Abstract : The purpose of this research is to test in Grade 5 (CM2) and Grade 6 (6ème) a new theoretical frame which is a combination of the theory of geometrical paradigms and the van Hiele levels theory. In primary school, geometry is basically spatio-graphic (G1): objects are representations of physical objects and validations are perceptive. The pupil must then master the 1st level of the van Hiele theory: identification-visualisation (N1). In secondary school, geometry tends to be more proto-axiomatic (G2): objects are theoretical and validations are based on hypothetic-deductive reasoning. The student is supposed to master the 4th of the van Hiele levels: informal deduction (N3).
The theoretical frame tested here assumes that the 3rd level from the van Hiele levels (N2: analysis) is the “linking level” between G1 and G2.
Pupils from Grades 5 and 6 were asked the same questions about triangles, quadrilaterals and circle in different ways: sorting drawings, tracing, analysis of drawings and of geometric figures; argumentations; explanations.
The analysis of the answers show that a pupil either in Grade 5 or Grade 6 can work within both geometrical paradigms and at different van Hiele levels, depending on the question he is asked. Analysis being the 3rd of the van Hiele levels has been proved as the “linking level” between the two paradigms G1 and G2. Activities at this van Hiele level in the context of either paradigm G1 or G2 can reduce the discontinuity between spatio-graphic geometry in primary school and proto-axiomatic geometry in secondary school.