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• Analysis of a problem in plane geometry discussed in an 11th grade group study session

  181-193

  Richárd Rakamazi

  Views:

  35

  The 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

  159-173

  Andreas Asperl

  Views:

  44

  Using 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

  259-269

  Csaba Sárvári

  Views:

  13

  The 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

  29-65

  Éva Szeredi

  Judit Török

  Views:

  52

  This 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

  207-216

  Eleonóra Stettner

  György Emese

  Views:

  88

  Tamás 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

• Nice tiling, nice geometry!?!

  269-280

  Emil Molnár

  Views:

  46

  The 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

  35-50

  Andreas Fest

  Ulrich Kortenkamp

  Views:

  47

  Combinatorial 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.

• On four-dimensional crystallographic groups

  391-404

  Eszter Horváth

  Views:

  15

  In his paper [12] S. S. Ryshkov gave the group of integral automorphisms of some quadratic forms (according to Dade [6]). These groups can be considered as maximal point groups of some four-dimensional translation lattices in E^4. The maximal reflection group of each point group, its fundamental domain, then the reflection group in the whole symmetry group of the lattice and its fundamental domain will be discussed. This program will be carried out first on group T. G. Maxwell [9] raised the question whether group T was a reflection group. He conjectured that it was not. We proved that he had been right. We shall answer this question for other groups as well. Finally we shall give the location of the considered groups in the tables of monograph [4]. We hope that our elementary method will be useful in studying linear algebra and analytic geometry. Futhermore, 4-dimensional geometry with some visualisation helps in better understanding important concepts in higher-dimensional mathematics, in general.

• The development of geometrical concepts in lower primary mathematics teaching: the square and the rectangle

  153-171

  Ibolya Szilágyiné Szinger

  Views:

  58

  Our research question is how lower primary geometry teaching in Hungary, particularly the concept of squares and rectangles is related to the levels formulated by van Hiele. Moreover to what extent are the concrete activities carried out at these levels effective in evolving the concepts of squares and rectangles.
  In the lower primary geometry teaching (classes 1-4) the first two stages of the van Hiele levels can be put into practice. By the completion of lower primary classes level 3 cannot be reached. Although in this age the classes of concepts (rectangles, squares) are evolved, but there is not particular relationship between them. The relation of involvement is not really perceived by the children.

• The unity of mathematics: a casebook comprising practical geometry number theory and linear algebra

  1-34

  Peter Hilton

  Jean Pedersen

  Views:

  33

  We give a sustained example, drawn largely from earlier publications, of how we may freely pursue a line of mathematical enquiry if we are not constrained, unnaturally, to confine ourselves to a single mathematical subdiscipline; and we draw conclusions from the study of this example which are relevant at many levels of mathematical instruction.
  We also include the statement and proof of a new result (Theorem 4.1) in linear algebra which is obviously fundamental to the geometrical investigation which constitutes the leit-motif of the paper.

• A role of geometry in the frame of competencies attainment

  41-55

  Ivan Anić

  Neda Bokan

  Tijana Šukilović

  Srdjan Vukmirović

  Views:

  43

  We discuss aspects of the Education Reform from teaching to educational system. In this context we recognize some problems in recognition of some competencies that students need to achieve and we present how we have developed the measurement method of spatial abilities and problem solving competence. Especially, we investigate how students use spatial visualization abilities in solving various problems in other mathematical course. We have tested how students use their spatial abilities previously developed in geometry courses based on conceptual approach to solve a test based on procedural concept in Mathematical Analysis course.

• The background of students' performance

  295-305

  Rita Nagy-Kondor

  Views:

  39

  The question to which we were seeking was: how can we reveal the students' strategies and mental process by following their work precisely and by finding out what correlation these have with their efficiency. Our aim was to understand the factors behind of students' achievement. We tried to follow up the process of problem solving by looking at the number of wrong turnings.

• Ein ungewöhnlicher Weg zu Jakob Steiners Umellipse eines Dreiecks und zur Steiner–Hypozykloide

  49-65

  Oswald Giering

  Views:

  26

  In real projective geometry of triangles two problems of collinear points are discussed. The problems differ only from the running through the vertices of a given triangle ABC. Resolving the problems we find two cubic curves kS and kT . Affine specialization leads to the circumscribed Steiner ellipse about the triangle ABC and shows us this ellipse in more general surroundings. Euclidean specialization leads to Steiners three-cusped hypocycloid.

