A Unified Framework for Bounded and Unbounded Numerical Estimation

Representations of numerical value have been assessed by using bounded (e.g., 0–1,000) and unbounded (e.g., 0-?) number-line tasks, with considerable debate regarding whether 1 or both tasks elicit unique cognitive strategies (e.g., addition or subtraction) and require unique cognitive models. To test this, we examined how well a mixed log-linear model accounted for 86 5- to 9-year-olds’ estimates on bounded and unbounded number-line tasks and how well it predicted mathematical performance. Compared with mixtures of 4 alternative models, the mixed log-linear model better predicted 76% of individual children’s estimates on bounded number lines and 100% of children’s estimates on unbounded number lines. Furthermore, the distribution of estimates was fit better by a Bayesian log-linear model than by a Bayesian distributional model that depicted estimates as being anchored to varying number of reference points. Finally, estimates were generally more logarithmic on unbounded than bounded number lines, but logarithmicity scores on both tasks predicted addition and subtraction skills, whereas model parameters of alternative models failed to do so. Results suggest that the logarithmic-to-linear shift theory provides a simple, unified framework for numerical estimation with high descriptive adequacy and yields uniquely accurate predictions for children’s early math proficiency.