The present study investigates in-service secondary teachers’ geometrical figure apprehension in relation to their ability to construct geometrical proofs and to predict didactically their students’ difficulties and mistakes. The theoretical framework of analysis is based on Duval’s geometrical figure apprehension which was taught to the teachers as part of an in-service training course in didactics of mathematics, offered by one of the researchers. As part of the course final assessment, a written test consisting of various geometrical tasks was constructed and administered to the sample. Participants’ answers in both solving and interpreting difficulties related to the tasks were analyzed. The results of the study indicate that theories and concepts of didactics of geometry can shed light to various facets of teaching and learning of geometry. A teacher who presents a correct solution at a task does not necessarily identify or understand the possible difficulties faced by students. Discussion concentrates on teaching implications about geometry and geometrical proof.

This study aims to investigate high school students’ geometry learning by focusing on mathematical creativity and its relationship with visualisation and geometrical figure apprehension. The presentation of a geometrical task and its influence on students’ mathematical creativity is the main topic investigated. The authors combine theory and research in mathematical creativity, considering Roza Leikin’s research work on Multiple-Solution Tasks with theory and research in visualisation and geometrical figure apprehension, mainly considering Raymond Duval’s work. The relations between creativity, visualization and geometrical figure apprehension are examined through four Geometry Multiple-Solution Tasks given to high school students in Greece. The geometrical tasks are divided into two categories depending on whether their wording is accompanied by the relevant figure or not. The results of the study indicate a multidimensional character of relations among creativity, visualization and geometrical figure apprehension. Didactical implications and future research opportunities are discussed.

This study aims to examine the relation between spatial ability and creativity in Geometry. Data was collected from 94 ninth graders. Three spatial abilities were investigated: spatial visualization, spatial relations and closure flexibility. As for students' creativity, it was examined through a multiple solution problem in Geometry focusing on three components of creativity: fluency, flexibility, and originality. The results revealed that spatial visualization predicted flexibility and originality while closure flexibility predicted all creativity components. Additionally, it was deduced that auxiliary constructions played an essential role in the problem-solution process. Finally, further study opportunities for the teaching and learning of Geometry are discussed.