• Classical theorems on hyperbolic triangles from a projective point of view

  175-181

  Zoltán Szilasi

  Views:

  47

  Using the Cayley-Klein model of hyperbolic geometry and the tools of projective geometry, we present elementary proofs for the hyperbolic versions of some classical theorems on triangles. We show, in particular, that hyperbolic triangles have no Euler line.

• GeoGebra in mathematics teaching

  101-110

  Zsuzsanna Papp-Varga

  Views:

  54

  GeoGebra is a dynamic mathematics software which combines dynamic geometry and computer algebra systems into an easy-to-use package. Its marvel lies in the fact that it offers both the geometrical and algebraic representation of each mathematical object (points, lines etc.). The present article gives a sample of the potential uses of GeoGebra for mathematics teaching in secondary schools.

• Herschel's heritage and today's technology integration: a postulated parallel

  419-430

  Kenneth Ruthven

  Views:

  32

  During the early 20th century, advocacy of a range of mathematical technologies played a central part in movements for the reform of mathematical education which emphasised ‘practical mathematics' and the ‘mathematical laboratory'. However, as these movements faltered, few of the associated technologies were able to gain and maintain a place in school mathematics. One conspicuous exception was a technology, originally championed by the mathematician Herschel, which successfully permeated the school mathematics curriculum because of its:
  • Disciplinary congruence with influential contemporary trends in mathematics.
  • External currency in wider mathematical practice beyond the school.
  • Adoptive facility of incorporation in classroom practice and curricular activity.
  • Educational advantage of perceived benefits outweighing costs and concerns.
  An analogous perspective is applied to the situation of new technologies in school mathematics in the early 21st century. At a general level, the cases of calculators and computers are contrasted. At a more specific level, the educational prospects of CAS and DGS are assessed.

• On an international training of mathematically talented students: assets of the 20 years of the “Nagy Károly Mathematical Student-meetings”

  77-89

  Sándor Kántor

  Tünde Kántor

  Views:

  43

  The focus of this paper is to present the gems of the "Nagy Károly Mathematical Student-meetings" in Rév-Komárom (Slovakia) from 1991 to 2010. During these 20 years there was done a lot of work to train mathematically talented students with Hungarian mother tongue and to develop their mathematical thinking, and to teach them problem solving and heuristic strategies for successful acting on the competitions. We collected the most interesting problems and methods presented by the trainer teachers.

• What does ICT help and does not help?

  33-49

  Enikő Jakab

  Views:

  126

  Year by year, ICT tools and related teaching methods are evolving a lot. Since 2016, the author of the present lines has been looking for a connection between them that supports the development of mathematical competencies and could be integrated into Transcarpathian minority Hungarian language education too. As a doctoral student at the University of Debrecen, I experienced, for example, how the interactive whiteboard revolutionized illustration in Hungarian mathematics teaching, and how it facilitated students' involvement. During my research of teaching in this regard, in some cases, the digital solution had advantageous effects versus concrete-manipulative representation of
  Bruner's too.
  At the same time, ICT "canned" learning materials (videos, presentations, ...) allow for a shift towards repetitive learning instead of simultaneous active participation, which can be compensated for by the "retrieval-enhanced" learning method.
  I have conducted and intend to conduct several research projects in a Transcarpathian Hungarian primary school. In the research so far, I examined whether, in addition to the financial and infrastructural features of the Transcarpathian Hungarian school, the increased "ICT-supported" and the "retrieval-enhanced" learning method could be integrated into institutional mathematics education. I examined the use of two types of ICT devices: one was the interactive whiteboard, and the other was providing one computer per student.
  In this article, I describe my experiences, gained during one semester, in the class taught with the interactive whiteboard on the one hand, and in the class taught according to the "retrieval-enhanced" learning method on the other hand.
  I compare the effectiveness of the classes to their previous achievements, to each other, and to a class in Hungary.

  Subject Classification: 97U70

• Notes on the representational possibilities of projective quadrics in four dimensions

  167-177

  Sándor Bácsó

  Zoltán Szilasi

  Views:

  16

  The paper deals with hyper-quadrics in the real projective 4-space. According to [1] there exist 11 types of hypersurfaces of 2nd order, which can be represented by 'projective normal forms' with respect to a polar simplex as coordinate frame. By interpreting this frame as a Cartesian frame in the (projectively extended) Euclidean 4-space one will receive sort of Euclidean standard types of hyper-quadrics resp., hypersurfaces of 2nd order: the sphere as representative of hyper-ellipsoids, equilateral hyper-hyperboloids, and hyper-cones of revolution. It seems to be worthwhile to visualize the "typical" projective hyper-quadrics by means of descriptive geometry in the (projectively extended) Euclidean 4-space using Maurin's method [4] or the classical (skew) axonometric mapping of that 4-space into an image plane.

