The application of arithmetic principles predicts mathematical achievement in college students

Number sense and arithmetic fluency are fundamental to early mathematical development. However, these capacities generally fail to predict mathematical achievement in older adolescents and adults. We propose that later mathematical development is driven by coming to understand the higher-order principles that bring structure to mathematics. To evaluate this proposal, we tested whether college students (n = 134) apply arithmetic principles – inverse, associativity, and commutativity – to efficiently verify arithmetic sentences mixing multiplication and division operations such as 18 × 7 ÷ 3 = 42. This was the case. People were more accurate and faster when verifying arithmetic sentences that could be simplified by the application of arithmetic principles compared to control problems. People found problems that required the associativity principle to be more difficult (i.e., they made more errors and took longer) than those that required the inverse principle, and problems that additionally required the commutativity principle to be more difficult still. Converging evidence for the use of these principles came from their strategy self-reports. Critically, individual differences in applying these principles predicted mathematical achievement even after controlling for number sense, arithmetic fluency, and verbal achievement. These findings have implications for theories of mathematical development and may point the way to future interventions for increasing the mathematical achievement of younger children.