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Writing a textbook – as we do it
185-201Views:64Recent 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. -
Report of conference XXXVIII. National Conference on Teaching Mathematics, Physics and Computer Sciences: August 25-27, 2014 Pécs, Hungary
281-303Views:88The XXXVIII. National Conference on Teaching Mathematics, Physics and Computer Sciences (MAFIOK) was held in Pécs, Hungary between 25 and 27 August, 2014 at the Pollack Mihály Faculty of Engineering and Information Technology. It was organized by the Engineering Mathematics Department. The 65 participants – including 4 invited lecturers and 53 lecturers – came from 2 countries and represented 14 institutions of higher education. -
Potential, actual and practical variations for teaching functions: cases study in China and France
157-166Views:167This contribution is based on two major hypotheses: task design is the core of teachers’ work, and variation is the core of task design. Taking into account variation in task design has a profound theoretical foundation in China and France. Developing my PhD with two co-supervisors, in China and France, I wish to seize this opportunity for constructing an analytic model of “teaching mathematics through variation” making profit of this theoretical diversity. This model distinguishes between potential variation and practical variation and is based on the process of transforming potential variation into actual variation, and of using practical variation for rethinking potential variation. The design of this model is based both on theoretical networking, and on case studies, in France and China. In this contribution, we will focus on a critical aspect in the two cases, from potential to practical variation.
Subject Classification: 97-06
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Analysing the effects of OOP helper application
65-75Views:94Nowadays 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. -
The mathematics textbook as an aid to differentiation: a first Hungarian example
35-53Views:81Differentiation is a way of teaching where each student is taught according to his/her personal needs. This technique is not widely used in Hungary yet, although this would be necessary due to the introduction of the two-level final examination and to a growing concern for equal opportunities and integrated teaching. One of the most significant aids to differentiation is an appropriate textbook, and that is why a group of professionals wrote a set of textbooks that supports this technique. The paper examines the requirements for a differentiated textbook, and the extent to which the textbook in question meets them. -
Psychology - an inherent part of mathematics education
1-18Views:233On the chronology of individual stations of psychology and their effect on mathematics education designed as working document for use in teacher training.
The article is structured as a literature survey which covers the numerous movements of psychology towards mathematics education. The current role of psychology in mathematics education documented by different statements and models of mathematics education should provide a basis for the subsequent investigations. A longitudinal analysis pausing at essential marks takes centre of the continuative considerations. The observed space of time in the chapter covers a wide range. It starts with the separation of psychology from philosophy as a self-contained discipline in the middle of the 19th and ends with the beginning of the 21st century. Each stop states the names of the originators and the branches of psychology they founded. These stops are accompanied by short descriptions of each single research objective on the one hand, and their contributions to mathematics education on the other hand. For this purpose, context-relevant publications in mathematics education are integrated and analysed. The evaluation of the influence of concepts of psychology on teaching technology in mathematics is addressed repeatedly and of great importance. The layout of this paper is designed for the use as a template for a unit in teacher-training courses. The conclusion of the article where the author refers to experiences when teaching elements of psychology in mathematics education courses at several universities in Austria is intended for a proof on behalf of the requested use.Subject Classification: 01A70, 01-XX, 97-03, 97D80
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Research studies in didactics of mathematics supported by the Operant Motive Test
153-173Views:112The present paper reports a case-study which took place within an EUsupported international program organized for research and development of multi-grade schools (NEMED, [16] [26]). One of the main goals of the research was to develop the connection between disadvantageous social situations and the efficiency (success or failure) in learning mathematics especially from the point of view of average and above-average (talented) students: Why does the talent of children with socially disadvantageous background remain undiscovered? How can we make school mathematics more aware of hidden talents?
The author was looking for a didactical solution that compensated for social disadvantages without restricting the development of "average" students by using sociological, educational, psychological and mathematical (experimental and theoretical) studies in interaction with a series of experimental (hypothesis testing and exploratory) investigations.
We constructed tools and methods for exploration and experimental teaching, adapted to Hungarian conditions (Curriculum Development, teacher training, materials, interviews, Kuhl's motivation test, Malara's "researchers and practicing teachers in cooperation" method, etc., see [18], [20]).
