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The appearance of the characteristic features of the mathematical thinking in the thinking of a chess player
201-211Views:242It 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. -
Development of high school students' geometric thinking with particular emphasis on mathematically talented students
93-110Views:262We 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. -
Mathematical gems of Debrecen old mathematical textbooks from the 16-18th centuries
73-110Views:119In the Great Library of the Debrecen Reformed College (Hungary) we find a lot of old mathematical textbooks. We present: Arithmetic of Debrecen (1577), Maróthi's Arithmetic (1743), Hatvani's introductio (1757), Karacs's Figurae Geometricae (1788), Segner's Anfangsgründe (1764) and Mayer's Mathematischer Atlas (1745). These old mathematical textbooks let us know facts about real life of the 16-18th centuries, the contemporary level of sciences, learning and teaching methods. They are rich sources of motivation in the teaching of mathematics. -
Teaching integral transforms in secondary schools
241-260Views:197Today, Hungarian students in the secondary schools do not know the idea of complex numbers, and they can not integrate except those ones who learn mathematics in advance level. Without this knowledge we can teach Fourier transform for students. Why should we teach Fourier transform (FT) or Wavelet transform (WT) for them? To teach image file formats like JPEG, (JPEG2000) we need to talk about integral transforms. For students who are good in computer programming, writing the program of 1D FT or 2D FT is a nice task. In this article we demonstrate how we can teach Fourier and Wavelet transform for students in secondary school. -
Sequenced problems for functional equations
179-192Views:193There are many possible methods to solve equations of the form H(f(x + y), f(x − y), f(x), f(y), x, y) = 0 (x, y 2 R), where H is a known function and f is the unknown function to be determined. Here we will create a sequence of problems for equations of type (1) (see on the next page). These sequenced problems are appropriate for the fostering of talented students on different level of mathematical education. -
The time spent on board games pays off: links between board game playing and competency motivation
119-131Views:427The impact playing has on the development of thinking is an important topic of psychology of learning, brain research and mathematics didactics.
Our research is also connected to the aforementioned topic. We investigated the effects of playing board games on competence motivation and the development of mathematical competencies.
In this paper, we present the results of an experiment carried out in a secondary school class.
The experimental group spent one of three weekly mathematics lessons playing board games.
Apart from the several advantages of playing games in general, we can conclude that, based on the results of the national competence measurement, the mathematical competence of the students developed properly.
The readiness and the progress of the pupils were compared on the basis of input and output tests and an initial knowledge measurement and, at the same time, we compared their level of mathematical competence with the results of the national competence
measurement.Subject Classification: 97C70, 97D40
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Game theory for managers and mechanical manager students
73-91Views:216In this article we describe the second part of a case study, in which 48 Mechanical Management students were involved. The participants of the case study were MSc level students at Szent István University, Gödöllő.
In the case study we looked for methods by which we can support the most important components of competence motivation and the development of mathematical and other key competences during the mathematics lessons and individual learning.
Another goal of our research was to get reliable information about students learning methods and their awareness of self-efficiency, furthermore their achievement in the subject of Engineering and Economic Mathematics.
Detailed assistance was provided for the students in the e-learning portal. Knowledge tests, questionnaire and personal interviews with the students were also used.
During the semester four topics have been discussed: linear programming, graph theory, game theory and differential equations. In this article I will describe the lesson preparations, the help for examinations and the students' achievement on game theory. -
Teaching fractions at elementary level in the light of Hungarian mathematics textbooks in Romania
149-159Views:247According to the new curriculum in Romania, fractions are introduced in the second grade. The present study analyses Hungarian elementary mathematics textbooks on the topic of fractions focusing on the types of tasks in the textbooks, the significance of representations and the proportion of word problems. Additionally, the paper presents a questionnaire-based research on teachers’ opinion regarding the adequacy and sufficiency of the digital materials and exercises related to fractions in the textbooks.
Subject Classification: 97F40, 97F80, 97U20, 97U50
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Conventions of mathematical problems and their solutions in Hungarian secondary school leaving exams
137-146Views:187Collecting and analyzing the conventions indispensable for interpreting mathematical problems and their solutions correctly assist successful education and objective evaluation. Many professional and didactic questions arose while collecting and analyzing these conventions, which needed clarification, therefore the materials involved concisely in the conventions enrich both the theory and practice of mathematics teaching. In our research we concentrated mainly on the problems and solutions of the Hungarian school leaving examinations at secondary level in mathematics. -
Fuzzy Datalog with background knowledge
257-281Views:214In this paper we give a possible model for handling uncertain information. The concept of fuzzy knowledge-base will be defined as a triplet of a background knowledge defined by the similarity of predicates and terms; a deduction mechanism: a fuzzy Datalog program, and a decoding set of the program, which help us to determine the uncertainty level of the results. -
Wichtige Momente aus der ungarischen Geschichte des Analysisunterrichts
57-76Views:219Törner et al. (2014) paper gives an outstanding review about teaching analysis at high school level in (Western) Europe. We tried to extend this paper with some results from the Hungarian Math History (Beke and Rátz 1897-1924, after second World War 1949-1960, the current situation-first of all based on schoolbooks, and we also included an experiment from 1984-1989 by E. Deák, which was interrupted and partially forgotten). In summary, this paper deals with the turning points of the brief history of teaching secondary school analysis in the XXth century in Hungary, including some conclusions at the end.
