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This study tested the hypothesis that intuitions about the effect of operations, e.g., 'addition makes bigger' and 'division makes smaller', are still present in educated adults, even after years of instruction. To establish the intuitive character, we applied a reaction time methodology, grounded in dual process theories of reasoning. Educated adult participants were asked to judge the correctness of statements about the effect of operations. Their accuracy and reaction times were measured. For items where the correct answer was not in line with the assumed intuitions, more mistakes were observed; moreover, we found longer reaction times for correct responses, indicating that these intuitions interfere in participants' reasoning on these tasks, even when they provide a correct response. [ABSTRACT FROM AUTHOR]

Inversion is a fundamental relational building block both within mathematics as the study of structures and within people's physical and social experience, linked to many other key elements such as equilibrium, invariance, reversal, compensation, symmetry, and balance. Within purely formal arithmetic, the inverse relationships between addition and subtraction, and multiplication and division, have important implications in relation to flexible and efficient computation, and for the assessment of students' conceptual understanding. It is suggested that the extensive research on arithmetic should be extended to take account of numerical domains beyond the natural numbers and of the difficulties students have in extending the meanings of operations to those of more general domains. When the range of situations modelled by the arithmetical operations is considered, the complexity of inverse relationships between operations, and the variability in the forms that these relationships take, become much greater. Finally, some comments are offered on the divergent goals and preoccupations of cognitive psychologists and mathematics educators as illuminated by research in this area. [ABSTRACT FROM AUTHOR]