Abstraction and embodiment: exploring the process of grasping a general

Abstract

This paper reports from a case study which explores kindergarten children’s mathematical abstraction in a teaching–learning activity about reflection symmetry. From a dialectical perspective, abstraction is here conceived as a process, as a genuine part of human activity, where the learner establishes “a point of view from which the concrete can be seen as meaningfully related” (van Oers & Poland Mathematics Education Research Journal, 19(2), 10–22, 2007, p. 13–14). A cultural-historical semiotic perspective to embodiment is used to explore the characteristics of kindergarten children’s mathematical abstraction. In the selected segment, two 5-year-old boys explore the concept of reflection symmetry using a doll pram. In the activity, the two boys first point to concrete features of the sensory manifold, then one of the boys’ awareness gradually moves to the imagined and finally to grasping a general and establishing a new point of view. The findings illustrate the essential role of gestures, bodily actions, and rhythm, in conjunction with spoken words, in the two boys’ gradual process of grasping a general. The study advances our knowledge about the nature of mathematical abstraction and challenges the traditional view on abstraction as a sort of decontextualised higher order thinking. This study argues that abstraction is not a matter of going from the concrete to the abstract, rather it is an emergent and context-bound process, as a genuine part of children’s concrete embodied activities.

Tangible connections within the mathematical horizon: Exploring the Dihedral Calculator

“…Symmetries admit other connections to object-process duality as well. Breive ( 2022 ) discusses a tension between how individuals perceive symmetry and how we describe it. Individuals can visually appreciate symmetry as a whole, but “the linearity of speech does not allow us to refer simultaneously to two different points, rather a symmetrical interaction between two entities is transformed into one entity that interacts with the other entity” (Breive, 2022 , p.318).…”

Theories and practical perspectives on fostering embodied abstraction in primary school geometry education

“…Most contemporary theoretical perspectives on abstraction relate to the mathematical topics of numbers and algebra, leaving the field of geometry underexposed (Breive, 2022). Where the Van Hiele level theory of geometric understanding (Van Hiele, 1986) does relate to abstraction, its applications primarily relate to the abstraction of formal shapes, forms and solids.…”