Didactical analysis of primary geometric concepts ii

Milosav M. Marjanoviĉ

Surveyed in historical perspective, the contents of school geometry can be sorted into the following three stages: intuitive, pre-Euclidean and Euclidean. Geometric topics usually found in primary school programs constitute the first stage of intuitive geometry. Although this geometry is related to the perception of the surrounding real world, both the coherence and order in its exposition are equally important. This paper is devoted to that didactical task. At this first stage, pairings of the real world appearances (objects and their collections) with geometric models (iconic signs and their configurations) make the main learning procedures. These pairings are interactive, meaning that the appearances ``re-animate'' the models and the latter serve to the intelligible conceiving of the former. In order to discriminate between the two sides of this process of pairing, we consider two levels of intuitive geometry. When conveyors of geometric meaning are real world appearances, we speak of inherent geometry and when such conveyors are configurations of iconic signs, we speak of visual geometry. These latter signs (called here ideographs) express the meaning of geometric concepts and their deliberate use leads to the assimilation of this meaning. Following Poincaré's views on genesis of geometric ideas, we formulate a cognitive principle stating that perception of solids in the outer space, in the way when all their physical properties are abstracted (ignored), leads to the creation of these ideas. In addition, an intelligent ignoring of extension leads, then to the creation of concepts: point, line and surface. In the final didactical analysis, a series of basic geometrical concepts will be discussed (and this paper includes: point, line, segment, ray and straight line).