How numbers mean: Comparing random walk models of numerical cognition varying both encoding processes and underlying quantity representations

Abstract

How do people derive meaning from numbers? Here, we instantiate the primary theories of numerical representation in computational models and compare simulated performance to human data. Specifically, we fit simulated data to the distributions for correct and incorrect responses, as well as the pattern of errors made, in a traditional "relative quantity" task. The results reveal that no current theory of numerical representation can adequately account for the data without additional assumptions. However, when we introduce repeated, error-prone sampling of the stimulus (e.g., Cohen, 2009) superior fits are achieved when the underlying representation of integers reflects linear spacing with constant variance. These results provide new insights into (i) the detailed nature of mental numerical representation, and, (ii) general perceptual processes implemented by the human visual system.

Keywords: Numerical architecture; Numerical cognition; Numerical distance; Physical similarity; Random walk; Simulation.

Fig. 1

Basic data and key models. (A) Summary data from a typical relative quantity task in which the quantity denoted by a probe digit is compared with “5.” RT is mean reaction time in ms. Open circles indicate human data and ‘+’s indicate the fit provided by the Welford function (Welford, 1960). Two robust effects are (i) the numerical distance effect (NDE) such that RTs are an inverse function of the numerical distance between the two numbers presented and (ii) the size effect (SZE) reflects a monotonically increasing function relating RTs and the quantity denoted by the probe. (B) The three alternative models of the representation of numerical quantities. The x-axis represents the psychological representation of quantity (from small to large); the y-axis represents density. The graphs, from left to right, describe the linear, logarithmic, and scalar variance theories. Please see the text for detailed descriptions.

Fig. 2

A visualization of the PDQs for the number symbols 3, 4, and 5. The x-axis is the psychological representation of quantity (e.g., the mental number-line). The distributions represent the frequencies that each symbol activates a particular psychological quantity. The overlap of the distributions determines the difficulty of distinguishing the quantities of the two symbols. The PDQ for 4 and 5 (A) overlap more than those of 3 and 5 (B). Therefore, 4 is more difficult to distinguish from 5 than 3.

Fig. 3

The random walk process and details of the behavioral data. (A) Schematic representation of the random walk process. The account assumes that both the standard (“S” e.g., “5”) and the probe digit “P” are represented in terms of corresponding Gaussian PDQs (left hand side of the figure). At each step in the walk the information associated with the probe is assessed relative to the distribution of the standard. Evidence accumulates once the stimulus is presented and a decision is made once either the upper or the lower threshold is reached. (B) The left and center panels display the 25th percentiles of the human data from the relative quantity task. The right-most panel displays the error count for each probe.

Fig. 4

The fit statistic, r² (BIC_Z and chi square provides the same results), of the Traditional Encoding and Encoding Error models for the three primary quantity representations. The simulations were simultaneously fit to the correct and error RTs as well as the proportion of error for each probe. The Encoding Error models outperformed the Tradition Encoding models and the Linear Encoding Error Model out performs all other models.

Fig. 5

The summary behavioral data broken down according to the fits of the Traditional Encoding model for the three quantity representations. Open circles indicate human data and filled circles indicate the fit provided by the model. No model fares well when simultaneously fit to the correct RT, error RT, and proportion of errors for each probe.

Fig. 6

The summary behavioral data broken down according to the fits of the Error Encoding model for the three quantity representations. Open circles indicate human data and filled circles indicate the fit provided by the model. All models fare well when simultaneously fit to the correct RT, error RT, and proportion of errors for each probe. The Linear Error Encoding model out performs all other models with an r² = 0.86.

Fig. 7

The importance of encoding in the size effect. (A) Left most panel. Schematic representation of the general framework for thinking about the derivation of number meaning. Encoding works in tandem with the comparison process and continually influences this process. Continuous encoding as shown in the rightmost panel is fundamental to the operation of the linear hybrid account. (B) The mean psychological sense of quantity (Ψ) by the actual quantity (Θ) for digits 1–9 stemming from the underlying Linear Theory representation and the confusions resulting from continuous encoding. The latter influence causes the shift from linearity to the negatively decelerating function present.