This paper is a part of more extensive research accomplished with students of an education faculty under the title „Estimation of mathematical education levels“ which the Scientific Society of Mathematicians Banja Luka is realising. Theoretical aspect includes questions about elementary geometry teachings, van Hiele's theory of understanding geometry, and students' undetstanding the roles of mathematical definitions. In this paper, using circle and circumfurence as examples, we demonstrate relationships between school knowledge of geometry, knowledge of geometry needed to any elemenary school teachers of mathematics, capabilities of construction of geometry knowledge in the elemnatary school classes (Didactic mathematics domain) and understanding of interaction between pupils and teachers during that construction (Research of Mathematics Education domain).