Dancing geometry: imagining auxiliary lines by reflecting on physical movement

Constructing auxiliary lines is an important component skill in solving geometry problems, and yet it is difficult to teach, precisely because these lines are ‘invisible’ until they are actually drafted. Is there any intuitive resource that geometry students could possibly draw on to develop this skill? Is there any domain of human activity where we all naturally entertain imaginary lines, even if we are not aware of doing this? And yet, if so, how would these tacit imaginary lines come forth to be geometrical auxiliary lines? It turns out that dancers spontaneously imagine linear structures, known as attentional anchors, to help them enact their movements. These attentional anchors are drawn out in the dancer's subjective perception and are, therefore, invisible to others. Notwithstanding, we have used an embodied design-based research framework to create a gridded floor mat where students can render their covert dance-oriented attentional anchors as overt geometry-oriented auxiliary constructions. Situated in the cultural context of traditional Balinese dance, this practitioner-oriented paper demonstrates several activities for global classroom use by way of sharing some empirical results from implementing this pedagogical approach with young learners. An appendix lists a set of additional activities for dance-based geometry exploration.