Figueiras, L. y Cañadas, M. C. (2010). Reasoning on transition from manipulative strategies to general procedures in solving counting problems. Journal of the British Society for Research into Learning Mathematics, 30(1), 89-96.
We describe the procedures used by 11- to 12-year-old students for solving basic counting problems in order to analyse the transition from manipulative strategies involving direct counting to the use of the multiplication principle as a general procedure in combinatorial problems. In this transition, the students sometimes spontaneously use tree diagrams and sometimes use numerical thinking strategies. We relate the findings of our research to recent research on the representational formats on the learning of combinatorics, and reflect on the didactic implications of these investigations.