Placing Numbers in Behavioral Space: Activity-Specific Interactions between Number and Space with a Single Response Button

Placing Numbers in Behavioral Space: Activity-Specific Interactions between Number and Space with a Single Response Button Tyler Marghetis* (tmarghet@ucsd.edu) Jasmeen Kanwal* (jkanwal@ucsd.edu) Benjamin K. Bergen (bkbergen@ucsd.edu) University of California, San Diego 9500 Gilman Drive, La Jolla, CA 92093 USA Abstract to reason about numbers as abstract concepts (Lakoff & Nunez, 2000; Nunez & Marghetis, to appear). Mathematical diagrams often associate numbers with particular locations. And number interacts with space in less explicit ways during the online performance of mathematical activities. In a seminal study, Dehaene and colleagues (1993) asked participants to judge the magnitude (greater or less than 5?) or parity (even or odd?) of single digit numbers. Participants were reliably faster to respond to smaller numbers when responding with a button in left space, and to larger numbers when responding in right space—as if they were spontaneously thinking of numbers along a left-to-right “mental number line.” This interaction between numerical magnitude and spatial location has been dubbed the “SNARC” effect. In the two decades since, the literature on such number- space associations has exploded (Hubbard et al., 2005; Wood et al., 2008). Similar effects have been found with bipedal responses (Schwarz & Muller, 2006) and saccades (Fischer et al., 2003; Schwarz & Keus, 2004), further reinforcing the genuinely spatial nature of this effect. While the particular direction of this “mental number-line” is quite flexible, shaped by such factors as habitual reading direction (Shaki, Fischer, & Petrusic, 2009) and recent experience (Fischer, Mills, & Shaki, 2010), there is a growing consensus that number and space are intimately related. But doubts remain. A number of authors have suggested that the effects have less to do with a stable spatial representation of number, and more to do with flexible or non-spatial associations (e.g., Fischer et al., 2010; Gevers et al, 2010; Santens and Gevers, 2006). Others have pointed out that most studies force participants to respond spatially, using buttons that are distinguished by their spatial location, and thus implicitly inject space in virtue of the experimental setup (Nunez, Doan & Nikoulina, 2011). Indeed, when other response modalities are used—e.g., responding with higher or lower pitches—participants exhibit interactions that are SNARC-like but non-spatial (Marghetis et al., 2011). Number-space associations, therefore, may be more flexible and context-sensitive than first assumed. Beyond these concerns, one additional question has been largely unaddressed: the nature of the “space” that sometimes, undeniably, interacts with number. Previous work has conflated the multiple egocentric frames of reference that we use to encode space, which include head-, How are we able to reason about abstract concepts that lie resolutely beyond the reach of perception? One strategy is to ground understanding in space. Numbers, for instance, are known to interact with egocentric space during rapid numerical judgments. A range of experimental results have demonstrated that, among literate Western people, this “mental number-line” goes from left to right, with smaller numbers associated with left space, and larger numbers with right space. But what is the nature of this “space”? Previous work has conflated multiple possible egocentric frames of reference—head-based, eye-based, action-based—leaving it unclear which space is interacting with number. In the present paper, two studies investigated whether a single centrally- located button, stationary in hand- and eye-based coordinates, can nevertheless exhibit different spatial properties in virtue of task-specific activity. In a go/no-go paradigm, participants judged the magnitude (Exp. 1) and parity (Exp. 2) of single- digit numbers. Crucially, they responded only with the index or middle finger of a single hand. While judging magnitude (Exp. 1), participants were faster to respond to smaller numbers with the more leftward finger, and larger numbers with the more rightward finger, regardless of the hand being used. This effect disappeared when judging parity (Exp. 2), replaced by finger-specific associations on the left hand only. In sum, in a task-sensitive way, participants associated numbers with egocentric space—but a behavioral space defined relative to embodied interaction rather than head- or eye-based reference frames. We discuss implications for number representation and the nature of “space” in embodied activity. Keywords: number; space; action; SNARC; embodiment; go/no-go; frames of reference; Merleau-Ponty Introduction 1 “[The body’s] spatiality is not, like that of external objects or like that of ‘spatial sensations’, a spatiality of position, but a spatiality of situation.”- Merleau-Ponty (1962, p. 114). How are we able to reason about abstract concepts that lie resolutely beyond the reach of perception? One strategy is to ground understanding in space. Numbers, for instance, are tightly linked with space across human activity. We recycle the language of space to talk about numbers, counting up to higher numbers and down to lower numbers, and use space JK and TM contributed equally to this work.