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Abstract The study of how children learn numbers has yielded one of the most productive research programs in cognitive development, spanning empirical and computational methods, as well as nativist and empiricist philosophies. This paper provides a tutorial on how to think computationally about learning models in a domain like number, where learners take finite data and go far beyond what they directly observe or perceive. To illustrate, this paper then outlines a model which acquires a counting procedure using observations of sets and words, extending the proposal of Piantadosi et al. (2012). This new version of the model responds to several critiques of the original work and outlines an approach which is likely appropriate for acquiring further aspects of mathematics. 

Children rely on their approximate number system (ANS) to guess quantities from a young age. Studies have shown that older children displayed better ANS performance. However, previous research did not provide an explanation for this ANS improvement. We show that children’s development in ANS is primarily driven by improved attentional control and awareness of peripheral information. Children guess the number of dots on a computer screen while being eye-tracked in our experiment. The behavioral and eye-tracking results provide supporting evidence for our account. Our analysis shows that children estimate better under the longer display-time condition and more visual foveation, with the effect of visual foveation mediating that of time. It also shows that older children make fewer underestimations because they are better at directing their attention and gaze toward areas of interest, and they are also more aware of dots in their peripheral vision. Our finding suggests that the development of children’s ANS is significantly impacted by the development of children’s nonnumerical cognitive abilities. 

People can identify the number of objects in small sets rapidly and without error but become increasingly noisy for larger sets. However, the cognitive mechanisms underlying these ubiquitous psychophysics are poorly understood. We present a model of a limitedcapacity visual system optimized to individuate and remember the location of objects in a scene which gives rise to all key aspects of number psychophysics, including error-free small number perception and scalar variability for larger numbers. We therefore propose that number psychophysics can be understood as an emergent property of primitive perceptual mechanisms — namely, the process of identifying and representing individual objects in a scene. To test our theory, we ran two experiments: a change-localization task to measure participants’ memory for the locations of objects (Experiment 1) and a numerical estimation task (Experiment 2). Our model accounts well for participants’ performance in both experiments, despite only being optimized to efficiently encode where objects are present in a scene. Our results demonstrate that the key psychophysical features of numerical cognition do not arise from separate modules or capacities specific to number, but rather from lower-level constraints on perception which are manifested even in non-numerical tasks 

Normative learning theories dictate that we should preferentially attend to informative sources, but only up to the point that our limited learning systems can process their content. Humans, including infants, show this predicted strategic deployment of attention. Here we demonstrate that rhesus monkeys, much like humans, attend to events of moderate surprisingness over both more and less surprising events. They do this in the absence of any specific goal or contingent reward, indicating that the behavioral pattern is spontaneous. We suggest this U-shaped attentional preference represents an evolutionarily preserved strategy for guiding intelligent organisms toward material that is maximally useful for learning.