Subject Classification: 97A30, 97C30, 97D30, 97E50, 97I20, 97I40, 97U20
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Analysing the effects of OOP helper application
65-75Views:216Nowadays students of secondary schools are familiar with the usage of computer very soon, lot of them are even capable of handling user applications very cleverly. This is satisfying for most of them. Those who imagine their future in programming or system developing, need to have deeper knowledge about object oriented programming, however, students do have it at very low level or not at all. We want to make sure whether this suppose is true, so different examinations have recently been made at Slovakian secondary schools with Hungarian teaching language. We have reached a conclusion that the students' knowledge of object oriented programming is deficient. We could achieve better results by using proper applications as a visual aid. In this paper we examine the efficiency of an application made by us. -
Teaching agile operation and leadership through linked university courses
1-32Views:253Agile software development methods, especially Scrum, are commonly used in software development companies. For this reason, our goal was that our undergraduate students gain experience as Scrum development team members and our master's students as agile leaders. To this end, we had redesigned and linked an undergraduate and a master's course, and launched the new course in the spring of 2021. The success of our approach was confirmed by a questionnaire survey of 86 undergraduate and 27 master's students. A/B testing was also performed. Our approach is a novelty compared to solutions where the Scrum Master is a course member, an instructor, or a university employee. In addition to being resource-efficient, it also offers master's students an unparalleled opportunity to develop agile leadership skills.
Subject Classification: 97U50
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The requirements in statistics education – comparison of PISA mathematical tasks and tasks from the mathematical textbooks in the field of statistics
263-275Views:216This work presents the results of the analysis of both PISA items and Croatian mathematical textbooks in the field of statistics.
The analysis shows that PISA's released statistics problems have in many ways different mathematical requirements from the requirements of textbook problems in the statistics chapters, with respect to the mathematical activities, complexity and in the forms of questions. The textbook analysis shows that mathematical examples and problems often require operation and interpretation skills on a reproductive or connections level. Statistics textbook problems are given in the closed-answer form. The results also show that while PISA puts strong emphasis on the statistics field, in the current Croatian curriculum this field is barely present. These discrepancies in requirements and portion of statistics activities surely affect the results of Croatian pupils on PISA assessment in the field of mathematical literacy. -
Research on IT language use at a company
203-219Views:224The aim of the research of the IT language, used in the written documents of a company, is to contribute to the creation of a (mono- or bilingual) dictionary or encyclopaedia available for the public on the Internet, serving, among others, as a reference tool for the unified, controlled and unambiguous use of IT terms for students at various educational levels. To this ongoing work, the participation and cooperation of a panel of experts of different competences, linguists as well as IT experts, is indispensable.
The methods of corpus linguistics were used to carry out the research. The IT terms were separated from the texts and then a concordance software was used to see the environment of the IT words and phrases in which they occur. So their morphological analysis became possible.
The results of the research showed that a great number of Hungarian morphological language use problems stem from the way the IT terms are used in the documents. This paper lists, groups, analyses these phenomena.
The conclusions of the author are: (1) If such an Internet dictionary is used generally and consulted when e.g. somebody wants to write a composition or essay, translate an article, write a newspaper article, a scientific publication or a textbook to be taught at schools of different types and levels, etc. most of the communication noises could be filtered out. (2) At the same time it could promote the use of adequate (both in linguistic and technical meaning) Hungarian terms eliminating the "Hunglish" usage. (3) It could also contribute to the prevailing use of the relevant Hungarian terminology. Such a dictionary would be indispensable, not only in educational and industrial environments but in the electronic and traditional media as well. Last but not least, it could raise the level of different teaching materials (textbooks, e-materials, etc.) used in public and higher education. -
Nice tiling, nice geometry!?!
269-280Views:208The 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. -
The development of geometrical concepts in lower primary mathematics teaching: the square and the rectangle
153-171Views:241Our 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. -
Engineering and Economic Mathematics for Engineering Management Students
35-50Views:225In this article we describe the first part of a case study, which was made with 48 Engineering Management students. The participants of the case study were MSc level students at the Szent István University, Gödöllő. We looked for methods by which we can support the most important components of competence motivation and the development of mathematical and other key competences during the mathematics lessons and individual learning. Another goal of our research was to get reliable information about students learning methods and their awareness of self-efficacy, furthermore their achievement in the subject of Engineering and Economic Mathematics. Detailed assistance was provided for the students in the e-learning portal. Knowledge tests, questionnaire and personal interviews with the students were also used. As an example we introduce one of the knowledge tests connected with the first half of the course about linear programming and graph theory. We detail its didactical background and show the results of the students. -
A computational thinking problem-thread for grade 7 students and above from the Pósa method
101-110Views:340Lajos Pósa has been developing his “learning through discovery” (Győri & Juhász, 2018) method since 1988. His weekend math camps are focused on fostering problem-solving skills and high-level mathematical-thinking skills in gifted students from grades 7 to 11. One of the core aspects of the method is the structure of the problems, all problems are part of a complex, intertwined, and rich network. In this article we analyze a computational thinking problem-thread and its role in the camps’s network of problems (Gosztonyi, 2019), and show some aspects of the method. The insights gained using this method can be useful in other contexts. The possible adaptation of the method to secondary and high schools is briefly discussed as well.
Subject Classification: 97D40
