I Got Rhythm—I Got Space: How Karim Generates an Epistemic Augmented-Reality Space While Exploring the Derivative

Abstract Spaces augmented by virtual objects (AR spaces) have entered the field of mathematics education, providing promising new ways of experiencing mathematical objects and fostering processes of conceptualization. Students’ learning in this new kind of space—which combines the virtual and real—calls for new theoretical and methodological tools. This paper proposes a conceptualization of the epistemic AR space, one that assumes space to be produced and reproduced by people and both time and space to be related by coherent processes built through spatial experiences and rhythmic repetition of meaning-making. This conceptualization is the basis for investigating how a student develops his view on the relationship between functions and their derivatives in an augmented reality environment by exploring virtual graphical representations using hand gestures. Video analyses linked to transcripts combine three coordinated perspectives: the analysis of spatial practices and epistemic actions and how both are rhythmically connected within and across episodes. The results show local and global mechanisms of producing an epistemic AR space. Two processes contribute locally to producing an epistemic AR space. Intentional spacing consists of spatial practices that provide the resources for epistemic experiences with the virtual elements, and these experiences are merged in processes of synthesizing . Globally, three epistemic phases are generated, whereby the key phase for knowledge construction is that of an in-depth and challenging exploration of a sine function that proceeds at a slow rhythmic pace. The findings may be used to inform future design and research of AR spaces, and teacher monitoring of students’ learning.