This paper describes the nature of children's solutions to Cartesian product problems according to their implied levels of combinatory reasoning in order to identify the learning process going on therein, and the nature of teaching intervention. Five nine-year-old students enrolled at the 3rd grade of a state elementary school were the participants. Data were collected during two individual sessions of problem solving. The researcher's teaching intervention was on the clinical-critical format. The qualitative analysis of videotape data resulted in the identification of the following solution levels: contextualized answer without signs of combination, the first approaches to combinatory solutions, the arising of some combinations, and the presence of combinatory solutions. The identified forms of teaching interventions were: orienting, re-orienting, questioning, and instigating. The discussion underlies the relevance of the described teaching interventions to the punctual progress of each child towards solutions with combinatory marks.
Cartesian product; combinatory reasoning; mathematical learning; teacher's interventions
