Mathematical Model of Developmental Changes in Number Cognition

The acuity of individuals’ approximate number system (ANS) is measured using a non-symbolic numerical discrimination task (e.g., Halberda & Feigenson, 2008) which is encapsulated by the Weber Fraction measure (Piazza & Dehaene, 2004). Children’s performance on this task is linked to symbolic arithmetic skills, standardized test scores and other long-term outcomes (e.g., Beddington et al., 2008; Libertus, Odic, & Halberda, 2012; Parsons & Bynner, 1997). To complete this task individuals must perceive relative quantity of non-symbolic numerosity and indicate their relative values. Numerosity is typically presented visually as sets of objects, though other modalities such as sound have been used (e.g., Jordan & Brannon, 2006). Discriminability of numerical stimuli is thought to depend solely on the ratio-difference between values being compared if other perceptual factors are controlled for. Thus comparison of the same ratio difference, such as 20:25 and 40:50 are equally discriminable and absolute value is not relevant. Thus individual performance on numerical discrimination in terms of accuracy and reaction time varies with the ratio differences and is independent of absolute value. This has been shown in a range of behavioral and neural data with adults (e.g., Barth, Kanwisher, & Spelke, 2003), children (e.g., Barth, Mont, Lipton, & Spelke, 2005) and non-human primates (e.g., Nieder & Miller, 2004). Research with adults contradicts the account that ratio is the sole predictor of discriminability in number comparison task. Though absolute value does not predict discriminability, the interaction of ratio difference and absolute value does (Prather, 2014). The interaction between absolute value is not typically reported because data including comparisons with small ratio differences and variations in absolute value are needed to clearly show the effect. Though there is initial evidence of the effect of absolute value on numerical discrimination it is unclear if such an effect would be present with children, and if the effect varies across development. The consequences of such an effect in children are directly relevant to measures of numerical acuity. Measures of numerical acuity are used to evaluate a correlation to performance on symbolic, and other educational outcomes. If children’s performance varies across absolute value, which is not typically accounted for, then it is unclear the reported numerical acuity is the “true” measure of participants’ accuracy. Inclusion of varying absolute values may change the degree to which non-symbolic discrimination predicts other outcomes. The current study employs behavioral experimentation and computational modeling to address the following questions: 1) is children’s numerical discrimination solely dependent on ratio difference or also an interaction between ratio difference and absolute value, 2) do the factors contributing to children’s numerical discrimination difficulty change with development, 3) how might the neural coding involved numerical perception account for these behavioral effects?