Walking the Number Line: Toward an Enactive Understanding of Integer Arithmetic

• In Event: In-Sight Out: Challenges and Opportunities in Learning Mathematics Through Negotiating Egocentric and Allocentric Perspectives

Abstract

Students often struggle to add and subtract positive and negative integers (Bossé et al., 2016; Hawthorne et al., 2022). This paper reports on preliminary results from the implementation of an innovative educational design that utilizes the number-line (NL) as a cognitive resource for early integer arithmetic, centering on movement-based calculation activities. Nurnberger-Haag (2018) found that students developed stronger integer fluency when utilizing a walking number line over other methods. However, the researcher did not enable students to link these body-scale embodied experiences with typical desk-scale classroom tasks. This design seeks to enable students the opportunity of coordinating their egocentric experience on a walking NL with an allocentric experience using a desk-scale NL.

First, students enact simple addition and subtraction problems on the body-scale NL (egocentric orientation; Figure 1). Next, they “walk” a figurine across a small, traditional NL (allocentric perspective; Figure 2).

Embodied designs (such as the Walking NL) typically allow for students to enact problems from an egocentric perspective, whereas ultimately they are required to adopt an allocentric perspective in utilizing standard materials, such as paper and pencil (Abrahamson, 2014). Our design solution for this perspectival reconciliation was to offer students resources for combining egocentric and allocentric elements: we invited students to reenact the walking NL experience upon the paper NL by “marking” the action of a proxy agent who experiences the paper NL from an egocentric perspective (cf. Kirsh, 2010).

Fifteen Grade 7 students participated in this pilot study. The activity proved pedagogically advantageous in that it elicited students’ implicit confusions surrounding the content of integer arithmetic. For example, multiple participants in this study, who had just excelled in solving problems on the body-scale NL, then faced challenges when they switched to the desk-top NL and reverted to previous erroneous strategies and “rules” they had learned in the classroom. However, when reminded that they should move the figurine as if it were their own body on the Walking NL, students were able to reconnect, reenact, and sustain this imaginatively mediated egocentric perspective, even as they were visually apprehending the figurine allocentrically; they thus arrived at correct solutions. Students’ initial faltering, along with their eventual “tuning in” to the figurine’s walking experience, suggest that they were proactively negotiating between two different spatial perspectives, thus achieving perspectival mutuality (i.e., using an alternate perspective to inform their own; Benally et al., 2022). Eventually, students may achieve perspectival synergy (a combination of two perspectives that is greater than each perspective alone; Benally et al., 2022). Perspectival synergy in this case would allow students to develop a linear, spatial–numerical mental number line (Mock et al., 2019), which would serve them in all tasks requiring integer arithmetic and fluency.

In summary, the constant availability of both the body-scale and desk-scale NL models created opportunities for students to coordinate across perceptual perspectives. This design allowed all students the opportunity to ground the abstract notion of a negative-integer arithmetic in concrete action (Varma & Schwartz, 2011).

Author

• Jacqueline Anton, University of California - Berkeley