Doing arithmetic by hand: Hand movements during exact arithmetic reveal systematic, dynamic spatial processing

Abstract

Mathematics requires precise inferences about abstract objects inaccessible to perception. How is this possible? One proposal is that mathematical reasoning, while concerned with entirely abstract objects, nevertheless relies on neural resources specialized for interacting with the world—in other words, mathematics may be grounded in spatial or sensorimotor systems. Mental arithmetic, for instance, could involve shifts in spatial attention along a mental “number-line”, the product of cultural artefacts and practices that systematically spatialize number and arithmetic. Here, we investigate this hypothesized spatial processing during exact, symbolic arithmetic (e.g., 4 + 3 = 7). Participants added and subtracted single-digit numbers and selected the exact solution from responses in the top corners of a computer monitor. While they made their selections using a computer mouse, we recorded the movement of their hand as indexed by the streaming x, y coordinates of the computer mouse cursor. As predicted, hand movements during addition and subtraction were systematically deflected toward the right and the left, respectively, as if calculation involved simultaneously simulating motion along a left-to-right mental number-line. This spatial–arithmetical bias, moreover, was distinct from—but correlated with—individuals' spatial–numerical biases (i.e., spatial–numerical association of response codes, SNARC, effect). These results are the first evidence that exact, symbolic arithmetic prompts systematic spatial processing associated with mental calculation. We discuss the possibility that mathematical calculation relies, in part, on an integrated system of spatial processes.

Keywords:

• Mental arithmetic
• Operational momentum
• Spatial–numerical association of response codes
• Spatial–operation association of responses
• Grounded cognition
• Neural recycling

Thanks to Brock Hazen Dumville, Esther Walker, and Bodo Winter for helpful discussions and to two anonymous reviewers for productive criticism. We are grateful to the undergraduate research assistants who helped with data collection: Luke Eberle, Alec Gasperian, Chau Nguyen, and Kendall Youngstrom.T.M. was supported by a doctoral fellowship from the Fonds de Recherche sur la Société et la Culture (Québec, Canada) and a Glushko Fellowship.

Notes

¹To illustrate: If an individual's mean MD for calculations with a solution of 3 was 0.35 for rightward responses and 0.3 for leftward responses, then their dMD for 3 would be 0.05, the difference between 0.35 and 0.3.

²We did not include a factor for the second term because it only ranged from 0 to 3.

³Giaquinto (2007) distinguishes between syntactic and semantic manipulation of symbols, which may relate to the use of space to simulate movement of the terms rather than to ground the calculation in meaningful spatial intuitions.