Investigating children's understanding of inversion using the missing number paradigm

The development of conceptual understanding in arithmetic is a gradual process and children may make use of a concept in some situations before others. Previous research has demonstrated that when children are given arithmetic problems with an inverse relationship they can infer that the initial and final quantities are the same (e.g. 15 + 8–8 = □). However, we do not know whether children can perform the complementary inference that if the initial and final quantities are the same there must be an inverse relationship (i.e. 15 + □ − 8 = 15 or 15 + 8 − □ = 15). This paper reports two experiments that presented inverse problems in a missing number paradigm to test whether children (aged 8–9) could perform both these types of inferences. Children were more accurate on standard inverse problems (a + b − b = a) than on control problems (a + b − c = d), and their performance was best of all on rearranged inverse problems (b − b + a = a). The children's performance on inverse problems was affected by the position of the missing number and also by the order of elements within the problem. This may be due to the different types of inferences that children must make to solve these kinds of inverse problems.