The powers of noise-fitting: reply to Barth and Paladino.

Barth and Paladino (2011) argue that changes in numerical representations are better modeled by a power function whose exponent gradually rises to 1 than as a shift from a logarithmic to a linear representation of numerical magnitude. However, the fit of the power function to number line estimation data may simply stem from fitting noise generated by averaging over changing proportions of logarithmic and linear estimation patterns. To evaluate this possibility, we used conventional model fitting techniques with individual as well as group average data; simulations that varied the proportion of data generated by different functions; comparisons of alternative models' prediction of new data; and microgenetic analyses of rates of change in experiments on children's learning. Both new data and individual participants' data were predicted less accurately by power functions than by logarithmic and linear functions. In microgenetic studies, changes in the best fitting power function's exponent occurred abruptly, a finding inconsistent with Barth and Paladino's interpretation that development of numerical representations reflects a gradual shift in the shape of the power function. Overall, the data support the view that change in this area entails transitions from logarithmic to linear representations of numerical magnitude.