Grounding Geometry as Movement Discourse: The Case of Auxiliary Constructions in Balinese Dance

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

Abstract

Engaging in geometrical reasoning and proof often relies on generating auxiliary lines (Palatnik & Dreyfus, 2018). Teaching how to generate auxiliary lines, though, can be challenging, in part because students must envision figural elements that, by definition, are not yet available for perception (Fan et al., 2017; Herbst & Brach, 2006). Notwithstanding, humans share an innate capacity to spontaneously generate attentional anchors—perceptual forms that tighten one’s grip on the environment (Abrahamson & Sánchez-García, 2016). Designed appropriately, we conjecture, geometry students’ natural ability to produce attentional anchors as movement solutions could be tapped as a means of training them to construct auxiliary lines as geometry solutions. In this study we introduce a gridded floor mat and marking accessories, collectively called GRiD (Geometry Resources in Dance), designed to serve students as a frame of reference for both eliciting and objectifying their tacit attentional anchors as explicit auxiliary lines, as they engage in solving a Balinese dance task.

A 10-year-old Balinese dancer in training, Anna (pseudonym), participated in a task-based semi-structured clinical interview, in which she practiced a dance posture called Tapak Sirang Pada (TSP), where the feet, touching at the heels, formed a 900 angle. Qualitative analysis focused on Anna’s multimodal justification of her posture.

When performing the posture, Anna became part of the angle incarnate, and saw the angle from a first-person (egocentric) perspective (Figure 3a; imagine looking down at your feet). The challenge is that, in traditional mathematics textbooks, a 900 angle is usually captured in a third-person (allocentric) perspective (Figure 3b). Therefore, Anna had to coordinate her perspectives.

Anna justified the positioning of her feet by focusing on the angle vertex, where her heels met. Using GRiD, she first gestured the imaginary point and lines that helped her determine she was forming a 900 angle (Figure 4a-c), and later placed three dot stickers on the tarp to mark her heels’ junction and the tips of her toes, respectively (Figure 4d-e).

First, Anna related metaphorically to the red dot as “a place like a center, where everybody comes to meet. It is also where my heels click.” Next, she ensured that her left and right feet constitute angle bisectors of the left and right 900 angles, so that the angle between her feet becomes the sum of twice 450: 900.

Once Anna had adjudicated the GRiD's auxiliary lines as effective attentional anchors for performing TSP, she was able to “transport” this geometrical construction when moving to different locations on the mat (see Figure 5). Anna exclaimed, “This triangle thing always stays with my feet like whenever I move around!” Anna was thus synergizing egocentric and allocentric perspectives (Benally et al., 2022)—her feet were forming a geometrically formal 900 angle (egocentric perspective), even as she could imagine the 900 angle markings (allocentric perspective) as affording TSP.

In summary, engaging in dance practice creates situated, authentic opportunities for students to negotiate allocentric and egocentric perspectives in order to objectify their movement-based attentional anchors in the form of auxiliary lines that construct mathematical reasoning.

Author

• Ratih Ayu Apsari, University of California - Berkeley