A primarily serial, foveal accumulator underlies approximate numerical estimation

Abstract

Significance The question of how people estimate numerical quantities is centrally important in cognitive psychology, neuroscience, and applied educational research. It is generally believed that estimation of numbers is rapid and occurs in parallel across a visual scene. Here, we show that people’s estimates are determined by a sequence of visual fixations, with both their mean estimates and their precision increasing as a function of how many points they foveate. This mechanism suggests that a considerable body of research which treats estimation as a purely numerical measure is likely to be missing an important part of the picture: Numerical estimation ability is closely tied to the mechanisms that control eye movements and attention. The approximate number system (ANS) has attracted broad interest due to its potential importance in early mathematical development and the fact that it is conserved across species. Models of the ANS and behavioral measures of ANS acuity both assume that quantity estimation is computed rapidly and in parallel across an entire view of the visual scene. We present evidence instead that ANS estimates are largely the product of a serial accumulation mechanism operating across visual fixations. We used an eye-tracker to collect data on participants’ visual fixations while they performed quantity-estimation and -discrimination tasks. We were able to predict participants’ numerical estimates using their visual fixation data: As the number of dots fixated increased, mean estimates also increased, and estimation error decreased. A detailed model-based analysis shows that fixated dots contribute twice as much as peripheral dots to estimated quantities; people do not “double count” multiply fixated dots; and they do not adjust for the proportion of area in the scene that they have fixated. The accumulation mechanism we propose explains reported effects of display time on estimation and earlier findings of a bias to underestimate quantities.