Even Early Representations of Numerical Magnitude are Spatially Organized: Evidence for a Directional Magnitude Bias in Pre-Reading Preschoolers

Even Early Representations of Numerical Magnitude are Spatially Organized: Evidence for a Directional Magnitude Bias in Pre-Reading Preschoolers John E. Opfer (opfer.7@osu.edu) Department of Psychology, 1835 Neil Avenue Columbus, OH 43210 USA Clarissa A. Thompson (thompson.1345@osu.edu) Department of Psychology, 1835 Neil Avenue Columbus, OH 43210 USA Abstract Adults’ Mental Number Line Previous work has suggested that an important tool of adult numeric competence is a “mental number line” that codes increasing numeric magnitudes spatially and with a directional bias. Partly because the direction of this bias (left- to-right versus right-to-left) is culture-specific, it has been assumed that a directionally-biased mental number line is a late development. Our results, in contrast, suggest that nearly half of all preschoolers possess a directional magnitude bias and that those who possess this bias demonstrate competence in representing large numbers, a competence that is otherwise lacking in same-aged peers. Further, this directional bias does not appear to be related to pre-reading abilities, but instead seems to build on already existing preferences in counting direction. Three types of evidence suggest that adults represent knowledge about numeric values using mental number lines. The first type of evidence comes from adults’ performance when comparing numerical magnitude (e.g., Which is more, 5 or 6? ), where minimizing the distance in value between two symbols (5 or 6 vs. 7 or 4) or increasing the value of the two symbols (8 or 9 vs. 1 or 2) reduces solution times logarithmically (Gobel, Walsh, & Rushworth, 2001; Moyer & Landauer, 1967). An important parallel exists in psychophysics, where reducing the distance in length between two lines or increasing their individual lengths also reduces comparison times logarithmically (Johnson, 1939). These parallel findings of logarithmic decrement in comparison times suggest that when adults compare numerals, they automatically simulate the numeric quantities as linear magnitudes, which increase logarithmically with numeric value (Banks & Hill, 1974; Dehaene, 1997). In support of this interpretation, focal transcranial magnetic stimulation of brain regions involved in visuospatial search also disrupt performance in comparing numerosity while sparing performance on other tasks (Gobel, Walsh, & Rushworth, 2001). A second type of evidence for adults’ use of mental number lines is provided by their performance on a parity task, where subjects are asked whether a given number is odd or even (Dehaene et al., 1993; Fischer, 2001, 2003). Although knowledge of numeric magnitude is unnecessary to solve the task, if knowledge of parity is stored in semantic memory with the mental number line, Dehaene and colleagues (1993) reasoned, representation of numerals on a mental number line might intrude on task performance. Specifically, they found that when subjects were asked whether large numbers (e.g., 8 or 9) were even, they responded most quickly with their right hand, whereas when they were asked about small numbers (e.g., 1 or 2), they responded most quickly with their left hand. Interestingly, the direction of this effect—the “Spatial-Numerical Association of Response Codes (SNARC) effect”—was not affected by handedness or hemispheric dominance, but by the direction of writing in the culture of the subjects. When Keywords: conceptual development, representation, numeric cognition, spatial cognition Development of the Mental Number Line Numeric competence is a fundamental prerequisite of many social, economic, and educational activities, and the importance of its acquisition is reflected in the considerable resources devoted to teaching children mathematics, to understanding the mental disorders that inhibit mathematical understanding, and to identifying the cognitive bases of mathematical understanding. A number of studies have indicated that an important tool of numeric competence is a “mental number line” (Dehaene, et al., 2004). Mental number lines are representations of numeric quantity in which numerals (e.g., integers) are associated with physical magnitudes and are ordered serially along the same dimension as the increasing physical magnitude. By mentally mapping a directionally-ordered list of integers onto a series of increasing linear magnitudes, people appear to gain proficiency in solving a number of quantitative tasks, including: (1) determining which of two integers denote greater magnitude (Gobel, Walsh, & Rushworth, 2001), (2) organizing semantic knowledge about the parity of numerals (Dehaene et al., 1993; Fischer, 2001), and (3) translating between numeric symbols and spatial magnitudes (Siegler & Opfer, 2003).