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In his contribution to this issue of the journal Stéphane Clivaz introduces Brousseau’s conception of teaching and learning in a didactical situation to model the learning situation for teachers participating in lesson study. This model is based on the Theory of Didactical Situations (TDS). His hypothesis is that teacher professional knowledge is developed through the interaction of the adidactical situations of action, formulation, validation and institutionalization around the milieu of the research lesson. Clivaz develops a diagrammatic model based on an amalgamation of Lewis’s lesson study schema (Lewis and Hurd, 2011) and Brousseau’s (1997) schema. Figure 1 here is based on the present writer’s interpretation of Clivaz’s model in which the research lesson—involving teacher, students and milieu—becomes the milieu with which the lesson study participants and the lesson study facilitator/researcher engage.
Given that it is not possible to understand a theory without comparison with another theory about the same phenomenon, Clivaz’s theorization of lesson study may be contrasted with the theorization of learning study. According to Ruthven et al. (2009), citing (Brousseau, 1997):
Piaget’s theory of “equilibration” was a crucial source for the idea of “a-didactical” adaptation in which students construct new knowledge through becoming directly engaged in solving a novel type of problem, refining their concepts and strategies in the light of feedback from a (material and social) milieu.
(Ruthven et al., 2009, p. 330)
Subsequent development of the design tool included the addition of the idea of the “epistemological obstacle” (from Bachelard) and “the (inescapable) intervention of the (mathematical) culture through the medium of the teacher” leading to “institutionalization.” Ruthven et al. (2009) include a classic example of a mathematical adidactical situation in their paper. In this situation, students work in groups to enlarge the six pieces of a square puzzle 11 × 11 cm² having been instructed to make a similar but larger puzzle using the information that the edge of one piece that measures 4 cm in the original puzzle should measure 7 cm in the larger one. Each student is instructed to work on a single piece of the puzzle. The situation is designed to invalidate an additive model of scaling (epistemological obstacle). When the feedback to the students from engagement with the milieu proves insufficient, the teacher draws on the...