The purpose of this study is to compare and evaluate the effectiveness of computer assisted teaching of geometry shapes and an interactionist approach to teaching geometry in kindergarten versus other more traditional teaching methods. Our research compares the improvement of the children's geometrical competence using two teaching approaches. The first one employs our ICT oriented learning method specifically targeting Realistic Mathematics Education (RME) and the van Hiele model for geometry shapes. The second method is based on the interactionist approach of teaching children in kindergarten. The sample consisted of 139 Greek kindergarten children from Crete. We conducted interventions based on the above two methods, in two experimental groups. The first experimental group consisted of 50 children who were taught geometry shapes according to the principles of Realistic Mathematics Education with the use of appropriate educational software. The second experimental group consisted of 45 children who were taught about geometry shapes with an interactionist approach without the use of computers. The control group consisted of 44 children and was taught about geometry shapes following the ways teachers chose to put the basic pedagogical principles of Curriculum for kindergarten into practice without the use of ICT. Children in all groups were pre-tested and post-tested for their geometry performance. The results showed that the teaching approaches of our interventions, the one with the additional use of ICT and the interactionist one, contribute significantly more to the development of kindergarten children's geometry competence as compared to other traditional methods.