• Forming the concept of congruence II.

  1-12

  Éva Szeredi

  Views:

  46

  This paper is a continuation of the article Forming the concept of congruence I., where I gave theoretical background to the topic, description of the traditional method of representing the isometries of the plane with its effect on the evolution of congruence concept.
  In this paper I describe a new method of representing the isometries of the plane. This method is closer to the abstract idea of 3-dimensional motion. The planar isometries are considered as restrictions of 3-dimensional motions and these are represented with free translocations given by flags.
  About the terminology: I use some important concepts connected to teaching of congruence, which have to be distinguished. My goal is to analyse different teaching methods of the 2-dimensional congruencies. I use the term 3-dimensional motion for the orientation preserving (direct) 3-dimensional isometry (which is also called rigid motion or rigid body move). When referring the concrete manipulative representation of the planar congruencies I will use the term translocation.

• Über ähnliche Aufsatzdreiecke einer Strecke

  337-348

  Oswald Giering

  Views:

  32

  In this article we investigate (with methods of school geometry) a figure (PQ,ABC) consisting of three given similar triangles PQA, PQB, PQC with side PQ in common (Figure 1). We combine other triangles with this figure such as triangle ABC which is proved to be similar to the given triangles. The incircles of three additional triangles adjacent to triangle ABC will be determined.

• Forming the concept of congruence I.

  181-192

  Éva Szeredi

  Views:

  19

  Teaching isometries of the plane plays a major role in the formation of the congruence-concept in the Hungarian curricula.
  In the present paper I investigate the way the isometries of the plane are traditionally introduced in most of the textbooks, especially the influence of the representations on the congruence concept, created in the teaching process.
  I am going to publish a second part on this topic about a non-traditional approach (Forming the concept of congruence II). The main idea is to introduce the isometries of the two dimensional plane with the help of concrete, enactive experiences in the three dimensional space, using transparent paper as a legitimate enactive tool for building the concept of geometric motion. I will show that this is both in strict analogy with the axioms of 3-dimensional motion and at the same time close to the children's intuitive concept of congruence.

• Straight line or line segment? Students’ concepts and their thought processes

  327-336

  Vlasta Moravcová

  Jana Hromadová

  Views:

  122

  The article focuses on students’ understanding of the concept of a straight line. Attention is paid to whether students of various ages work with only part of a straight line shown or if they are aware that it can be extended. The presented results were obtained by a qualitative analysis of tests given to nearly 1,500 Czech students. The paper introduces the statistics of students’ solutions, and discusses the students’ thought processes. The results show that most of the tested students, even after completing upper secondary school, are not aware that a straight line can be extended. Finally, we present some recommendations for fostering the appropriate concept of a straight line in mathematics teaching.

  Subject Classification: 97C30, 97D70, 97G40

• Learning and teaching combinatorics with Sage

  389-398

  István Vajda

  Views:

  56

  Learning Mathematics is not an easy task, since this subject works with especially abstract concepts and sophisticated deductions. Many students lose their interest in the subject due to lack of success. Computer algebra systems (CAS) provide new ways of learning and teaching Mathematics. Numerous teachers use them to demonstrate concepts, deductions and algorithms and to make learning process more interesting especially in higher education. It is an even more efficient way to improve the learning process, if students can use the system themselves, which helps them to practice the curriculum.
  Sage is a free, open-source math software system that supports research and teaching algebra, analysis, geometry, number theory, cryptography, numerical computation, and related areas. I have been using it for several years to aid the instruction of Discrete Mathematics at Óbuda University. In this article I show some examples how representations provided by this system can help in teaching combinatorics.

• Besondere Punkte der Euler-Geraden

  145-157

  Oswald Giering

  Views:

  32

  In the following article the concepts "Euler line of a triangle" and "radical centre of three circles" are connected. In this way we could find some relations between special points of a triangle (orthocentre H, centre of gravity S, circumcentreM, midpoint F of the nine-point circle) and the radical centres of special triples of circles.