We aim to clarify Matthew Lipman's definition of critical thinking. The research project was conducted during a complete school year among eight groups of pupils from three cultural contexts: Australia, Mexico and Quebec. We worked in an inductive manner, inspired by the grounded theory approach, and analysed transcripts of the pupils' exchanges while they used a philosophical approach adapted to mathematics. Analysis of the transcripts was performed according to the characteristics of a grid that we developed elsewhere, which represented the manifestations of 'dialogical critical thinking' in pupils aged 10 to 12. From the analysis, the Lipmanian criteria (sensitivity to context, thinking governed by criteria and self-correction) do not seem representative of dialogical critical thinking, although the modalities of thinking that he formulates did manifest themselves in pupils engaged in the research project.