The teaching materials and methodological guidelines are based on Bruner's representation theory (see [5]). The empirical research took place in 16 multi-grade schools located in different parts of the country. The author co-operated with nearly 250 students and 25 teachers for 3 years. In this paper we try to demonstrate how an Operant Motive Test can be involved in this research (see [18]). -
Veranschaulichung der Lehrstoffstruktur durch Galois-Graphen
217-229Views:143In this article we compare the process diagram with the Galois-graph, the two hierarchical descriptions of the curriculum's construction from the point of didactics. We present the concrete example through the structure of convex quadrangles. As a result of the analysis it is proved that the process diagram is suitable for describing the activity of pupils, still the Galois-graph is the adequate model of the net of knowledge. The analysis also points out that in teaching of convex quadrangles the constructions of curriculum based only on property of symmetry and only on metrical property are coherent. Generalizing concept is prosperous if the pupils' existing net of knowledge lives on, at most it is amplified and completed. Teaching of convex quadrangles in Hungarian education adopts this principle. -
Conventions of mathematical problems and their solutions in Hungarian secondary school leaving exams
137-146Views:94Collecting 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. -
The single-source shortest paths algorithms and the dynamic programming
25-35Views:123In this paper we are going to present a teaching—learning method that help students look at three single-source shortest paths graph-algorithms from a so called "upperview": the algorithm based on the topological order of the nodes, the Dijkstra algorithm, the Bellman-Ford algorithm. The goal of the suggested method is, beyond the presentation of the algorithms, to offer the students a view that reveals them the basic and even the slight principal differences and similarities between the strategies. In order to succeed in this object, teachers should present the mentioned algorithms as cousin dynamic programming strategies. -
Report of the conference "Connecting Tamás Varga’s Legacy and Current Research in Mathematics Education": November 6-8, 2019, Budapest, Hungary
5-8Views:326On 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.
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Teaching of old historical mathematics problems with ICT tools
13-24Views:149The aim of this study is to examine how teachers can use ICT (information and communications technology) tools and the method of blended learning to teach mathematical problem solving. The new Hungarian mathematics curriculum (NAT) emphasizes the role of history of science, therefore we chose a topic from the history of mathematics, from the geometry of triangles: Viviani's Theorem and its problem field. We carried out our teaching experiments at a secondary school with 14-year-old students. Students investigated open geometrical problems with the help of a dynamic geometric software (GeoGebra). Their research work was similar to the historical way. -
Difference lists in Prolog
73-87Views:109Prolog is taught at Bradford University within the two-semester module Symbolic and Declarative Computing/Artificial Intelligence. Second year undergraduate students are taught here the basics of the functional and the logic programming paradigms, the latter by using the Linux implementation of SWI Prolog [6]. The topic 'Difference lists' is mentioned in traditional textbooks such as [2] and [5] but it was felt that the available texts do not quite serve our purposes. We present here a lecture handout and a laboratory sheet for the teaching sessions on Difference lists. It is believed that the lectures and lab sessions together with the handouts shown here are a gentle, self-contained and reasoned introduction into the topic. The figures here shown to illustrate the concepts are considered a special feature of the handouts which in this form do not seem to be well known. -
Mathematician Judita Cofman (1936–2001)
91-115Views:149Judita Cofman was the first generation student of mathematics and physics at Faculty of Philosophy in Novi Sad, Serbia, and the first holder of doctoral degree in mathematical sciences at University of Novi Sad. Her Ph.D. thesis as well as her scientific works till the end of 70's belong to the field of finite projective and affine planes and the papers within this topic were published in prestigious international mathematical journals. She dedicated the second part of her life and scientific work to didactic and teaching of mathematics and to work with young mathematicians. -
Cooperative learning in teaching mathematics: the case of addition and subtraction of integers
117-136Views:99In the course of teaching and learning mathematics, many of the problems are caused by the operations with integers. My paper is a presentation of an experiment by which I tried to make the acquisition of these operations easier through the use of cooperative methods and representations. The experiment was conducted in The Lower-Secondary School of Paptamási from Romania, in the school year 2009-2010. I present the results of the experiment. -
Report of Conference XXXIX. National Conference on Teaching Mathematics, Physics and Computer Science-August 24-26, 2015 Kaposvár, Hungary
309-331Views:70The XXXIX. National Conference on Teaching Mathematics, Physics and Computer Sciences (MAFIOK) was held in Kaposvár, Hungary between 24 and 26 August, 2015 at the Faculty of Economic Sciences of Kaposvár University. It was organized by the Department of Mathematics and Physics. The 67 participants – including 5 invited lecturers and 54 lecturers – came from 5 countries and represented 16 institutions of higher education. -
Nice tiling, nice geometry!?!
