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David Barner; Constructing exact number approximately: a case study of mental abacus representations. Journal of Vision 2009;9(8):15. doi: https://doi.org/10.1167/9.8.15.
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Exact numerical representation is usually accomplished through linguistic representations. However, an alternative route for accomplishing this task is through the use of a “mental abacus”—a mental image of an abacus (a device used in some cultures for keeping track of exact quantities and doing arithmetic via the positions of beads on a rigid frame). We investigated the nature of mental abacus representations by studying children ages 7–15 who were trained in this technique. We compared their ability to read the cardinality of “abacus flashcards” (briefly presented images of abacuses in different configurations) with their ability to enumerate sets of dots after similarly brief, masked presentation. We conducted five studies comparing abacus flashcards to: (1) random dot enumeration, (2) spatially proximate dot enumeration, (3) enumeration of dots arranged in an abacus configuration without the abacus frame, (4) enumeration of dots on a rotated abacus, (5) enumeration of dots arranged on an abacus. In all conditions, participants were faster and more accurate in identifying the cardinality of an abacus than they were in enumerating the same number of beads, even when the display was physically identical. Analysis of errors suggested that children in our studies viewed the abacus as a set of objects with each separate row of beads being a single object, each with its own independent magnitude feature. Thus, the “mental abacus” draws on pre-existing approximate and exact visual abilities to construct a highly accurate system for representing large exact number.
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