Walking the number line: towards an enactive understanding of integer arithmetic

Abstract Early mathematics education presents middle-school students with the challenge of adding and subtracting negative integers. This paper reports on results from the experimental implementation of a proposed educational design for integer arithmetic that utilized the number-line (NL) form as a resource for students to enact simple addition and subtraction problems under two conditions: (1) a body-scale floor-based NL, where arithmetic operations are enacted by walking; and (2) a regular desk-based NL supplemented with an action-figure for re-enacting the floor-based solutions. This design is the first iteration of a design-based research project and was developed based on the experience of the first author’s five years teaching in this topic. 15 Grade 7 students participated in the project’s pilot study that centered on how students coordinate procedurally analogous calculation activities across the large and small NL. The activity elicited students’ implicit confusions surrounding integer subtraction, thus creating opportunities for corrective intervention. Analyses also generated operative inferences shaping the subsequent design iteration. Implications are drawn more broadly for enactive mathematics pedagogy, particularly through the lens of comparing students’ egocentric orientations toward immersive instantiations of cultural–historical mathematical forms to their allocentric perceptual orientations toward the normative forms of the same concepts. As Extended Reality (XR, e.g. virtual reality, augmented reality) experiences enter mathematics classrooms, it may become vital to develop pedagogical methodologies in support of coordinating conceptually complementary perceptual perspectives.