The Role of Implicit Models in Solving Verbal Problems in Multiplication and Division

Arithmetical operations were assumed to remain attached to primitive behavioral models that influence tacitly the choice of an operation even after the learner has had solid formal-algorithmic training. The model for multiplication was conjectured to be repeated addition, and two primitive models (partitive and quotative) were seen as linked to division. A total of 623 pupils enrolled in 13 Italian schools (Grades 5, 7, and 9) were asked to choose the operation needed to solve 26 multiplication and division word problems. Violations by the numerical data of the constraints imposed by the assumed tacit models (for instance, when the operator was a decimal number) constituted particular sources of difficulty at all three grade levels. The findings seemed to confirm the impact of the repeated addition model on multiplication and of the partitive model on division. The quotative division model influenced the pupils' choices only at the ninth-grade level.