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Geometry expressions: an interactive constraint based symbolic geometry system
303-310Views:25Dynamic geometry systems such as Geometers' SketchPad or Cabri are productive environments for the exploration of geometric relationships. They are, however, strictly numeric, and this limits their applicability where the interplay between geometry and algebra are being studied. We present Geometry Expressions – a dynamic symbolic geometry environment. While retaining the ease of use of a typical dynamic geometry environment, Geometry Expressions diverges by using constraints rather than constructions as the primary geometry specification mechanism and by working symbolically rather than numerically. Constraints, such as distances and angles, are specified symbolically. Symbolic measurements for quantities such as distances, angles, areas, locus equations, are automatically computed by the system. We outline how these features combine to create a rich dynamic environment for exploring the interplay between geometry and algebra, between induction and proof. -
Applications of methods of descriptive geometry in solving ordinary geometric problems
103-115Views:31The importance of descriptive geometry is well-known in two fields. Spatial objects can be mapped bijectively onto a plane and then we can make constructions concerning the spatial objects. The other significance of descriptive geometry is that mathematical visual perception of objects in three-dimensional space can be improved by the aid of it. The topic of this paper is an unusual application of descriptive geometry. We may come across many geometric problems in mathematical competitions, in entrance examinations and in exercise books whose solution is expected in a classical way, however, the solution can be found more easily and many times more general than it is by the standard manner. We demonstrate some of these problems to encourage to use this geometric method. Understanding the solution requires very little knowledge of descriptive geometry, however, finding a solution needs to have some idea of descriptive geometry. -
The effect of augmented reality assisted geometry instruction on students' achiveement and attitudes
177-193Views:60In this study, geometry instruction's academic success for the students and their attitudes towards mathematics which is supported by education materials of Augmented Reality (AR) and its effect on the acceptance of AR and its usage by teachers and students have been researched. Under this research, ARGE3D software has been developed by using augmented reality technology as for the issue of geometric objects that is contained in the mathematics curriculum of 6th class of primary education. It has been provided with this software that three-dimensional static drawings can be displayed in a dynamic and interactive way. The research was conducted in two different schools by an experiment and control group. In the process of data collection, Geometry Achievement Test (GAT), Geometric Reasoning Test (GRT), Attitudes Scale for Mathematics (ASM), students' math lecture notes, semi-structured interviews with teachers and students and observation and video recordings were used. Results showed that geometry instruction with ARGE3D increased students' academic success. In addition, it was found that geometry instruction with ARGE3D became more effective on students' attitudes that had negative attitudes towards mathematics and it also provided support to reduce fear and anxiety. -
Outstanding mathematicians in the 20th century: András Rapcsák (1914-1993)
99-110Views:27In this paper we commemorate the life and work of András Rapcsák on the occasion of the centenary of his birth. He was an outstanding professor and a scholar teacher. He was head of the Department of Geometry (1958-1973) and the director of the Institute of Mathematics at the University of Debrecen (Hungary). He played an important role in the life of the University of Debrecen. He was the rector of this university between 1966 and 1973.
At the beginning of his career he taught at secondary schools in several towns. He wrote mathematical schoolbooks with coauthors. He also taught at Teacher's College in Debrecen and in Eger.
He became to interested in differential geometry under the influence of Ottó Varga. The fields of his research were line-element spaces and related areas. He was elected an Ordinary Member of the Hungarian Academy of Science in 1965. He wrote 21 papers, 8 school and textbooks and 3 articles in didactics of mathematics. -
Hungarian mathematicians in the twentieth century: Ottó Varga (1909–1969)
109-120Views:30In this article we want to present life and work of Ottó Varga on occasion of the centenary of his birth. He was an outstanding geometer, the head of the Mathematical Seminar / Department (1942-1959) and the first dean of the Faculty of Sciences of the University of Debrecen (Hungary).
His area of research was differential geometry. The Debrecen school of differential geometry emerged due his activity. He wrote 57 papers. -
Packings in hyperbolic geometry
209-229Views:22I am becoming older. That's why I am returning to my youth sins. "On revient toujours á ses premiers amoures". This sin was the noneuclidean hyperbolic geometry – especially the Poincaré model. I was teaching this kind of geometry over many years as well in highschool (Gymnasium) as for beginners at the university too.
