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Psychology - an inherent part of mathematics education
1-18Views:284On 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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Report of conference XXXVIII. National Conference on Teaching Mathematics, Physics and Computer Sciences: August 25-27, 2014 Pécs, Hungary
281-303Views:143The 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. -
Programming theorems on enumerator
89-108Views:197This paper deals with the examination of the programming patterns best known by programmers: the programming theorems. It is a significant issue that in what way these patterns can be formulated in order to solve a relatively broad spectrum of problems using a small number of patterns. In this paper, the well known programming theorems are applied to the processing of enumerators. To this end, the robustness of patterns gained this way will be presented, and it will be also pointed out how the programs thus constructed can be implemented in the modern object-oriented programming environments: in language C++, Java and C#. -
Interdisciplinary Secondary-School Workshop: Physics and Statistics
179-194Views:170The paper describes a teaching unit of four hours with talented students aged 15-18. The workshop was designed as a problem-based sequence of tasks and was intended to deal with judging dice whether they are regular or loaded. We first introduced the students to the physics of free rotations of rigid bodies to develop the physics background of rolling dice. The highlight of this part was to recognise that cubes made from homogeneous material are the optimal form for six-sided objects leading to equal probabilities of the single faces. Experiments with all five regular bodies would lead to similar results; nevertheless, in our experiments we focused on regular cubes. This reinsures that the participants have their own experience with the context. Then, we studied rolling dice from the probabilistic point of view and – step-by-step – by extending tasks and simulations, we introduced the idea of the chi-squared test interactively with the students. The physics and the statistics part of the paper are largely independent and can be also be read separately. The success of the statistics part is best described by the fact that the students recognised that in some cases of loaded dice, it is easier to detect that property and in other cases one would need many data to make a decision with small error probabilities. A physical examination of the dice under inspection can lead to a quick and correct decision. Yet, such a physical check may fail for some reason. However, a statistical test will always lead to reasonable decision, but may require a large database. Furthermore, especially for smaller datasets, balancing the risk of different types of errors remains a key issue, which is a characteristic feature of statistical testing.
Subject Classification: F90, K90, M50, R30
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The role of computer in the process of solving of mathematical problems (results of research)
67-80Views:161We would like to present results of an almost two years investigations about the role computer in the process of solving of mathematical problems. In these investigations took part 35 students of the secondary school (generalists) in the age 17–19 years. Each of these students solved following problem:
Find all values of the parameter m so that the function
f(x) = |mx + 1| − |2x − m| is:
a) bounded,
b) bounded only from the bottom,
c) bounded only from above,
first without a computer and next with a special computer program. We would like to show results of these researches. -
Nice tiling, nice geometry!?!
269-280Views:148The 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. -
Report of Meeting Researches in Didactics of Mathematics and Computer Sciences, April 1-3, 2022 Baja, Hungary
135-155Views:295The meeting Researches in Didactics of Mathematics and Computer Sciences was held in Baja, Hungary, at Eötvös József College, from the 1st to the 3th of April, 2022. It was organized by the Doctoral School of Mathematical and Computational Sciences of the University of Debrecen and by Eötvös József College. The 62 participants - including 18 PhD students - came from 8 countries and represented 26 institutions of higher and secondary education. There were 3 plenary and 40 session talks in the program.
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Radio Frequency Identification from the viewpoint of students of computer science
241-250Views:151This paper aims at creating the right pedagogical attitudes in term of teaching a new technology, Radio Frequency Identification (RFID) by evaluating the social acceptance of this new method. Survey of future teachers, students of teacher master studies and students from informatics oriented secondary schools were surveyed comparing their attitudes in terms of RFID to other recent technologies. Consequences of this survey are incorporated into the curriculum of the new RFID course at our institution. -
Using the computer to visualise graph-oriented problems
15-32Views:142The 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. -
Report of Meeting Researches in Didactics of Mathematics and Computer Sciences: February 1-3, 2019 Stúrovo, Slovakia
105-129Views:327The meeting Researches in Didactics of Mathematics and Computer Sciences was held in Sturovo, Slovakia from the 1st to the 3th of February, 2019. It was organized by the Doctoral School of Mathematical and Computational Sciences of University of Debrecen. The 63 participants – including 17 PhD students – came from 7 countries, 22 cities and represented 36 institutions of higher and secondary education. There were 4 plenary, 42 session talks and 7 poster presentations in the program.
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How to teach testing?
215-232Views:221Testing methodology is an important part of IT education. It is desired to show the beginner programmer students the advantage of testing by having them do only a small amount of work. In this paper, we will show how to make testing as a part of programming in simple exercises. These exercises are solved with the analogous programming technique, which is based on programming theorems over enumerators. We have elaborated grey-box test cases for the programs which have been developed based on programming theorems. These test cases can be taught together with the programming theorems, and they can serve as a standard testing procedure for programmers. We also suggest a test tool to automatize test runs, and we will discuss its usage in a short case study. -
New style in teaching word processing
417-426Views:136Teaching word processing is confined to looking through some menus and showing some functions of a word processor program, although technology presents just a small part of forming layouts. This fact causes that people who are writing documents spend a lot of time by trying to form, e.g., title pages or inner pages.
