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• Integral part problems derived from a solution of an in mum problem

  43-53

  Paweł Pasteczka

  Árpád Száz

  Views:

  3

  In this paper, we solve the following two integral part problems:
  Find all r ϵ R satisfying r^2 = [r]*([r]+1), resp. r^2≤[r]*([r]+1).
  These problems have been mainly motivated by a solution of an infimum problem of Z. Boros and Á. Száz.

• "On the way" to the function concept - experiences of a teaching experiment

  17-39

  Gyöngyi Szanyi

  Views:

  67

  Knowing, comprehending and applying the function concept is essential not only from the aspect of dealing with mathematics but with several scientific fields such as engineering. Since most mathematical notions cannot be acquired in one step (Vinner, 1983) the development of the function concept is a long process, either. One of the goals of the process is evolving an "ideal" concept image (the image is interrelated with the definition of the concept). Such concept image plays an important role in solving problems of engineering. This study reports on the beginning of a research aiming the scholastic forming of the students' function concept image i.e. on the experiences of a "pilot" study. By the experiment, we are looking for the answer of the following question: how can the analysis of such function relations be built into the studied period (8th grade) of the evolving process of the function concept that students meet in everyday life and also in engineering life?

  Subject Classification: D43, U73

• Some problems of solving linear equation with fractions

  339-351

  Gordana Stankov

  Views:

  10

  The aim of this paper is to offer some possible ways of solving linear equations, using manipulative tools, in which the "−" sign is found in front of an algebraic fraction which has a binomial as a numerator. It is used at 8th grade.

• Frequency-based dynamic models for the analysis of English and Hungarian literary works and coursebooks for English as a second language

  53-70

  Mária Csernoch

  Views:

  12

  We examined the characteristics of how word types are introduced in English and Hungarian literary works as well as in English coursebooks written for second language learners. By subdividing the texts into small segments of equal length, we were able to pinpoint subtle changes in the narrative. Based on the frequency of the word types in the original text a model was generated, and applying the model artificial texts were created. By comparing the original and the artificial texts, the places where these changes within the narrative occurred, could be located. Studying coursebooks we found that their vocabulary and how they introduce word types resembled those of randomly collected and concatenated short stories. According to our observations writers of the coursebooks forget that not only should the number of word types be carefully planned, but their repetition, in sufficient number, should also be cared for.

• Some thoughts concerning power sums

  303-308

  Gábor Nyul

  Views:

  4

  In this note we present an elementary way to derive directly closed-form expressions for power sums. Applying this method, we deduce some general results on power sums with arbitrary exponents. Finally, we give an outlook on higher mathematical connections between power sums, Stirling and Bernoulli numbers.

• Dynamic geometry systems in teaching geometry

  67-80

  Rita Nagy-Kondor

  Views:

  12

  Computer 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.

• What can we learn from Tamás Varga’s work regarding the arithmetic-algebra transition?

  39-50

  Gyöngyi Szanyi

  Zsuzsanna Jánvári

  Views:

  67

  Tamás Varga’s Complex Mathematics Education program plays an important role in Hungarian mathematics education. In this program, attention is given to the continuous “movement” between concrete and abstract levels. In the process of transition from arithmetic to algebra, the learner moves from a concrete level to a more abstract level. In our research, we aim to track the transition process from arithmetic to algebra by studying the 5-8-grader textbooks and teacher manuals edited under Tamás Varga's supervision. For this, we use the appearance of “working backward” and “use an equation” heuristic strategies in the examined textbooks and manuals, which play a central role in the mentioned process.

  Subject Classification: 97-01, 97-03, 97D50

• The theory of functional equations in high school education

  345-360

  Dorottya Bessenyei

  Views:

  12

  In this paper, we are going to discuss some possible applications of the theory of functional equations in high school education. We would like to line up some problems, the solution of which by functional equations are mostly not new results – they have also been treated in [1] and [2] –, although their demonstrations in high school can show a new way in teaching of talented students. The area of the rectangle, the calculating method of compound interest, binomial coefficients, Euler's formula, the scalar product and the vector product of vectors – we are looking for the reasons behind the well-known formulas. Finally, we are going to give a functional equation in connection with mean values. It can be understood easily, but its solution is beyond the high school curriculum, so we advise this part only to the most talented students.

• Linear clause generation by Tableaux and DAGs

  109-118

  Gergely Kovásznai

  Views:

  10

  Clause generation is a preliminary step in theorem proving since most of the state-of-the-art theorem proving methods act on clause sets. Several clause generating algorithms are known. Most of them rewrite a formula according to well-known logical equivalences, thus they are quite complicated and produce not very understandable information on their functioning for humans. There are other methods that can be considered as ones based on tableaux, but only in propositional logic. In this paper, we propose a new method for clause generation in first-order logic. Since it inherits rules from analytic tableaux, analytic dual tableaux, and free-variable tableaux, this method is called clause generating tableaux (CGT). All of the known clause generating algorithms are exponential, so is CGT. However, by switching to directed acyclic graphs (DAGs) from trees, we propose a linear CGT method. Another advantageous feature is the detection of valid clauses only by the closing of CGT branches. Last but not least, CGT generates a graph as output, which is visual and easy-to-understand. Thus, CGT can also be used in teaching logic and theorem proving.

