Use of representations and generalisation of the multiplication principle

We have examined the problem-solving protocols of 50 students with the goal of analysing the construction of the multiplication principle as a basic scheme for solving counting problems. In particular, we investigated how induction and different representations are used to move from exhaustive enumeration to the generalisation of this principle. The analysis revealed the existence of two processes of generalisation: 1) on the dimension of the problem, and 2) the number of elements involved in each factor. We show how both processes are related to the effective use of tree diagrams that students generated spontaneously, and suggest possible educational implications. Moreover, the analysis of our data has generated the need to investigate the connection between textual representations and other types of representations, evaluating its functionality. © 2011 Fundación Infancia y Aprendizaje.