Embodied Continuity Explorer: An Action-Based Embodied Design For Enacting Continuity Dynamically

Abstract Continuity is considered a key concept when dealing with functions in school and beyond, across STEM studies. At the same time, students are highly challenged to build up viable conceptual understandings that go beyond intuitive — and misleading — simplifications, such as “drawing a line without gaps.” Extending their comprehension of continuity and encountering the Weierstrass definition of continuity they encounter a variety of obstacles related to the complexity of the definition and mismatch with their prior intuitive understanding. This paper acknowledges the disembodied and non-intuitive nature of the $$\epsilon -\delta$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ϵ</mml:mi> <mml:mo>-</mml:mo> <mml:mi>δ</mml:mi> </mml:mrow> </mml:math> definition mainly presented as static and suggests an embodied approach to experiencing its structural relationships. Drawing from radical embodiment and an action-based embodied design approach, we present design and development of a learning opportunity for enacting the $$\epsilon -\delta$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ϵ</mml:mi> <mml:mo>-</mml:mo> <mml:mi>δ</mml:mi> </mml:mrow> </mml:math> definition of continuity, called the Embodied Continuity Explorer (ECE). Created as a motor control problem incorporating constraints, it enables students to explore the $$\epsilon -\delta$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ϵ</mml:mi> <mml:mo>-</mml:mo> <mml:mi>δ</mml:mi> </mml:mrow> </mml:math> definition through perception–action loops and aiming for the phenomenalization of relations between the various elements of the $$\epsilon -\delta$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ϵ</mml:mi> <mml:mo>-</mml:mo> <mml:mi>δ</mml:mi> </mml:mrow> </mml:math> definition. We show how theoretical conjectures inform the design to lead towards the desired learning outcomes and present a case study involving a high school student to evaluate the design against the student’s process of phenomenolizing continuity. By analyzing parts of a learning trajectory of a 16-year-old student with no prior knowledge of continuity, it has been found that correct relations of elements of the $$\epsilon -\delta$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ϵ</mml:mi> <mml:mo>-</mml:mo> <mml:mi>δ</mml:mi> </mml:mrow> </mml:math> definition can be established, resulting in a meaningful conceptualization at a pre-formal level.