• The use of e-tests in education as a tool for retrieval practice and motivation

  59-76

  Gabriella Papp

  Views:

  67

  In many studies we can read about what techniques are used in the educational process to deepen knowledge, and what can motivate students to learn. We aimed to give our students (who will be a teacher) a practical demonstration of learning techniques. We carried it within the framework of a course, at the end of which we also examined how much it motivates students if they write an e-test as a retrospective in order to deepen the material of the lesson. In the paper, we will present the results of the research as well as students’ opinions regarding the motivating effect of the tests.

  Subject Classification: 97-01, 97D40, 97I10

• Using the computer to visualise graph-oriented problems

  15-32

  Erika Gyöngyösi

  Views:

  9

  The computer, if used more effectively, could bring advances that would improve mathematical education dramatically, not least with its ability to calculate quickly and display moving graphics. There is a gap between research results of the enthusiastic innovators in the field of information technology and the current weak integration of the use of computers into mathematics teaching.
  This paper examines what exactly the real potentials of using some mathematics computer software are to support mathematics teaching and learning in graph-oriented problems, more specifically we try to estimate the value added impact of computer use in the mathematics learning process.
  While electronic computation has been used by mathematicians for five decades, it has been in the hands of teachers and learners for at most three decades but the real breakthrough of decentralised and personalised micro-computer-based computing has been widely available for less than two decades. And it is the latter facility that has brought the greatest promise for computers in mathematics education. That computational aids overall do a better job of holding students' mathematical interest and challenging them to use their intellectual power to mathematical achievement than do traditional static media is unquestionable. The real question needing investigation concerns the circumstances where each is appropriate.
  A case study enabled a specification of advantages and obstacles of using computers in graph-oriented questions. Individual students' interviews revealed two less able students' reactions, difficulties and misinterpretations while using computers in mathematics learning.
  Among research outcomes is that the mathematical achievement of the two students observed improved and this makes teaching with computers an overriding priority for each defined teaching method.
  This paper may not have been realised without the valuable help of the Hungarian Eötvös State Grant.

• Development of high school students' geometric thinking with particular emphasis on mathematically talented students

  93-110

  Ákos Győry

  Eszter Kónya

  Views:

  4

  We carried out research using Zalman Usiskin's test (1982) and also a modified version of his test to see how the geometric approach of secondary school students (Grades 8-10) specialized in mathematics had changed. We observed two groups of students for several years. Our aim was to find a relation between the change of the mean of the van Hiele level of the students and the structure of the geometry syllabus. We also observed if there was a change in the geometric approach of the students during the summer holidays and if so, in what way it changed.

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

  205-230

  E. Herendiné Kónya

  Views:

  14

  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.

• Report of the conference "Connecting Tamás Varga’s Legacy and Current Research in Mathematics Education": November 6-8, 2019, Budapest, Hungary

  5-8

  Michèle Artigue

  Katalin Gosztonyi

  Eszter Kónya

  Ödön Vancsó

  Views:

  73

  On the occasion of the 100th anniversary of the birth of the Hungarian mathematics educator, didactician and reform leader Tamás Varga, a conference on mathematics education has been organized in November 2019 and held at the Hungarian Academy of Science.

• Decision based examination of object-oriented programming and Design Patterns

  83-109

  Szabolcs Márien

  Views:

  6

  On the basis of our examination experience of Design Patterns the existing interpretations and descriptions of Design Patterns do not realise a clear and understandable answer for their aims. The reason for this is that the existing interpretation of the object-oriented paradigms is used for their description and formulation. In order that clear answers could be found for the aims of using Design Patterns, a new conception of their interpretation has to be established. In order to create a new conception, we have to analyze object-oriented paradigms.
  According to our new conception the object-oriented methodology is based on the elimination of decision repetition, thus sorting the decisions to class hierarchy, with the help of which the data structure and methodology of decision options can be determined by the subclasses of the given class. Sorting the decisions and decision options to a class and its subclasses only the first decision case will be executed, which will be archived and enclosed by instantiation of one of the subclasses. For the following decision cases the archived decision result can be used without knowledge of which decision option was used, so to say which subclass was instantiated, because it is enclosed by using the type of the parent class.
  The aim of the object-oriented technology is the elimination of decision repetition, which can be realized by sorting the decisions. The derivations are the abstract definitions of decisions, so the derivations can be interpreted as decision abstractions. The Design Patterns offer recipes for sorting the decisions. With the help of the decision concept the aim of Design Patterns can be cleared and a more natural classification of Design Patterns can be realized.

• The background of students' performance

  295-305

  Rita Nagy-Kondor

  Views:

  14

  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.

