The ERMEL team (IFé, ENS-Lyon) in France, builds complete engineering to teach math in elementary school. For several years, we have been experimenting with teaching situations on space and geometric learning for 5-8 years old students. In this paper, we focus on the results about the relationship between actions on objects, graphs and first geometric properties about curved line. Our methodology involves an analysis of the student's way of solving problems and thus their abilities, but it also allows us to make a hypothesis on what it is that teachers need in order to carry these learning situations out. Issue of this paper Our research takes place in the French context of geometry teaching in primary school. Our goal is to build a proven, complete and reliable teaching engineering and thus to improve geometry teaching. In CERME 9, we reported our results about the knowledge of straight line for the same kind of students, not as an element of geometrical figure but as a usual or new component of the pupil's practice. Several point of view in the WG4 showed the similar approach and questions were shared:  The relation between everyday and geometrical concept, perception, language and manipulation  How to start with low level, and long planning time  What kind of tools for the research and tools for teaching geometry are there? (How to help teacher to know what student are able to do) In this paper, we aim to make our contribution to the three first topics of the WG4 group: what is doing, learning and teaching geometry at school? Why working on geometry teaching? A starting point of our research is a finding of deficiencies in the geometry teaching practices in the early grades of elementary school. A short analysis of easily available French resources for teaching geometry to young pupils shows that two mains goals are pursued: learning geometric words and drawing abilities. Spatial activities also exist but without much problem solving, and also unrelated to the pupil's initial knowledge or mathematical activity. This research is also based on the idea that students' abilities are insufficiently taken into account in geometry teaching in primary school. Thus we have to identify the knowledge at stake in this learning and take the students' already acquired knowledge into account.