Towards a curriculum for neural networks to simulate symbolic arithmetic

Understanding and operating on multi-digit numbers is a critical step in the development of mathematical skills and concepts. Empirical and computational modeling evidence suggests that multi-digit numbers are processed decomposed into units, tens, hundreds, etc. for magnitude comparison. Yet, there is currently no computational model to simulate multi-digit number arithmetic. Accordingly, we developed a neural network model with three interconnected modules reflecting single-digit additions, recognition of a carry-over, and decision-making. We then compared model performance after following two training curricula: i) a \textit{ step-by-step} curriculum, where single-digit addition are trained before progressing to multi-digit problems and ii) an \textit{all-at-once} curriculum, where all modules of the model were trained simultaneously. Our results indicated that only the \textit{step-by-step} curriculum made the model learn multi-digit addition successfully as reflected by replicating empirical effects of carry-over and problem size. These findings highlight the importance of structured, incremental learning in both cognitive modeling and education.