This study investigated the promotion of children's understanding and acquisition of arithmetic concepts and the effects of inhibitory skills. Children in Grades 3, 4, and 5 solved two sets of three-term addition and subtraction problems (e.g., 3 + 24 - 24, 3 + 24 - 22) and completed an inhibition task. Half of the participants received a demonstration of conceptually-based shortcuts between problem sets. All participants increased their use of the inversion shortcut (stating that the answer to a problem of the form a + b - b was the first number without any calculations) across a problem set but only the participants who received the demonstration increased their associativity shortcut use (subtracting first and then adding on a problem of the form a + b - c), particularly if they evaluated the shortcut as being better than a traditional left-to-right computational algorithm (adding then subtracting). Four clusters of participants using varying degrees of conceptually-based shortcuts were identified. Participants with weak inhibition skills were more likely to use a strategy that mixed conceptual knowledge and a computational algorithm suggesting that although they had conceptual knowledge, they had difficulties inhibiting a well-learned computational algorithm. (Contains 4 tables and 2 figures.)