What do Balinese dancing and geometrical proof have in common? Turns out that in both of these disparate practices, we imagine lines to solve the common problem of perceptually engaging the environment to perform sensorimotor tasks. In dance, these lines have been called “attentional anchors.” In geometry, these lines are called “auxiliary constructions.” But whereas dancing may come naturally to people and bear cultural identity and personal meaning, geometry is often viewed as unnatural and having nothing to do with the body. What if we created blended dance~geometry spaces where children could draw on their natural attentional anchors to generate auxiliary lines?

Figure 1. Students are working on a dancing-geometry problem on the GRiD

Geometry Resources in Dance (GRiD) is a gridded floor mat designed to bridge the discourse of mathematics and dance. Students attend to, and foreground, particular lines on the mat in order to coordinate their dance movements. These tacit attentional anchors then become explicit auxiliary lines for geometric practice. We have been collecting data from fifth-grader students (9th – 13th years old) in both Bali (Indonesia) and California (United States) with various backgrounds in Balinese dancing.

## Rationale

Creative geometry problem-solving and reasoning often require constructing auxiliary lines or other elements. However, as studies have found, teaching how to construct auxiliary lines is not easy because students get confused about which lines or elements will be helpful for them. So we asked: What would be a situation where the lines “come” intuitively as one’s strategy to solve a problem? As Ratih, the lead researcher in our project, sought the answer, she reflected on her experiences as both a mathematics educator and a Balinese dancer. Auxiliary lines are intuitive for her: they help her navigate daily tasks and solve mathematical problems. It started with her Balinese dance practice, where she spontaneously imagined various lines around her to organize complex dance movements (see the following video for an example of a Balinese dance lesson).

We would like to foster similar intuitive strategies in students, so how do we bridge the professional discourse across the practices, so that, just like Ratih, all children can learn mathematics through dancing and perhaps dance through mathematics? We proposed GRiD, a gridded floor mat developed within the embodied design research framework (Abrahamson, 2014). GRiD is designed to create opportunities for students to objectify the imaginary perceptual constructions, which they spontaneously generate to coordinate their dance movements, in the form of actual auxiliary elements used in geometrical reasoning.

Figure 2. GRiD – Geometry Resources in Dance

## Activities

We designed dancing geometry problems where students perform a Balinese dance maneuver on GRiD. We were interested in understanding whether and how dancing on GRiD could lead to geometrical reasoning. Here are some examples of the problems.

## Sample Problem

Tapak Sirang Pada, a basic Balinese dance posture: heels meet, and toes open diagonally at ninety degrees. Viewed from above, our feet will form this shape.

Figure 3. Illustration of feet configuration in Tapak Sirang Pada

We ask students to enact the movement, evaluate if they are performing it correctly, and then explain how they can tell it is correct. Next, using GRiD lines, students re-perceive the dance form geometrically to justify their reasoning. Finally, they move to different locations on GRiD, per the choreography sequence, yet all the while locating the appropriate line formations.

## Sample responses

Figure 4. Examples of student’s reasoning about Tapak Sirang Pada on GRiD

Figure 4 presents an example of how students gauge their Tapak Sirang Pada form. Working with GRiD, they shift to a geometrical discourse—they construct auxiliary lines (blue and yellow) to argue that their feet’s aperture is a ninety-degree angle (green lines). Then, as they relocate their Tapak Sirang Pada form to other areas on GRiD, per the dance sequence, they construct a new attentional anchor (see purple triangles).

### Project Team

Ratih Ayu Apsari, Arianna De Los Angeles Castro, Emma Nerea Kelly, Tiffany Hoang, Dor Abrahamson.

Figure 5. Ratih, Arianna, Emma, and Tiffany (from left to right) during the CalTeach poster presentation session in Fall 2023

### Acknowledgment

Ratih Ayu Apsari expresses her deep gratitude to the Indonesia Endowment Fund for Education (LPDP) as the scholarship donor of her Ph.D.

## Publications and Presentations

Castro, A.D.L.A., Kelly, E.N., Hoang, T., & Apsari, R.A. (2024). Elicitation of multimodal mathematical reasoning in Balinese dance through verbal and non-verbal justification. Poster presented for CalTeach Poster Session, University of California, Berkeley.

Apsari, R.A. (2024, April). Grounding geometry as movement discourse: The case of auxiliary constructions in Balinese dance. In D. Abrahamson (Chair) & S. Gerofsky (Discussant), In-sight out: Challenges and opportunities in learning mathematics through negotiating egocentric and allocentric perspectives. Symposium presented for the SIG Research in Mathematics Education at the annual meeting of the American Educational Research Association, Philadelphia, April 11–14.

Apsari, R.A. (2023). Grounding auxiliary geometrical constructions as semiotic articulation of tacit proprioception: The case of basic posture in Balinese dance. Poster prepared for Research Day, Graduate School of Education, University of California, Berkeley.