Increasing the use of conceptually-derived strategies in arithmetic: using inversion problems to promote the use of associativity shortcuts

Conceptual knowledge of key principles underlying arithmetic is an important precursor to understanding algebra and later success in mathematics. One such principle is associativity, which allows individuals to solve problems in different ways by decomposing and recombining sub-expressions (e.g. ‘a + b − c’ = ‘b − c + a’). More than any other principle, children and adults alike have difficulty understanding it, and educators have called for this to change. We report three intervention studies that were conducted in university classrooms to investigate whether adults' use of associativity could be improved. In all three studies, it was found that those who first solved inversion problems (e.g. ‘a + b − b’) were more likely than controls to then use associativity on ‘a + b − c’ problems. We suggest that ‘a + b − b’ inversion problems may either direct spatial attention to the location of ‘b − c’ on associativity problems, or implicitly communicate the validity and efficiency of a right-to-left strategy. These findings may be helpful for those designing brief activities that aim to aid the understanding of arithmetic principles and algebra.