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Cognitive conditions of the geometric learning: Developing visualisation, distinguishing various kinds of reasoning and co-ordinating their running. (Les conditions cognitives de l’apprentissage de la géométrie: Développement de la visualisation, différenciation des raisonnements et coordination de leurs fonctionnements.) (French)
Ann. Didact. Sci. Cogn. 10, 5-53 (2005).
Geometry is a kind of knowledge area that requires the cognitive joining of two representation registers: on one hand the visualisation of shapes in order to represent the space and on the other hand the language for stating some properties and for deducing from them many others. The troubles of learning first come from the fact these two registers are used in a way which is opposite to their cognitive use apart from mathematics. The way of seeing a geometrical figure depends on the activity for what it is used. Thus it can run in an iconic way or in a non-iconic way. The non-iconic visualisation involves that the first recognised shapes would be visually deconstructed. There are three kinds of shape deconstruction: deconstruction by using tools in order to construct any figure, the heuristic breaking down in order to solve problems and the dimensional deconstruction. This one is the central process of geometrical visualisation. The analysis of language in geometry requires that three levels of discursive operations would be separated: verbal designation, property statement and deduction. This separation is important because the relation between visualisation and language changes completely from one level to the other. This variation conceals the most important cognitive phenomenon: the dimensional hiatus. Comings and goings between visualisation and language involve a jump into the number of dimensions in order to recognise the knowledge objects that are represented within each register. Becoming aware of the dimensional deconstruction of shapes and understanding the process of the various discursive operations are the conditions for succeeding in making the two registers run in synergy. There are the crucial thresholds for learning in geometry. Is that really taken into account in the teaching?.
Classification: C30 G10