UC Berkeley

Constructing auxiliary lines—supplementary diagrammatic elements added to geometric figures to reveal latent relational properties—often leads to elegant insights by enabling problem solvers to perceive a configuration in new ways. Yet this skill remains elusive to teach and learn because it requires students to envision figural elements that, by definition, are not yet overtly present. Espousing an enactivist perspective, this design-based research project investigates whether culturally grounded movement practice can serve as a generative resource for geometric reasoning. Learning new dance movements frequently elicits the spontaneous formation of attentional anchors, imaginary perceptual structures that support complex bodily coordination. This dissertation examines how, given appropriate semiotic resources and instructional guidance, students can objectify these tacit attentional anchors from dance practice as explicit auxiliary lines in geometry. To this end, I designed a set of learning materials situated in Balinese traditional dance, collectively called Geometry Resources in Dance (GRiD). Across three interrelated papers, the dissertation analyzes both the design rationale and the learning processes emerging from its implementation, aiming to contribute to educational practice, to the development of humble theory, and to the advancement of design-based research methodology. The first paper introduces the GRiD design and reports findings from teaching experiments conducted in individual and small-group settings, demonstrating how students’ generated attentional anchors in proto-geometry dance-based tasks can be surfaced, negotiated, and stabilized as auxiliary constructions for geometry reasoning. The second paper presents fine-grained microgenetic analyses of the socio-cognitive-semiotic processes unfolding during collaborative movement problems. These analyses led to the formulation of the theoretical construct of sign–tool hybridity to articulate students’ inventive appropriation of material artifacts as both semiotic means and cognitive instruments. The paper also accounts for technical elements of the activity design that enabled the emergence of this unique behavior, which, I argue, is desirable as an epistemic practice and, therefore, should be explicitly fostered through a dedicated design framework. The third paper steps back to present an autoethnographic case study of culturally situated embodied design in mathematics exemplified through analyses of Balinese topography, architecture, spirituality, language, and traditional dance. The paper specifies how tacit cultural knowledge can be transformed into epistemic resources for learning design. Overall, this dissertation contributes to the learning sciences by examining how geometric reasoning can be grounded in tacit spatial practice through culturally situated embodied design.