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.

Gamble on gaze: Eye movements reflect the numerical value of blackjack hands

“…Moreover, in the case of mental arithmetic specifically, spatial biases may be driven by semantic associations between space and arithmetic operations (e.g., left-subtraction/right-addition; Hartmann, Mast, & Fischer, 2015), not by shifts of attention along the MNL toward the outcome of such operations (i.e., solution magnitude; cf. Marghetis et al, 2014). Thus, previous research, though suggestive, has not provided unambiguous evidence for spatial representations of number along which movement is mentally simulated.…”

“…Mentally summing the values of successive cards, and thereby computing the total value of one's hand, is a prerequisite for making this decision wisely. Research suggests that such mental arithmetic may rely on an internal spatial representation of numerical magnitude (e.g., Knops, Viarouge, & Dehaene, 2009;Marghetis, Nuñez, & Bergen, 2014;McCrink, Dehaene, & DehaeneLambertz, 2007; for a review, see Fischer & Shaki, 2014). For example, when pointing to an arithmetic solution on a visually presented number line, participants are biased leftward on subtraction problems and rightward on addition problems (Pinhas & Fischer, 2008).…”

“…For example, when pointing to an arithmetic solution on a visually presented number line, participants are biased leftward on subtraction problems and rightward on addition problems (Pinhas & Fischer, 2008). Such spatial biases suggest that mental arithmetic may induce dynamic shifts of attention, or simulated movement, along a left-to-right-oriented mental number line (Knops, Dehaene, Berteletti, & Zorzi, 2014;Marghetis et al, 2014;Masson & Pesenti, 2014;McCrink et al, 2007), building on earlier findings from number classification tasks in which smaller and larger numbers yielded faster left-and right-side responses, respectively (e.g., Dehaene, Bossini, & Giraux, 1993).…”

“…spatial nature of the experimental procedure (Santiago & Lakens, 2015). In many studies, participants respond along a particular spatial axis-for example, by pressing left-and right-side response keys (e.g., Dehaene et al, 1993;Pinhas & Fischer, 2008) or by making leftward and rightward head or arm movements (e.g., Loetscher, Schwarz, Schubiger, & Brugger, 2008;Marghetis et al, 2014;Wiemers, Bekkering, & Lindemann, 2014;Winter & Matlock, 2013). In other studies, the numerical stimuli themselves are presented at different spatial locations (e.g., Knops, Viarouge et al, 2009;Masson & Pesenti, 2014).…”

“…Since both the stimuli and responses were centrally located and eye movements were task-irrelevant, the task had no overt spatial component, thus precluding any interpretation based on polarity correspondence. Moreover, spatial biases in mental arithmetic are typically based on a comparison of addition and subtraction operations (e.g., Marghetis et al, 2014;Pinhas & Fischer, 2008;Wiemers et al, 2014). In our task, operation was held constant (blackjack requires addition, but not subtraction, of values); thus, any relation between gaze patterns and hand value would reflect spatial-numerical mappings, not semantic associations based on operation alone (cf.…”