Rethinking Affordances and Feedback in AR Environments to Foster Richer Mathematical Inquiry: Lessons from Touch the Derivative
Katherine Riding, Yang Yang, Alison Clark-Wilson, Manolis Mavrikis, Eirini Geraniou, Cosette Crisan
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Published: Sep 5, 2025
DOI: 10.1007/s40751-025-00179-8
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Abstract Augmented reality (AR) technologies offer unique opportunities to explore fundamental ideas in calculus by blending digital and physical worlds, yet realising the full potential of this hybrid reality requires a degree of creativity as we grapple with existing theoretical constructs and seek new ones. This study centres on a single-participant case: Karim, a 15-year-old secondary-school student who used an AR prototype “Touch the Derivative” to investigate relationships between functions and their derivatives. In this paper, we examine how AR technologies can support rich mathematical inquiry by rethinking two interconnected elements: affordances and feedback. We analyse the crucial role of the physical world within AR environments through two intersecting perspectives: the spatial affordances enabled by six degrees of freedom (6DoF), and the physical, cognitive, and contextual dimensions of AR. We then examine how AR facilitates various forms of feedback—through Karim’s interactions with both the researcher and the AR environment itself—highlighting the role of feedback to support understanding and engagement. We conclude by exploring how intentionally designed feedback mechanisms—enabled by analytics and automation—can amplify the affordances of AR and provide more impactful, inquiry-based learning experiences.
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