We explore a recurrent neural network model of counting based on the differentiable recurrent attentional model of Gregor et al. (2015). Our results reveal that the model can learn to count the number of items in a display, pointing to each of the items in turn and producing the next item in the count sequence at each step, then saying ‘done’ when there are no more blobs to count. The model thus demonstrates that the ability to learn to count does not depend on special knowledge relevant to the counting task. We find that the model’s ability to count depends on how well it has learned to point to each successive item in the array, underscoring the importance of coordination of the visuospatial act of pointing with the recitation of the count list. The model learns to count items in a display more quickly if it has previously learned to touch all the items in such a display correctly, capturing the relationship between touching and counting noted by Alibali and DiRusso. In such cases it achieves performance sometimes thought to result from a semantic induction of the ‘cardinality principle’. Yet the errors that it makes have similarities with the patterns seen in human children’s counting errors, consistent with idea that children rely on graded and somewhat variable mechanisms similar to our neural networks.

Researchers exploring mathematical abilities have proposed that humans and animals possess an approximate number system (ANS) that enables them to estimate numerosities in visual displays. Experimental data shows that estimation responses exhibit a constant coefficient of variation (CV: ratio of variability of the estimates to their mean) for numerosities larger than four, and a constant CV has been taken as a signature characteristic of the innate ANS. For numerosities up to four, however, humans often produce error-free responses, suggesting the presence of estimation mechanisms distinct from the ANS specialized for this ‘subitizing range’. We explored whether a constant CV might arise from learning in generic neural networks using widely-used neural network learning procedures. We find that our networks exhibit a flat CV for numerosities larger than 4, but do not do so robustly for smaller numerosities. Our findings are consistent with the idea that estimation for numbers larger than 4 may not require innate specialization for number, while also supporting the view that a process different from the one we model may underlie estimation responses for the smallest numbers.