A lot of results concerning packings in hyperbolic geometry are proved by the Hungarian school around László Fejes Tóth. In this paper we construct very special packings and investigate the corresponding densities. For better understanding we are working in the Poincaré model. At first we give a packing of the hyperbolic plane with horodisks and calculate the density. In an analogous way then the hyperbolic space is packed by horoballs. In the last case the calculation of the density is a little bit difficult. Finally it turns out that in both cases the maximal density is reached. -
Hyperbolische 5-Rechtecke
111-123Views:25The main topic of this paper is the investigation of 5-pentagons whose interior angles are all right angles within the hyperbolic geometry (so-called 5-rectangles). Some knowledge of elementary hyperbolic geometry is required.
At first the existence of such a polygon is shown by construction within the Kleinmodel. Then two formulas due to D. M. Y. Sommerville [3] are proved. This means to juggle with trigonometric formulas of hyperbolic geometry.
In the last years a big number of papers concerning hyperbolic geometry was published. This proves that the interest in this nice discipline is growing again. -
Dynamic geometry systems in teaching geometry
67-80Views:30Computer drawing programs opened up new opportunities in the teaching of geometry: they make it possible to create a multitude of drawings quickly, accurately and with flexibly changing the input data, and thus make the discovery of geometry an easier process. The objective of this paper is to demonstrate the application possibilities of dynamic geometric systems in primary and secondary schools, as well as in distance education. A general characteristic feature of these systems is that they store the steps of the construction, and can also execute those steps after a change is made to the input data. For the demonstration of the applications, we chose the Cinderella program. We had an opportunity to test some parts of the present paper in an eighth grade primary school. -
Visualisation in geometry education as a tool for teaching with better understanding
337-346Views:165In primary and secondary geometry education, some problems exist with pupils’ space thinking and understanding of geometric notions. Visualisation plays an important role in geometry education, and the development of pupils’ visualisation skills can support their spatial imagination. The authors present their own thoughts on the potential of including visualisation in geometry education, based on the analysis of the Hungarian National Core Curriculum and Slovak National Curriculum. Tasks for visualisation are also found in international studies, for example the Programme for International Student Assessment (PISA). Augmented reality (AR) and other information and communication technology (ICT) tools bring new possibilities to develop geometric thinking and space imagination, and they also support mathematics education with better understanding.
Subject Classification: 97U10, 97G10
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Compositions of dilations and isometries in calculator-based dynamic geometry
257-266Views:32In an exploratory study pre-service elementary school teachers constructed dilations and isometries for figures drawn and transformed using dynamic geometry on calculators. Observational and self assessments of the constructed images showed that the future teachers developed high levels of confidence in their abilities to construct compositions of the geometric transformations. Scores on follow-up assessment items indicated that the prospective teachers' levels of expertise corresponded to their levels of confidence. Conclusions indicated that dynamic geometry on the calculator was an appropriate technology, but one that required careful planning, to develop these future teachers' expertise with the compositions. -
Miscellaneous topics in finite geometry: in memory of Professor Dr. Ferenc Kárteszi (1907-1989)
255-275Views:33The article starts with a short introduction to finite (K,L)-geometry. Then a lot of counting propositions is given and proved. Finally the famous theorem of Miquel is investigated in classical and in finite geometry. At the end of the article there is a call to all readers: Don't forget (finite) geometry and don't forget the outstanding geometer Prof. Dr. F. Kárteszi! -
Dynamic methods in teaching geometry at different levels
1-13Views:37In this paper we summarize and illustrate our experiences on DGS-aided teaching geometry of the courses "Computer in mathematics" and "Mathematical software" held for students at Juhász Gyula Teacher Training College of University of Szeged. Furthermore, we show examples from our grammar school experiences too. The figures in this paper were made by using Cinderella ([19]) and Euklides ([21]). -
Béla Kerékjártó: (a biographical sketch)
231-263Views:30Kerékjártó published more than 70 scientific papers mainly in the field of topology. He achieved his most important results in the classical transformation topology and in the theoretical research of the continuous groups. He was the author of three books: Vorlesungen über Topologie; Euclidean geometry; Study on the projective geometry. -
Teaching geometry using computer visualizations
