Investigation of gifted students' geometric reasoning within the framework of mathematical working space

The objective of this study was to examine the mathematical work-based geometric reasoning of gifted 8th grade students in figurless geometry problems analyzed within a real-life context. The Theory of Figural Apprehension was used to address the quality of geometric reasoning. The study's methodology entailed an examination of geometric reasoning processes based on the components of the Mathematical Working Space (MWS) as well as the semiotic, instrumental, and discursive dimensions defined in the theory. The case study was employed in the analysis. The participants of the study were selected from the students enrolled in the ÖYG2 program (Special Skills Development Program) attending BİLSEM (Science and Arts Center) through the use of the purposeful sampling method. In the preparation process of the study, supportive training sessions were conducted with the students to instruct them in the use of mathematical tools and technology. Ten separate sessions were conducted in each of the three groups for ten problems that were developed in accordance with the study's objectives. The data collection instruments employed in this study encompassed practice problems, camera and audio recordings obtained during problem-solving sessions, observation notes, student worksheets, group interviews with students following the sessions, and researcher field notes. Analysis framework based on MWS components were used for the a priori and posteriori analyses of problem solving sessions, with the analyses further deepened through the application of Figural Apprehension Theory. It was determined that gifted students drew numerous shapes, managed the shape, and did not consider the quality of the shape in the reasoning process related to the solutions of geometry problems in which geometric shapes were not provided. The study's findings indicated that students engaged in a mathematical inquiry characterized by the predominance of semiotic and discursive formations with minimal instrumentation. The analysis revealed that operative apprehension was activated in response to variations in representation, instrumentation, and reference. These elements, constituting the epistemological plane, facilitated significant interplanar circulation. The findings of this study suggest that by empowering gifted students to create their own geometric shapes during the geometric reasoning process, their mathematical studies can be enriched.