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 to objectify tacit attentional anchors for movement coordination into auxiliary lines for geometric practice. 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, Dor Abrahamson

Research Assistants

Surya Ayu Audina, Sekar Nareswari Setyanto

Undergraduate Mentored Researcher (EDSTEM C122)

Fall 2023: Arianna De Los Angeles Castro, Emma Nerea Kelly, Tiffany Hoang

Spring 2024: Sierra Nicole Hahn, Olivia Kathryn Kulchin, Jacob Antonio Perez, Xumo Huang, Alma Denisse Tolentino, Justin Yan

Figure T.1. Ratih, Arianna, Emma, and Tiffany (from left to right) during the CalTeach poster presentation session in Fall 2023

### Project Collaborations

Digital GRiD: Real-time movement tracer for embodied geometry learning

Project Team: Ratih Ayu Apsari, Yangyang Yang

Digital GRiD was funded by the Jacobs Institute for Design Innovation at the University of California, Berkeley, in Spring 2024.

We designed a real-time movement tracer for students to make sense of geometrical concepts, such as the concept of a circle, through full-body engagement.

Figure C.1. Yangyang illustrates drawing with a colored object through real-time color detection using webcam input

Figure C.2. Ratih with one of the visitors of Jacobs Spring Design Showcase trying the DIgital GRiD in Spring 2024

Figure C.3. Ratih (left) and Yangyang (right) during Jacobs Spring Design Showcase

### GRiD Workshop at EDaMMLA

Organized by the University of California Berkeley and the University of Oslo, the International Workshop on Embodied Learning and Multimodal Learning Analytics (EDaMMLA) aims to explore the analysis of multimodal learning in embodied learning. During the GRiD session, participants engaged in a participatory presentation on the connection between traditional Balinese dancing and mathematics teaching and learning.

Figure C.4. Ratih made the invisible reference lines she used for performing Balinese dance posture and movement perceptible by arranging colorful yarns. Other participants from UC Berkeley helped to hold the yarns.

During the event, attendees had the opportunity to take part in a traditional Balinese dance lesson led by renowned Balinese dancer and teacher, I Made “Cat” Suteja. This activity allowed us to delve into the possibilities of multimodal learning analysis in the realms of dance, mathematics, and rhythm. Some participants utilized full-body motion-capture gear to collect multimodal bio-kinematic sensor data on their movement dynamics, while others opted for physiological sensor vests.

Figure C.5. From left: Ratih Ayu Apsari (PhD Student in Learning Sciences and Human Development), Ami Shulman (US-based Movement Director, Feldenkrais Method practitioner), I Made “Cat” Suteja (internationally renowned Balinese dancer and instructor), Dr. Bilge Serdar Göksülük (dancer and researcher, RITMO, Norway)

In this session, we also tested the Digital GRiD prototype to explore various emerging strategies for using the body to understand geometric objects.

Figure C.6. Alik Palatnik, EDaMMLA participant from The Hebrew University of Jerusalem, trying the real-time movement tracker of Digital GRiD

### 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. (2023). 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. (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.

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. (2024). Grounding Auxiliary Geometrical Constructions as Semiotic Articulation of Tacit Attentional Anchors: The Case of Balinese Dance. Poster presented for Research Day, Berkeley School of Education, University of California, Berkeley.

Hahn, S., Tolentino, A., Kulchin, O., Huang, X., Perez, J., Yan, J., Apsari, R.A. (2024). Geometry in Balinese Dance. Poster presented for EDSTEM C122 Spring Semester, University of California, Berkeley.

### Grants

Jacobs Institute Innovation Catalysts, Spring 2024