During the school year 2015/2016, the National Institute for Certified Educational Measurements carried out a nation-wide testing of the 5th-grade pupils. The test of mathematics within T5-2015 also included the task from geometry. In our paper, we present the results of the selected items from geometry, their analysis and examples of some students´ solutions.

Příspěvek pojednává o dílčích výsledcích testování žáků čtvrtých ročníků ZŠ v oblasti rozvoje představ žáků o pojmu čtverec. Zabývá se nedostatky v pojmotvorném procesu žáků pro pojem čtverec.

In geometry, it is very important that pupils‘ initial first steps with geometry should be of an intuitive kind, connected with their experiences. The axis of symmetry of the pair of points, which is the basis for the special structure – Voronoi tessellation, is understood as the basic geometric construction. The contribution concerns the solutions of the task about the Voronoi tessellation.

This paper aims to explain a way of teaching quadrilaterals by using paper folding. The reason for choosing paper folding is the characteristics it has that enables us to address symmetry, which, in turn, provides a sound conceptual base for the classification of quadrilaterals.

The authors of the article present their research on the understanding of the selected basic concepts in school geometry by the first grade students of the upper secondary grammar school in Pardubice, Mozartova 449, Czech Republic. There was used in this experimental teaching the methodological approach of Franz Močnik.

In this article we are focusing on creating electronic course for preparing future mathematics teachers. The course is created in system LMS Moodle. We will deal with creating eLearning lesson from Graph theory. We will show some possible ways of working with software GeoGebra in Graph theory. In GeoGebra we created new applets for the work with graphs.

Spatial abilities and their measurement are in the forefront of methodical research in mathematics since decades. This paper is a short survey of our recent results in this field, with special emphases on classical tests results and gender differences in Hungarain higher education institutions.

This contribution presents three series of activities using combinatory as an instrument for developing geometrical abilities of children and for the further deepening of (pre-)geometrical notions. The analysis of obtained data is illustrated by several cases of casuistic (children aged 5 to 10 years).

Children learn how to recognize basic planar geometrical shapes as early as kindergarten. By the time the children start grade school, they should be able to comfortably identify circles, triangles, squares and rectangles, although this usually holds only for traditional models of these shapes, i.e. in their standard horizontal-vertical configuration. Very little attention is being paid to this area within primary education, yet at the same time pupils are expected to be able to identify these shapes regardless of their dimensions and positioning. Our research focuses on exploring the ideas and models of planar shapes in 3rd and 4th-graders. This article will present our findings regarding rectangles in 3rd-graders exclusively.

The GeoGebra DGS is suitable to visualise the main models of the non-Euclidean geometries. We will introduce the idea of modelling by a model by inversion of the Euclidean Geometry, and present some of the possibilities to visualize some of the most known models, like the Beltrami–Klein model, the Poincare model, and the Upperhalf plane model. The Geogebra files which are produced can be used to let students to experiment with them, and discover some of the properties. Keywords: Non-Euclidean Geometry, Beltrami-Klein model, Poincare model, Half-plane model. MESC: G90, G50, U60, R20

This article represents an attempt to analyse geometry as a pedagogical problem. The first part is an introduction. The second part contains a brief description of two approaches to teaching mathematics: instruction as a transmission of knowledge and instruction as a process of knowledge formation. The third part investigates geometry as a beautiful area of human axtivity. The forth part focuses on the concept of metric space and its interpretations. The fifth part deals with historical approaches to geometrical axiomatics. The sixth part focuses on the connection between geometry and poetry. The last part is devoted to the concept of didactical structure and historical conceptions of geometry.

The aim of the research was to find out which component of a figural concept, visual or conceptual one, a primary pre-service teacher uses in solving tasks about basic geometrical concepts. We present the results of the research conducted on primary pre-service teachers (N=74) studying in Slovenia. Our results show that more than 70% of pre-service teachers had false or poor images of basic geometrical concepts and perceived boundary points only as common points in solving tasks. The results also show that the degree of unlimitedness is not the cause for mismatch of an evoked concept image and a formal definition, which raises questions for further research. Results were contrasted with former research regarding square as a figural concept. We argue that teaching geometry at primary level could be impeded by common content knowledge primary teachers’ possess and urge to restructured teacher training programmes in the area of geometry education.

Imagination is what makes our sensory survivals understandable. It allows us an anachronistic interpretation on which we can take the traditional conclusions ourselves, or we can create something new, original and unique. Geometry can effectively help us in solving arithmetic problems. We list a few examples of solutions, we are looking for, when using geometry.

E-learning can be used to support teaching process. Learning Management Systems represents a tool for this support. LMS Moodle is currently used very often. LMS Moodle allows its users to use a great variety of modules. One of these modules is GeoGebra, the software for supporting teaching geometry.

Geometry is an integral part of mathematical education in primary stage of education. A significant space in the teaching of geometry is dedicated to plane geometric figures. Teachers at the primary stage have a significant share in creating a correct understanding of plane geometric figures by pupils. In this context, it is therefore necessary to secure that the geometrical knowledge of prospective teachers is at the appropriate level and any such misconceptions should be identified and corrected in the course of their undergraduate studies.

