Embodied geometry across technological ecologies

Introduction The technologies such as artificial intelligence (AI) and immersive virtual reality (VR) are increasingly entering mathematics classrooms, yet their pedagogical role remains largely confined to enhancing visualization and feedback. This study addresses this limitation by reconceptualizing technology integration through an ecological–embodied perspective, examining how different technological ecologies reorganize the perception–action system underlying geometry learning. Methods Grounded in 4E cognition, ecological dynamics, and Embodied Design, a design-based research (DBR) study was conducted with sixth-grade students. Three learning environments were implemented: (1) material bricolage construction, (2) AI-mediated bodily classification using Teachable Machine, and (3) immersive 3D modeling in Gravity Sketch VR. Multimodal data, including video, interaction logs,and artifacts, were analyzed through a microgenetic, theory-informed approach focusing on agent–task–environment dynamics. Results Findings show that geometric invariants, such as closure, symmetry, convergence, and spatial coherence, emerged through iterative cycles of perturbation, recalibration, and stabilization. Each technological ecology redistributed affordances and constraints in distinct ways: material environments foregrounded structural stability, AI-based systems emphasized categorical discrimination, and immersive VR supported volumetric oherence. Across cases, geometric understanding did not precede action but arose from embodied coordination within distributed systems. Discussion The results suggest that the educational potential of AI and VR lies not in representational enhancement but in their capacity to reorganize perception–action loops. Technologies functioned as constitutive elements of cognitive systems rather than external tools, enabling geometry to emerge as a relational and embodied achievement. This study contributes to shifting the discourse on technology integration from tool adoption toward the ecological design of learning environments that support the stabilization of mathematical invariants.