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Η διδασκαλία της γεωμετρίαςTHE TEACHING OF GEOMETRY. The aim of the present research is to investigate the problems encountered in the transition of Practical to Theoretical Geometry, the views and disposition of the teachers towards it, the necessity for its teaching and the best way for its approach.The advancement of the Geometric thought is presented with well-documented analysis from the stone age down to our days. The presence of Geometry in the practical and scientific life is underlined. The overall reversal of thought with the appearance of the Greek Philosophy and Science is studied. Achievements of the representatives of various Philosophical Schools along with the work of certain notable commentators are discussed.A brief review is made of Geometry from Middle-Ages down to Modern Times.We present the efforts to prove the V postulation, the non euclιdιan geometries and certain research fields of the two last centuries, with impact on foundation, the teaching approach or the mode of mathematics presentation.At the same time the political and social conditions are touched allusively.The evolution of Geometry teaching is examined using the French Revolution as point of reference. A special mention is made to Greece up to 1821. The contents of school geometry is built on the basis of two significant French books on "Elements of Geometry" by Alexis Clairaut, and by Andrien Legendre. The latter was used as a school textbook for more than a century. The study of school mathematics textbooks in the 20th century is separated into before and after the momentous introduction of New Mathematics.The pedagogical value of Geometry is investigated, also "when" and "how" it should be taught.A short historical cut-back to the pedagogical concepts on Geometry is made and data on the modern pedagogical approaches are given, like problem solving, modeling, construction mathematics, the Van Hiele levels, the Alan Hoffer theory and teaching with the aid of computers.The field research, (53 questions) and the statistical methodology (descriptive analysis, crosstabulation, hierarchical cluster) are quoted.The data analysis shows that the mathematicians of the sample are divided in four categories on the basis of their respective stance on Geometry teaching.The basic conclusions of the research include:•Geometry assists the cognitive representation of abstract concepts and the mental development of children, advances their critical thought. It should be taught as an independent subject and never be substituted by others.•Modern pedagogical approaches, cultivation of positive emotional stance, alternative ways of pupils’ appraisal, historical data, intersubjective approach, everyday life applications and the use of technology information assist the teaching of Geometry.•Teaching of Geometry at junior high schools has to be practical and be complemented with certain theoretical principles.•The training of the teachers of mathematics is essential and necessary. Thoughts are presented on Geometry with proposals for more productive teaching, the drafting of a new curriculum and issues of future research orientations.
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Η διδασκαλία της γεωμετρίαςTHE TEACHING OF GEOMETRYThe aim of the present research is to investigate the problems encountered in the transition of Practical to Theoretical Geometry, the views and disposition of the teachers towards it, the necessity for its teaching and the best way for its approach.The advancement of the Geometric thought is presented with well-documented analysis from the stone age down to our days. The presence of Geometry in the practical and scientific life is underlined. The overall reversal of thought with the appearance of the Greek Philosophy and Science is studied. Achievements of the representatives of various Philosophical Schools along with the work of certain notable commentators are discussed.A brief review is made of Geometry from Middle-Ages down to Modern Times.We present the efforts to prove the V postulation, the non euclιdιan geometries and certain research fields of the two last centuries, with impact on foundation, the teaching approach or the mode of mathematics presentation.At the same time the political and social conditions are touched allusively.The evolution of Geometry teaching is examined using the French Revolution as point of reference. A special mention is made to Greece up to 1821. The contents of school geometry is built on the basis of two significant French books on "Elements of Geometry" by Alexis Clairaut, and by Andrien Legendre. The latter was used as a school textbook for more than a century. The study of school mathematics textbooks in the 20th century is separated into before and after the momentous introduction of New Mathematics.The pedagogical value of Geometry is investigated, also "when" and "how" it should be taught.A short historical cut-back to the pedagogical concepts on Geometry is made and data on the modern pedagogical approaches are given, like problem solving, modeling, construction mathematics, the Van Hiele levels, the Alan Hoffer theory and teaching with the aid of computers.The field research, (53 questions) and the statistical methodology (descriptive analysis, crosstabulation, hierarchical cluster) are quoted.The data analysis shows that the mathematicians of the sample are divided in four categories on the basis of their respective stance on Geometry teaching.The basic conclusions of the research include:•Geometry assists the cognitive representation of abstract concepts and the mental development of children, advances their critical thought. It should be taught as an independent subject and never be substituted by others.•Modern pedagogical approaches, cultivation of positive emotional stance, alternative ways of pupils’ appraisal, historical data, intersubjective approach, everyday life applications and the use of technology information assist the teaching of Geometry.•Teaching of Geometry at junior high schools has to be practical and be complemented with certain theoretical principles.•The training of the teachers of mathematics is essential and necessary. Thoughts are presented on Geometry with proposals for more productive teaching, the drafting of a new curriculum and issues of future research orientations.
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Secondary School Students’ Meaning and Learning of Circle GeometryThe purpose of this study is to understand and gain insight into the types of meanings secondary school students hold in learning circle geometry, the role these meanings play in students’ learning of circle geometry concepts, the role of formal instruction in shaping and reshaping these meanings, and the implications for teaching geometry. Meanings, framed in a constructivist perspective, are interpreted as the students’ ways of making sense of the concepts learned and the formal instruction is interpreted as the instruction that students experience in their regular classrooms on topics of circle geometry based on the curriculum and teaching approaches used by the teacher. 
 A qualitative case study approach is employed, with data sources consisting of pre- and post-intervention interviews to capture students’ meanings, a contextual circle geometry task to capture meanings through problem solving, and classroom observations of the sessions on the circle geometry unit to capture the nature of the instruction and students’ participation in it. The participants are 20 students from a single Grade 9 mathematics classroom. Data analysis involves a thematic approach based on coding the data to identify students’ meanings and the impact of the instruction on them.
 Findings indicate that the students held several pre-instruction meanings that were influenced mainly by their real-world experiences and prior learning of school geometry. Many of these meanings did not reflect the mathematical meanings of the concepts or support problem solving involving contextual circle geometry problems. While the formal instruction resulted in positive changes in these meanings, for the most part the post-instruction meanings were focused on definitions of the concepts and applications to contextual circle geometry problems in an instrumental way. The findings suggest that students’ pre-instruction meanings provide a basis for their learning of new concepts and could impact this learning both positively and negatively, so it is important for teachers to consider and unpack the students’ initial meanings when introducing new concepts. The study contributes to our understanding of these meanings and how they should be dealt with in teaching and learning of geometry and mathematics teacher education, with implications for further research.