Non-symbolic estimation of big and small ratios with accurate and noisy feedback

Abstract

The ratio of two magnitudes can take one of two values depending on the order they are operated on: a ‘big’ ratio of the larger to smaller magnitude, or a ‘small’ ratio of the smaller to larger. Although big and small ratio scales have different metric properties and carry divergent predictions for perceptual comparison tasks, no psychophysical studies have directly compared them. Two experiments are reported in which subjects implicitly learned to compare pairs of brightnesses and line lengths by non-symbolic feedback based on the scaled big ratio, small ratio or difference of the magnitudes presented. Results of Experiment 1 showed all three operations were learned quickly and estimated with a high degree of accuracy that did not significantly differ across groups or between intensive and extensive modalities, though regressions on individual data suggested an overall predisposition towards differences. Experiment 2 tested whether subjects learned to estimate the operation trained or to associate stimulus pairs with correct responses. For each operation, Gaussian noise was added to the feedback that was constant for repetitions of each pair. For all subjects, coefficients for the added noise component were negative when entered in a regression model alongside the trained differences or ratios, and were statistically significant in 80% of individual cases. Thus, subjects learned to estimate the comparative operations and effectively ignored or suppressed the added noise. These results suggest the perceptual system is highly flexible in its capacity for non-symbolic computation, which may reflect a deeper connection between perceptual structure and mathematics.