Angling the Stars: A Geometry Design Reconciling Indigenous and Colonial Perceptions

• In Event: In-Sight Out: Challenges and Opportunities in Learning Mathematics Through Negotiating Egocentric and Allocentric Perspectives

Abstract

Mainstream mathematics is grounded in European epistemology and for Diné students, it is a colonial imposition that fails to recognize cultural knowledge and practices of this land. In particular, Indigenous epistemic patterns are not elicited, accepted, or recognized in classroom mathematics. The following research aims to rematriate (Tuck, 2011) mathematics curriculum by investigating Diné land-based ethnomathematics and designing an enactive, intersubjective activity in discourse with Western approaches. Diné geometrical reasoning encompasses the individual within the land and sky environment in practice and language and fosters egocentric perspectives (D’Ambrosio, 2001; Pinxten et al., 1983) while traditional pencil and paper classroom mathematics requires an allocentric perspective. The project builds an epistemologically pluralistic learning environment (Turkle & Papert, 1991) that restructurates common mathematical practices (Wilensky & Papert, 2010) as emulating and extending embodied immersion in the natural world (Dimmel & Milewski, 2019).


STARR (Students Tracking Angular Rotation Recorder) is a design-based instructional activity meant to tackle students’ “absence of meaning” in the notion of an angle (Thompson, 2013). Implementing enactivist theory, we operationalize Thompson’s (2008) cognitive analysis of the angle concept by grounding it in perceptual–motor enactment, particularly in the action of “opening” the angle’s interior with the forearms (cf. Hardison, 2019, p. 361, on attentional motions; see Figure 6).


Dynamic Geometry Environments (DGE) offer design solutions for exploring angles interactively, however those technologies enable only single-person allocentric perspectives (Crompton, 2015; Smith et al., 2014). The embodied-enactive-egocentric approach elicits a perspective that is compatible with Diné epistemology, where being the angle has historically regulated ecological practices related to personal orientation and navigation. In contrast, the dualist Cartesian ontology that is inherent in Euclidean geometry portrays an allocentric orientation and perspective.


In STARR, students work together to create various angles that mimic the angles in a projected star constellation. Student A (see Figure 7, on the left) coordinates their egocentric and allocentric perspectives interpersonally while creating the angle with their arms; Student B coordinates these perspectives intrapersonally while watching the movement of their peer. As such, students engage in mixed-media task-based collaborative experiences, through which they are to reconcile two complementary embodied perspectives on angle: (a) egocentric—“being” (Gerofsky, 2011; Student A as the angle vertex); and (b) allocentric—“seeing” the angle as it is portrayed in the constellation and as it is being enacted by another person (Benally et al., 2022; Student B).

This project’s astronomy design comprises both concrete and digital elements (see Figure 8) that recruit a variety of full-body multimodal interactions (i.e., visual, kinesthetic, proprioceptive, etc.). The angle interior is to emerge as students’ attentional anchor (their perceptual solution for a motor-coordination problem; Abrahamson & Sánchez-Garcìa, 2016; see red curve in top of Figure 1). Through collaborative action, students objectify the interior space as a novel discursive ontology and can begin to perform operations on the interior of the angle (e.g., adding, equi-partitioning, measuring etc.). Project evaluation is investigating the co-constructed language used in the learning environment and whether the activity enables students to ground Euclidean geometric understanding in Diné epistemology.

Author

• Jessica K. Benally, University of California - Berkeley