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• The hyperbola and Geogebra in high-school instruction

  277-285

  Eugen Ljajko

  Milan Mihajlović

  Zlatka Pavličić

  Views:

  35

  In this article the results of teaching/learning hyperbola and its characteristics in high-school using computers and GeoGebra are shown. Students involved in the research attend Engineering School "Nikola Tesla" in Leposavic, Serbia. The aim of the research was to define ways and volume of computer and GeoGebra usage in mathematics instruction in order to increase significantly students' mathematical knowledge and skills.

• "How to be well-connected?" An example for instructional process planning with Problem Graphs

  145-155

  Eszter Varga

  Views:

  96

  Teachers’ design capacity at work is in the focus of didactical research worldwide, and fostering this capacity is unarguably a possible turning point in the conveyance of mathematical knowledge. In Hungary, the tradition hallmarked by Tamás Varga is particularly demanding towards teachers as they are supposed to be able to plan their long-term processes very carefully. In this contribution, an extensive teaching material designed in the spirit of this tradition will be presented from the field of Geometry. For exposing its inner structure, a representational tool, the Problem Graph is introduced. The paper aims to demonstrate that this tool has potential for analyzing existing resources, helping teachers to reflect on their own preparatory and classroom work, and supporting the creation of new designs.

  Subject Classification: 97D40, 97D50, 97D80, 97G10, 97U30

• Report of meeting Researches in Didactics of Mathematics and Computer Sciences: January 20 - January 22, 2012, Levoča, Slovakia

  205-230

  E. Herendiné Kónya

  Views:

  27

  The meeting Researches in Didactics of Mathematics and Computer Sciences was held in Levoca, Slovakia from the 20th to the 22th of January, 2012. The 66 participants – including 54 lecturers and 25 PhD students – came from 6 countries, 20 cities and represented 33 institutions of higher and secondary education. The abstract of the talks and the posters and also the list of participants are presented in this report.

• The "Teaching Mathematics and Computer Science" Journal logo's mathematical background

  55-65

  Róbert Oláh-Gál

  Júlia Salamon

  Views:

  3

  In the present contribution we give an elementary technology for drawing the geodesics, paracycles and hypercycles on the pseudosphere.

• Reflecting upon reflections

  1-12

  H. Zeitler

  D. Camp

  Views:

  8

  This paper considers many applications of reflections in geometry. It begins with a few motivational problems for the classroom and goes on to consider the formal application to cases involving reflections across one line, two lines and three lines. It wraps up with a summary of results for reflections in higher orders.
  All this stuff was treated in German and American schools too – so the paper is a typical example of German-American didactics.
  "Thinking is one of the greatest pleasure of mankind." – Galileo Galilei

• Report on the "English Language Section of Varga Tamás Days 2009"

  169-175

  Krisztina Barczi

  Views:

  31

  The 9th English Language Section as a part of the Varga Tamás Days was organised by the Department of Mathematics Education at the Teacher Training Institute of the Eötvös Loránd University. We report on the talks and the following discussions in this section.

• Is it possible to develop some elements of metacognition in a Mathematics classroom environment?

  123-132

  Márton Kiss

  Eszter Kónya

  Views:

  89

  In an earlier exploratory survey, we investigated the metacognitive activities of 9th grade students, and found that they have only limited experience in the “looking back” phase of the problem solving process. This paper presents the results of a teaching experiment focusing on ninth-grade students’ metacognitive activities in the process of solving several open-ended geometry problems. We conclude that promoting students’ metacognitive abilities makes their problem solving process more effective.

  Subject Classification: 97D50, 97G40

• Decomposition of triangles into isosceles triangles II: complete solution of the problem by using a computer

  275-300

  József Kosztolányi

  Zoltán Kovács

  Erzsébet Nagy

  Views:

  29

  We solve an open decomposition problem in elementary geometry using pure mathematics and a computer programme, utilizing a computer algebra system.

• Thoughts on Pólya’s legacy

  157-160

  Alan Schoenfeld

  Views:

  132

  There is a saying, "the older I get, the smarter my parents become." What it means, of course, is that the more we learn, the more we appreciate the wisdom of our forebears. For me, that is certainly the case with regard to George Pólya.

  There is no need to elaborate on Pólya's contributions to mathematics – he was one of the greats. See, for example, Gerald Alexanderson's (2000) edited volume The Random Walks of George Pólya, or Pólya's extended obituary (really, a
  53-page homage) in the November 1987 Bulletin of the London Mathematical Society (Chung et al., 1987). Pólya was one of the most important classical analysts of the 20th century, with his influence extending into number theory, geometry, probability and combinatorics.

• The appearance of the characteristic features of the mathematical thinking in the thinking of a chess player

  201-211

  Anita Misetáné Burján

  Views:

  34

  It is more and more important in 21st century's education that not only facts and subject knowledge should be taught but also the ways and methods of thinking should be learnt by students. Thinking is a human specificity which is significant both in mathematics and chess. The exercises aimed at beginner chess players are appropriate to demonstrate to students the mathematical thinking of 12-14 year-old students.
  Playing chess is an abstract activity. During the game we use abstract concepts (e.g. sacrifice, stalemate). When solving a chess problem we use logical quantifiers frequently (e.g. in the case of any move of white, black has a move that...). Among the endgames we find many examples (e.g. exceptional draw options) that state impossibility. Affirmation of existence is frequent in a mate position with many moves. We know there is a mate but the question in these cases is how it can be delivered.
  We present the chess problem on beginners' level although these exercises appear in the game of advanced players and chess masters too, in a more complex form. We chose the mathematical tasks from arithmetic, number theory, geometry and the topic of equations. Students encounter these in classes, admission exams and student circles. Revealing the common features of mathematical and chess thinking shows how we can help the development of students' mathematical skills with the education of chess.

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

  419-430

  Kenneth Ruthven

  Views:

  26

  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:

  33

  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:

  114

  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:

  12

  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:

  31

  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.