259-277Views:20In this work we study the development of students' creativity using computer-aided-teaching during IT classroom. Teaching geometry in Bolyai Grammar School specialized natural science classes is not an easy task. Here is introduced a new didactic means of teaching geometry which nevertheless requires the same effort to understand the material, but uses a different more active method to familiarize students with the topics. Traditional methods, and the use of compasses and rulers are not omitted either, as they develop the students' motor skills. -
On an analogy between spreadsheets and dynamic geometry environments
281-288Views:31There is a strong analogy between the fundamental way of operation of spreadsheet programs (SP) and dynamics geometry environments (DGE). We explain this analogy, demonstrate it in examples and consider didactical consequences. -
Comparative geometry on plane and sphere: didactical impressions
81-101Views:4Description of experiences in teaching comparative geometry for prospective teachers of primary schools. We focus on examples that refer to changes in our students' thinking, in their mathematical knowledge and their learning and teaching attitudes. At the beginning, we expected from our students familiarity with the basics of the geographic coordinate system, such as North and South Poles, Equator, latitudes and longitudes. Spherical trigonometry was not dealt with in the whole project. -
Different approaches of interplay between experimentation and theoretical consideration in dynamic geometry exploration: An example from exploring Simson line
63-81Views:31Dynamic geometry environment (DGE) is a powerful tool for exploration and discovering geometric properties because it allows users to (virtually) manipulate geometric objects. There are two possible components in the process of exploration in DGE, viz. experimentation and theoretical consideration. In most cases, there is interplay between these two components. Different people may use DGE differently. Depending on the specific mathematical tasks and the background of individual users, some approaches of interplay are more experimental whereas some other approaches of interplay are more theoretical. In this paper, different approaches of exploring a geometric task using Sketchpad (a DGE) by three individual participants will be discussed. They represent three different approaches of interplay between experimentation and theoretical consid- eration. An understanding of these approaches may contribute to an understanding on the mechanism of exploration in DGE. -
Central axonometry in engineer training and engineering practice
17-28Views:22This paper is concerned with showing a unified approach for teaching central and parallel projections of the space to the plane giving special emphasis to engineer training. The basis for unification is provided by the analogies between central axonometry and parallel axonometry. Since the concept of central axonometry is not widely known in engineering practice it is necessary to introduce it during the education phase. When teaching axonometries dynamic geometry software can also be used in an interactive way. We shall provide a method to demonstrate the basic constructions of various axonometries and use these computer applications to highlight their similarities. Our paper sheds light on the advantages of a unified approach in such areas of engineering practice as making hand drawn plans and using CAD-systems. -
Analysis of a problem in plane geometry discussed in an 11th grade group study session
181-193Views:27The main aim of this paper is to show those strategies and proof methods we try to teach in secondary maths education through an interesting geometric problem: Find a relation for the sides of a triangle where an angle is the double of another angle. Is the converse also true? Is it possible to generalize the problem? We try to answer these questions while discussing the upcoming difficulties in detail and presenting more possible solutions. Hopefully the paper can be successfully used in study group sessions and problem solving seminars in secondary schools. -
Eine geometrische Interpretation der Ausgleichsrechnung
159-173Views:27Using real examples of applied mathematics in upper secondary school one has do deal with inaccurate measures. This will lead to over constrained systems of linear equations. This paper shows an instructive approach which uses methods of descriptive and computer aided geometry to get a deeper insight into the area of calculus of observations. Using a qualified interpretation one can solve problems of calculus of observations with elementary construction techniques of descriptive geometry, independent of the norm one uses. -
Report on "The Computer Algebra and Dynamical Geometry Systems, as the catalysts of the Mathematics education": Conference, 6-7 June, 2003, Pécs, Hungary
259-269Views:10The Department of Mathematics of the University of Pécs, Pollack Mihály Engineering Faculty organized in the year 2003 a conference on the role of CAS and DGS in the Mathematics education. We discuss – based on the authors' abstracts – the conference's activities. -
Teaching polygons in the secondary school: a four country comparative study