Teacher to achieve pedagogy success in learning mathematics, should have substantive knowledge, it means: good understanding of math concepts which are to be described and explained, knowledge how children acquire concepts from psychological point of view, methodical skills. The article presents part of results from research on students of pedagogy – future teachers in early childhood education – in terms of their knowledge of the quadrilaterals.

One of the important mathematical skills, which pupils need to develop, is the ability of geometrical thinking. Pupils need stimulating activities, which enable them to solve geometric problems. Development of geometrical thinking takes time and therefore the pupils require of geometrical terms and their characteristics. An example might be various geometrical places of points or also sets of points of given property. It starts already at primary school, respectively in kindergarten, where there is a development of intuitive perception of the world around. We can show pupils, at the higher school levels, known geometrical objects as sets of points, even in less common situations. In this paper, we show at several geometrical places of points of conic sections, what are the possibilities for the development of pupil geometrical thinking, but especially how to use the tools of dynamic geometry effectively, for their own discovering and visualization and making them their discovery process easier and accelerates checking acquired knowledge.

The study deals with designing, executing and evaluationg of the research with pre-school children about their thinking about geometric figures. The work describes the concept of the research tool, the analysis and the interpretation of the results, in terms of the importance of the position of a plane figure, its shape and colour diversity. Position, colour and shape are all important atributes in the process of identyfing shapes of pre-school children. The interpretation of the results has been carried out in the context of the van Hiele´s levels of geometric thinking.

The article is focused on the study of geometrical models and their connection to the conceptions about fractions. We have used the fraction test according to Sfards´ theory of the reification, created by Marilena Pantziara and George Phillipou. The three chosen tasks have been solved by 113 university students of Pre-school and Elementary Education study program. We have investigated the ability of the students to solve the graduated tasks at the three hierarchical stages of the fraction conception and the usage of geometrical representations of the part of a whole.

The contribution deals with the application of manual and mental manipulations with objects in solving tasks, using geometrical imagination. The manual manipulation employs the task learning, when students (respectively pupils) help themselves in task solving by various models. It leads to confirmation and strengthening of their imaginations. The mental manipulation includes solving the task in the mind of the pupil, only without any model at all. When the imagination is not enough, the manual manipulation can be used for task solving. The contribution includes suggestions and tasks, which can support the development of geometrical and spatial imagination of pupils and students.

In recent years, a lot of space is given to the discussion about the teaching of geometry in lower educational levels. It emphasises the importance of geometry, not only in math education, but also in the general education of students. In this paper I present the preliminary results of research conducted among students of the first grade of primary school. Students helped Winnie the Pooh to lay a new floor with colorful tiles. This task was to show how the 6-7 year-old children cope with arranging the plane.

Researchers are of the common opinion that the formation of geometrical concepts is heading in a different direction than the concepts of arithmetic. Perception is recognised as the primary source for creating geometrical concepts. M. Hejný showed (2000) the importance of perceptual transfer in a pupil’s mind when they are grasping a processually perceived situation conceptually or conceptually perceived situation processually. And it is the latter of the two directions that is much more frequent in geometry than in arithmetic. Acts of perception are important but they are not a sufficient source of geometrical cognition. The problem of building dynamic geometrical reasoning can be analyzed in different ways. One of them, close to my interest, is a dynamic understanding of geometric transformations. Here I present some findings from the research among students at various educational levels.

According to the theory of Pierre and Dina van Hiele, students progress through levels of thought in geometry. Thinking develops from a Gestalt--like visual level through increasingly sophisticated levels of description, analysis, abstraction, and proof. Geometry and spatial reasoning are strongly interrelated, and most mathematics educators seem to include spatial reasoning as part of the geometry curriculum (Usiskin, 1982). The aim of the article is to present and analyze the results of the test tasks for pupils 9th class of primary school. Highlight the geometric misconception among students about quadrilaterals.

If we emit raylight onto a given curve from a certain direction, then the raylights reverberating from the points of the curve formulate an array of curves. The curve covered with this array of curves is called a caustic curve. We have met the caustic curves of some of the classical curves in secondary school. In this presentation the caustic curve of the most common free formed curve- the Bézier curve, and some of its geometric features are presented.

In our contribution, we focus on education supporting space and geometry imagination. We discuss the importance and necessity of dealing with these areas already at kindergarten school, when it is genetically proper period in children age for this development. In this way, we can prepare good basis for teaching geometry in primary and secondary education. Because of this, we prepared life-long education of kindergarten teachers dealing with space and geometry imagination. In the paper, we analyse teachers’ opinions on the tasks supporting cognitive process in geometry, both in the questionnaires, and in the final works prepared by teachers. This analysis showed us which tasks teachers consider as proper for supporting of cognitive process in geometry and also we have found out, that our life-long learning gives teachers inspirations and materials for using and creating such tasks.

This contribution is focused on the connection of modern technologies, approach CLIL and geometry lessons. Modern technologies have become a part of life for most of us. The current trend in the Czech Republic is an integration of tablets in education. But a lot of teachers lack applications in Czech language. In this situation, using of approach CLIL – integration of foreign language to non-language subject, can help. This contribution is about these possibilities and CLIL using together with tablets.