• Writing a textbook – as we do it

  185-201

  István Czeglédy

  András Kovács

  Views:

  5

  Recent surveys studying mathematics teaching show that there is a great variety in the level of mathematics teaching in Hungary. To increase efficiency (and decrease differences between schools) it is essential to create textbooks with new attitudes. The experiment we started after the PISA survey of 2000, produced a textbook that is new, in some sense even unusual in its attitude and methods. This paper presents the experiences we gained in the course of this work.

• The formation of area concept with the help of manipulative activities

  121-139

  Eszter Herendiné-Kónya

  Views:

  10

  Examining the performance of Hungarian students of Grades 4-12 in connection with area measurement, we found many deficiencies and thinking failures. In the light of this background, it seems reasonable to review the educational practice and to identify those teaching movements that trigger the explored problems and to design a teaching experiment that tries to avoid and exclude them. Based on result we make recommendations for the broad teaching practice. In our study we report on one part of a multi-stage teaching experiment in which we dealt with the comparison of the areas of figures, the decomposition of figures and the special role of the rectangle in the process of area concept formation. The conclusion of the post-test is that manipulative activities are important and necessary in Grades 5 and 6, more types of equidecomposition activities are needed and the number of measuring tasks with grid as a tool should also be increased.

• Examining relation between talent and competence through an experiment among 11th grade students

  17-34

  Veronika Árendás

  Views:

  12

  The areas of competencies that are formable, that are to be formed and developed by teaching mathematics are well-usable in recognizing talent. We can examine the competencies of a student, we can examine the competencies required to solve a certain exercise, or what competencies an exercise improves.
  I studied two exercises of a test taken by students of the IT specialty segment of class 11.d of Jedlik Ányos High School, a class that I teach. These exercises were parts of the thematic unit of Combinatorics and Graph Theory. I analysed what competencies a gifted student has, and what competencies I need to improve while teaching mathematics. I summarized my experience about the solutions of the students, the ways I can take care of the gifted students, and what to do to the less gifted ones.

• ICT methodology

  3-24

  László Zsakó

  Katalin Juhász

  Zoltán Kátai

  Views:

  13

  This article is an overview discussing what ICT subject methodology, ICT didactics and ICT methodology are, what the philosophy of ICT is, and finally what ICT pedagogy is.

• Application of a color education software to improve color aptitude

  267-285

  Erika Perge

  Views:

  2

  The ability to handle colors smoothly and consciously may be vital to professionals in various fields, including engineers specializing in architecture or design. Education in sciences related to colors and a developed color aptitude are essential. In our experience, many secondary school graduates in Hungary exhibit inadequate competence and need to be trained in both fields by college instructors, thereby laying the foundations for their future professional work. In our paper, we introduce a computer-based method to teach color theory using a self-developed interactive educational software. We also demonstrate the results of a test measuring the efficiency of the software. Our method was shown to be capable of familiarizing students with the basic fields of visual computing, e.g. graphics and image processing.

• Mathematics in Good Will Hunting II: problems from the students perspective

  3-19

  Gábor Horváth

  József Korándi

  Csaba Szabó

  Views:

  7

  This is the second part of a three paper long series exploring the role of mathematicians and of the mathematical content occurring in popular media. In particular we analyze the drama film Good Will Hunting. Here we investigate the mathematical content of the movie by considering the problems appearing in it. We examine how a mathematician or a mathematics student would solve these problems. Moreover, we review how these problems could be integrated into the higher education of Hungary.

• Die Methode von Prof. Tibor Szele im Unterricht begabter Schüler

  143-151

  Sándor Kántor

  Views:

  8

  Prof. Tibor Szele' has attempted to develop the mathematical problemsolving, creativity include the use of investigations and host of other devices beyond the classroom, i.e. in "mathematical circles" for talented students in secondary schools. This paper of the author – who himself has taken part in Seles1s mathematical circles – quotes from these activities according his earlier notes. This description illustrates the didactic method of Prof. T. Szele.

• Our duties in talent management in the light of the results of the International Hungarian Mathematics Competition of 2017

  55-71

  Erika Veres

  Katalin Fried

  Views:

  8

  The 4th International Hungarian Mathematics Competition held in Transcarpathia, Beregszász between April 28 and May 1, 2017, was organized by the Hungarian Carpathian Hungarian Teachers' Association (KMPSZ) and the Ferenc Rákóczi II. Transcarpathian Hungarian Institute (II. RFKMF).
  The venue for the competition was the building of the Ferenc Rákóczi II. Transcarpathian Hungarian Institute. 175 students participated in the competition from Hungary, Romania, Serbia, Slovakia and Transcarpathia.
  In this article, we are going to deal with the problems given in the two rounds to students in grades 5 and 6, and, in the light of expectations and performance, we make some suggestions for a more effective preparation of talented students on after-school lessons.

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

  175-181

  Zoltán Szilasi

  Views:

  16

  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.