Investigation of Sixth-Grade Students' Cognitive Processes in a Learning Trajectory Designed for Basic Geometric Constructions

Abstract

In this study, it is aimed to design a learning trajectory that supports the conceptual infrastructure and dynamic ways of thinking in the process of basic geometric constructions of sixth-grade students and to examine the cognitive development of the students in this process. In this qualitative study, which is a design based research, data were collected through individual and group worksheets, video recordings of courses, field notes, homework sheets and clinical interviews with focus participants. The focus participants were determined by criterion sampling through an open-ended test developed for the concepts of point, line, line segment and ray, which were considered as prerequisites for the basic geometric constructions. The study was conducted in three phases which are preparation and design, teaching experiment and retrospective analysis. During the preparation and design phase, the initial hypothetical learning trajectory has been designed with the literature review performed to allow comprehensive analysis in epistemological and didactic aspects. Formative evaluations were made by micro analyzes during the teaching experiment carried out for the implementation and evaluation of the learning trajectory. Finally, comparative analysis was conducted on students' thinking processes and actions between the assumed learning trajectory and the actual learning trajectory. Consequently, the revised learning trajectory and students' developmental progress are presented in an interpretive framework. In the most general sense, supporting the cognitive actions such as realizing a construction in different ways and directions, taking into account the changing and unchanging aspects of the geometric structures during the construction process, interpreting the variability of the compass opening, revealing all of the possible points as a circle or part of the required points as an arc, making changes in the route of the steps and interpreting or defending these changes are important in building dynamic geometric constructions. It was noted that the students who could follow the algorithmic steps only operationally without making defense with mathematical justifications were able to put forward more static thinking processes. It was seen that learning environments limited only by analysis and construction phases might be insufficient in realizing constructions with strengthened conceptual infrastructure, and therefore, the importance of designing learning environments that provide opportunity for proof and discussion phases that support dynamic processes of thinking was revealed. In addition, it has been concluded that a learning trajectory which provides opportunity to take into consideration the characteristic features of the geometric structures in the constructions and to make connections with other structures or constructions makes important contributions in strengthening of the conceptual infrastructure.