29-65Views:38This study presents the analysis of four sequences of videotaped lessons on polygons in lower secondary schools (grades 7 and 8) taught by four different teachers in four different countries (Belgium, Flanders, England, Hungary and Spain). Our study is a part of the METE project (Mathematics Educational Traditions in Europe). The aims and methodology of the project are described briefly in the introduction. In the next section of this paper we describe various perspectives on teaching and learning polygons which were derived from the literature, concerning the objectives, conceptual aspects and didactic tools of the topic. The next two sections introduce the main outcomes of our study, a quantitative analysis of the collected data and a qualitative description linked to the perspectives on teaching polygons. We conclude by discussing some principal ideas related to the theoretical and educational significance of this research work. -
The tools for developing a spatial geometric approach
207-216Views:74Tamás Varga writes about the use of tools: "The rational use of tools - the colored bars, the Dienes set, the logical set, the geoboard, and some other tools - is an element of our experiment that is important for all students, but especially for disadvantaged learners." (Varga T. 1977) The range of tools that can be used well in teaching has grown significantly over the years. This paper compares spatial geometric modeling kits. Tamás Varga uses the possibilities of the Babylon building set available in Hungary in the 1970s, collects space and flat geometry problems for this (Varga T. 1973). Similarly, structured kits with significantly more options have been developed later, e.g. ZomeTool and 4D Frame. These tools are regularly used in the programs of the International Experience Workshop (http://www.elmenymuhely.-hu/?lang=en). Teachers, schools that have become familiar with the versatile possibilities of these sets, use them often in the optional and regular classes. We recorded a lesson on video where secondary students worked with the 4D Frame kit. We make some comments and offer some thoughts on this lesson.
Subject Classification: 97G40, 97D40
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Nice tiling, nice geometry!?!
269-280Views:38The squared papers in our booklets, or the squared (maybe black and white) pavements in the streets arise an amusing problem: How to deform the side segments of the square pattern, so that the side lines further remain equal (congruent) to each other? More precisely, we require that each congruent transformation of the new pattern, mapping any deformed side segment onto another one, leaves the whole (infinitely extended) pattern invariant (unchanged).
It turns out that there are exactly 14 types of such edge-transitive (or so-called isotoxal) quadrangle tilings, sometimes with two different forms (e.g. black and white) of quadrangles (see Figure 2). Such a collection of tiling can be very nice, perhaps also useful for decorative pavements in streets, in flats, etc.
I shall sketch the solution of the problem that leads to fine (and important) mathematical concepts (as barycentric triangulation of a polygonal tiling, adjacency operations, adjacency matrix, symmetry group of a tiling, D-symbol, etc). All these can be discussed in an enjoyable way, e.g. in a special mathematical circle of a secondary school, or in more elementary form as visually attractive figures in a primary school as well.
My colleague, István Prok [11] developed an attractive computer program on the Euclidean plane crystallographic groups with a nice interactive play (for free download), see our Figures 3-5.
A complete classification of such Euclidean plane tilings (not only with quadrangles) can be interesting for university students as well, hopefully also for the Reader (Audience). This is why I shall give some references, where you find also other ones.
Further problems indicate the efficiency of this theory now. All these demonstrate the usual procedure of mathematics and the (teaching) methodology as well: We start with a concrete problem, then extend it further, step-by-step by creating new manipulations, concepts and methods. So we get a theory at certain abstraction level. Then newer problems arise, etc.
This paper is an extended version of the presentation and the conference paper [7]. The author thanks the Organizers, especially their head Professor Margita Pavlekovic for the invitation, support and for the kind atmosphere of the conference. -
Teaching graph algorithms with Visage
35-50Views:29Combinatorial optimization is a substantial pool for teaching authentic mathematics. Studying topics in combinatorial optimization practice different mathematical skills, and because of this have been integrated into the new Berlin curriculum for secondary schools. In addition, teachers are encouraged to use adequate teaching software. The presented software package "Visage" is a visualization tool for graph algorithms. Using the intuitive user interface of an interactive geometry system (Cinderella), graphs and networks can be drawn very easily and different textbook algorithms can be visualized on the graphs. An authoring tool for interactive worksheets and the usage of the build-in programming interface offer new ways for teaching graphs and algorithms in a classroom.