269-280Views:100The 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. -
Die Stichprobe als ein Beispiel dafür, wie im Unterricht die klassische und die bayesianische Auffassung gleichzeitig dargestellt werden kann
133-150Views:104Teaching statistics and probability in the school is a new challenge of the Hungarian didactics. It means new tasks also for the teacher- and in service-teacher training. This paper contains an example to show how can be introduced the basic notion of the inference statistics, the point- and interval-estimation by an elementary problem of the public pole. There are two concurrent theories of the inference statistics the so called classical and the Bayesian Statistics. I would like to argue the importance of the simultaneously introduction of both methods making a comparison of the methods. The mathematical tool of our elementary model is combinatorial we use some important equations to reach our goal. The most important equation is proved by two different methods in the appendix of this paper. -
Correction to Gofen (2013): "Powers which commute or associate as solutions of ODEs?", Teaching Mathematics and Computer Science 11 (2013), 241-254.
245Views:115In the article "Powers which commute or associate as solutions of ODEs?" by Alexander Gofen (Teaching Mathematics and Computer Science, 2013, 11(2), 241–254. https://doi.org/10.5485/TMCS.2013.0347), there was an error in Conjecture 1 (p. 250), and consequently, in the References (p. 254).
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Why some children fail? Analyzing a test and the possible signs of learning disorders in an answer sheet: dedicated to the memory of Julianna Szendrei
251-268Views:154Teachers and educators in mathematics try to uncover the background of the mistakes their students make for their own and their students' benefit. Doing this they can improve their teaching qualities, and help the cognitive development of their pupils. However, this improvement does not always support their students with learning disorders, since their problem is not caused by wrong attitude or lack of diligence. Therefore, it is the interest of a conscientious teacher to recognize whether the weaker performance of a student is caused by learning disorders, so the helping teacher can give useful advices. Although the teacher is not entirely responsible for the diagnosis, but (s)he should be be familiar with the possible symptoms in order to make suggestions whether or not to take the necessary test of the learning disorders.
In this article, through examining a test and the answer sheet of a single student, I show some signs that might be caused by learning disorders. -
Longest runs in coin tossing. Teaching recursive formulae, asymptotic theorems and computer simulations
261-274Views:122The coin tossing experiment is studied, focusing on higher education. The length of the longest head run can be studied by asymptotic theorems ([3]), by recursive formulae ([10]) or by computer simulations . In this work we make a comparative analysis of recursive formulas, asymptotic results and Monte Carlo simulation for education. We compare the distribution of the longest head run and that of the longest run (i.e. the longest pure heads or pure tails) studying fair coin events. We present a method that helps to understand the concepts and techniques mentioned in the title, which can be a useful didactic tool for colleagues teaching in higher education. -
Examining relation between talent and competence through an experiment among 11th grade students
17-34Views:120The 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. -
Realizing the problem-solving phases of Pólya in classroom practice
219-232Views:285When teaching mathematical problem-solving is mentioned, the name of Pólya György inevitably comes to mind. Many problem-solving lessons are planned using Pólya's steps and helping questions, and teachers often rely on his heuristics even if their application happens unconsciously. In this article, we would like to examine how the two phases, Making a plan and Looking back, can be realized in a secondary school mathematics lesson. A case study was designed to observe and analyse a lesson delivered using cooperative work.
Subject Classification: 97B10, 97C70, 97D40, 97D50
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How to use our own program evaluation system to streamline teaching computer programming
73-80Views:117During computer programming contests the use of automatic evaluation systems is becoming more and more frequent. In said systems the contestants are allowed to submit their source code that will be evaluated with the results reported back to them. According to this report the contestant can realise for what test cases his program works properly and for what cases does it fail. This kind of on-line evaluation system is used for example in the International Olympiad in Informatics (IOI), in the final round of the Nemes Tihamér National Programming Competition, and in the Selection Competition for IOI in Hungary. A contest management system can be used for other purposes apart from this singular example. A well-developed evaluation system can foster not only the teaching of computer programming and the preparation of students for programming contests but the teacher's work as well. -
Teaching Gröbner bases
57-76Views:113In this article we offer a demonstration of how the StudentGroebner package, a didactic oriented Maple package for Gröbner basis theory, could assist the teaching/learning process. Our approach is practical. Instead of expounding on deep didactic theory we simply give examples on how we imagine experimental learning in classroom. The educational goal is to prepare the introduction of two sophisticated algorithms, the division algorithm and Buchberger's algorithm, by gathering preliminary knowledge about them.
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