The comprehension and effective application of geometric concepts present significant challenges for students engaged in geometric problem-solving tasks. This challenge arises from the intricate coordination of visualization, construction, and reasoning processes during the transition from perceptual understanding to a mathematically informed perspective on geometric figures. Raymond Duval delineates various dimensions of cognitive apprehension integral to these processes, encompassing perceptual, sequential, discursive, and operative apprehension. Consequently, optimal geometry instruction necessitates educators to embrace a qualitative and reflexive approach that aligns with these cognitive dimensions.
This research thoughtfully examines students' cognitive apprehension and geometrical thinking levels while implementing the Geometrical Thinking Program (DGT), designed to enhance geometry teaching and learning by prioritizing cognitive processes and apprehension. Additionally, the study delves into the interplay between students' cognitive apprehension types during geometric problem-solving and their geometrical thinking levels, classified according to the van Hiele model.
The study, utilizing a quasi-experimental design and collaborative action research methods, engages four in-service teachers and four seventh-grade classes from primary schools in Croatia, resulting in a sample of 88 students. The experimental group, exposed to the DGT program, is compared with a control group maintaining conventional instructional methods. Throughout the study, the experimental group teachers implemented and assessed changes in their practices aligned with the Duval framework.
Quantitative and qualitative methods, which were based on the analysis of questionnaires, interviews, observations of teachers, and tests for students, reveal noticeable improvements in the experimental group in the areas of van Hiele's levels of geometric thinking and Duval's cognitive understanding. An improvement in van Hiele levels was observed in 52.5% of the students in the experimental group compared to 34.2% in the control group. Additionally, 57.5% of the experimental group students showed an improvement in Duval's cognitive understanding of geometric figures, in contrast to 21.1% in the control group.
The DGT program contributed to significant changes in teachers' attitudes and methods and promoted a high-quality mathematical environment. Teachers shifted to dynamic problem-solving activities, leveraging cognitive understanding to analyse geometric shapes. The positive reactions of the students increased the enthusiasm of the teachers, who recognized the potential of the program for transferability to different grades of schooling.
This research significantly contributes to mathematics education by offering a fresh perspective on geometrical thinking development through cognitive apprehension. It extends the knowledge base not only within the Croatian mathematics education community but also internationally. Collaboration between researchers and educators emerges as a vital force for educational improvements, with action research serving as a potent instrument for fostering student achievement in geometry education and enhancing problem-solving skills across various domains. It proposes a new model for translating theoretical principles into practice, with implications for teacher preparation, classroom practices, and greater teacher involvement in research.
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