Problem Difficulty in Arithmetic Cognition: Humans and Connectionist Models

In mathematical cognition, problem difficulty is a central variable. In the present study, problem difficulty was operationalized through five arithmetic operators --- addition, subtraction, multiplication, division, and modulo --- and through the number of carries required to correctly solve a problem. The present study collected data from human participants solving arithmetic problems, and from multilayer perceptrons (MLPs) that learn arithmetic problems. Binary numeral problems were chosen in order to minimize other criteria that may affect problem difficulty, such as problem familiarity and the problem size effect. In both humans and MLPs, problem difficulty was highest for multiplication, followed by modulo and then subtraction. The human study found that problem difficulty was monotonically increasing with respect to the number of carries, across all five operators. Furthermore, a strict increase was also observed for addition in the MLP study.