Dancing Figures

Dancing Figures is a group of STEAM learning activities that links trigonometric functions with music and dance. Students select or compose periodic functions, tune parameters (amplitude and frequency) in GeoGebra, and drive a MaLT2 “dancing” figure so its motion synchronizes with a chosen song.The prototype evolved across three versions: from simple sine/cosine control of movement, to trigonometric polynomials that better capture melody and energy, and finally to an embodied duet where one figure follows the function while a second figure is controlled live by the student’s hand via MaLT-Dance.The activity targets upper-secondary learners (11th–12th grade) and foregrounds concepts such as periodicity, rate of change, and coordination between analytic and embodied representations of change.Students work iteratively: they listen to a given song, choose a function in GeoGebra, port coefficients values into MaLT2 to animate a figure, and adjust until the movement aligns with musical criteria (rhythm, melody, intensity, etc.); in the embodied version (v3), one figure is function-driven, while the second one responds to horizontal hand motion.The technologies used are GeoGebra, MaLT2, and (optionally) MaLT2Dance, with example materials provided alongside the prototype. By pressing this link https://youtu.be/Jv3wlTECnRE, you accept that your browser will send information to YouTube to view the video. YouTube privacy policy Additional Information for Researchers Dancing Figures explores how students coordinate periodic functions with music/dance to create aesthetically-guided meaning for the mathematical concepts of periodic functions and covariation. Versions progress from simple trigonometric functions (sin/cos) to trigonometric polynomials (to capture melody/energy, not just tempo) and finally to embodied interaction with MaLT2Dance, where one figure follows a function and a second figure is controlled by the student’s hand.Theoretical lenses used for its design: Constructionist design with open, creative artefact-making (Papert & Harel, 1991) Embodied interaction/learning via MaLT2Dance (function ↔ movement coupling) (Co)variational reasoning around periodic change (Thompson & Carlson, 2017) Examples of potential research questions: How does embodied control change strategies for coordinating amplitude/frequency/phase? Do trigonometric polynomials promote talk about rate of change and melody shape more than single sinusoids? What kind of covariational reasoning do students develop while planning and performing a ‘hybrid’ mathematical-body choreography? Suggested data sources: Screen capture, keystroke/slider logs (GeoGebra/MaLT2), audio track features noted by learners, hand-tracking traces (v3), think-alouds and discussions through voice recordings, post-hoc interviews and questionnaires Papert, S., & Harel, I. (1991). Situating constructionism. constructionism, 36(2), 1-11.Thompson, P. W., & Carlson, M. P. (2017). Variation, covariation, and functions: Foundational ways of thinking mathematically. Compendium for research in mathematics education, 421. Publications Karavakou, M., & Kynigos, C. (under review). Choreo-graphing Trigonometric Functions: Aesthetically Driven Covariational Reasoning. Submitted to Educational Studies in Mathematics. https://dx.doi.org/10.21203/rs.3.rs-7324448/v1 Karavakou, M., & Kynigos, C. (2025, February). Dancing with trigonometric functions: aesthetically guided covariational reasoning while creating periodic dynamic models. In Fourteenth Congress of the European Society for Research in Mathematics Education (CERME14). Karavakou, M., & Kynigos, C. (2024, October). Aesthetic experiences with mathematics and music through digital technology. In 10th WAAE Global Arts Education Summit: Arts, Nature, Technology, Education: Harmony in Unity. Karavakou, M., & Kynigos, C. (2024). Exploring students’periodic covariational reasoning while creating dancing figural animations. In 17th ERME Topic Conference Mathematics Education in the Digital Age (MEDA4). Karavakou, M., & Kynigos, C. (2023, June). Sine, you think, it can dance? An Aesthetically Driven Mathematical Activity for Meaning Making on Trigonometric Functions. In Sixteenth International Conference on Technology in Mathematics Teaching (ICTMT16). p. 151-158. Additional Information for Educators Mathematics Educators can use the guidance below to run Dancing Figures as a ready-to-use classroom activity, covering step-by-step setup for GeoGebra and MaLT2, lesson flow, grouping, materials access, and quick assessment tips. Resources needed for implementing the activity: Worksheet framed as ‘Dance Challenge 1’ and ‘Dance Challenge 2’:https://transeet.eu/wp-content/uploads/2025/12/dancing-figures-worksheet.pdf GeoGebra (web): pick/adjust the function graph: For v.1: https://www.geogebra.org/calculator/gfpmhpx8 For v.2: https://www.geogebra.org/calculator/pxgdqfzf MaLT2 (desktop/web): animate a “dancing figure” with your chosen function by following the instruction from the worksheet: For v.1: http://etl.ppp.uoa.gr/malt2/?challenge1_move1or http://etl.ppp.uoa.gr/malt2/?challenge1_move2 For v.2: http://etl.ppp.uoa.gr/malt2/?challenge2_move1or http://etl.ppp.uoa.gr/malt2/?challenge2_move2 Songs with diverse temporal and melodic structures Suggested class flow (90–120 minutes; Grades 11–12): Warm-up (10–15′) Listen to a short excerpt (30–60s). Ask: “What in the music changes steadily? When does it peak? Where are repetitions?” Version 1 — Trig functions (sin/cos) (35’–45′) Suggested activity for students: In GeoGebra, choose a sinusoid and vary α (amplitude) and β (frequency) until the graph “matches” the song’s beat/phrasing. Note down the selected function formula in the worksheet table Transfer those parameters values to MaLT2 and animate the figure by constantly pressing the right arrow key, while listening to the song. Repeat this procedure for another five rounds, in order for the movement to better fit your musical criteria. Goal: match tempo/basic periodicity. C. Version 2 — Trigonometric polynomials (45’–60′) Suggested activity for students: In GeoGebra, choose a trigonometric polynomial by adjusting αi and βi coefficients until the graph “matches” the song’s structure. Note down the selected function formula in the worksheet table Transfer those parameters values to MaLT2 and animate the figure by constantly pressing the right arrow key, while listening to the song. Repeat this procedure for another five rounds, in order for the movement to better fit your musical criteria. Goal: match tempo/basic periodicity as well as melodic pattern, intensity, and more. Orchestration: Students can work in pairs: One would take the role of the “Graph tuner” (GeoGebra focus) and the other that of the “Animator” (MaLT2/MaLT-Dance focus); then after each round they can switch roles. Assessment focus: Students explain how A/ω/φ changed the animation. Students justify graphical properties (e.g., regarding its height, period, slope, monotony, curvature, extrema, turning points, etc.) in relation to melody features Reflection: “Where did your animation feel most in sync? Why? Learning Goals To explore trigonometric functions in order to match spatial movement to music and create dancing animations. Disciplines Mathematics Physics Topics Trigonometric functions Learner Levels: Upper Secondary Target Audience: Educators and Researchers Communities of Interest CoI 3 – Augmented Reality in STEAM Education Emerging Technologies: Hand-Motion captured technology