geometric construction teaching with the designed learning trajectories is effective in improving students' geometric reasoning. In addition, when the geometric thinking levels were examined, it was seen that the students at the 1st level could not perform the geometric construction tasks in general, and they could not perform valid reasonings because visual perception directed their reasoning processes. It was observed that the students at the 2nd level made successful constructions, but exhibited natural reasoning behaviors in mathematical justification, turned to experimental verification at the point of verification and performed reasoning based on measurement. It was observed that verbal perception managed the problem-solving processes of these students, and functional and sequential perception processes were carried out effectively. On the other hand, it was observed that the students at the 3rd level exhibited theoretical reasoning behaviors rather than natural reasoning and successfully realized the construction processes, and although they did not reach a deductive inference level, they made appropriate logical inferences according to their level based on mathematical principles.