The present paper deals with a design of an online course on word processing that fits better the needs of many users. The online course is designed for teaching (LA)TEX by leading the students to the technical issues of the typesetting system through layout and grammar rules: demonstrates the most important basic recommendations of typography and grammar rules through samples, and shows how to program the currently displayed layout in the (LA)TEX programming languages. This methodology suits better the common working habit, and can be a useful help in word processing documents. -
Teaching puzzle-based learning: development of transferable skills
245-268Views:343While computer science and engineering students are trained to recognise familiar problems with known solutions, they may not be sufficiently prepared to address novel real-world problems. A successful computer science graduate does far more than just program and we must train our students to reach the required levels of analytical and computational thinking, rather than hoping that it will just 'develop'. As a step in this direction, we have created and experimented with a new first-year level course, Puzzle-based Learning (PBL), that is aimed at getting students to think about how to frame and solve unstructured problems. The pedagogical goal is increase students' mathematical awareness and general problem solving skills by employing puzzles, which are educational, engaging, and thought provoking. In this paper we continue sharing our experiences in teaching such a course. Whereas a brief discussion on our pedagogical objectives were covered in the first paper together with the material of the first of two lectures on pattern recognition, this follow-up paper presents the material of the second of two lectures, in which additional exercises are discussed to reinforce the lesson. Along the way we provide a glimpse of some foundational ideas of computer science such as incomputability and general system development strategies such as incremental and iterative reasoning. This paper discusses the outcomes of PBL courses, which include expected improvement in the overall results achieved by students who have undertaken PBL courses, compared to those students who have not. -
Visualisation in geometry education as a tool for teaching with better understanding
337-346Views:369In primary and secondary geometry education, some problems exist with pupils’ space thinking and understanding of geometric notions. Visualisation plays an important role in geometry education, and the development of pupils’ visualisation skills can support their spatial imagination. The authors present their own thoughts on the potential of including visualisation in geometry education, based on the analysis of the Hungarian National Core Curriculum and Slovak National Curriculum. Tasks for visualisation are also found in international studies, for example the Programme for International Student Assessment (PISA). Augmented reality (AR) and other information and communication technology (ICT) tools bring new possibilities to develop geometric thinking and space imagination, and they also support mathematics education with better understanding.
Subject Classification: 97U10, 97G10
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Decision based examination of object-oriented programming and Design Patterns
83-109Views:153On 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. -
Central axonometry in engineer training and engineering practice
17-28Views:147This paper is concerned with showing a unified approach for teaching central and parallel projections of the space to the plane giving special emphasis to engineer training. The basis for unification is provided by the analogies between central axonometry and parallel axonometry. Since the concept of central axonometry is not widely known in engineering practice it is necessary to introduce it during the education phase. When teaching axonometries dynamic geometry software can also be used in an interactive way. We shall provide a method to demonstrate the basic constructions of various axonometries and use these computer applications to highlight their similarities. Our paper sheds light on the advantages of a unified approach in such areas of engineering practice as making hand drawn plans and using CAD-systems. -
Methods of teaching programming
247-257Views:127Programming methodology is one of the oldest fields of IS education, and thus various methods have evolved for its teaching. While some of them could be used effectively in primary or secondary education, others are more suited for students in higher education. The methods themselves determine the structure and curricula of courses such as Programming methodology, Data types and algorithms, Programming technology. -
Reappraising Learning Technologies from the Viewpoint of the Learning of Mathematics
221-246Views:170Within the context of secondary and tertiary mathematics education, most so-called learning technologies, such as virtual learning environments, bear little relation to the kinds of technologies contemporary learners use in their free time. Thus they appear alien to them and unlikely to stimulate them toward informal learning. By considering learning technologies from the perspective of the learner, through the analysis of case studies and a literature review, this article asserts that the expectation of these media might have been over-romanticised. This leads to the recommendation of five attributes for mathematical learning technologies to be more relevant to contemporary learners' needs: promoting heuristic activities derived from human history; facilitating the shift from instrumentation to instrumentalisation; facilitating learners' construction of conceptual knowledge that promotes procedural knowledge; providing appropriate scaffolding and assessment; and reappraising the curriculum. -
Teaching puzzle-based learning: development of basic concepts
183-204Views:405While computer science and engineering students are trained to recognise familiar problems with known solutions, they may not be sufficiently prepared to address novel real-world problems. A successful computer science graduate does far more than just program and we must train our students to reach the required levels of analytical and computational thinking, rather than hoping that it will just 'develop'. As a step in this direction, we have created and experimented with a new first-year level course, Puzzle-based Learning (PBL), that is aimed at getting students to think about how to frame and solve unstructured problems. The pedagogical goal is increase students' mathematical awareness and general problem solving skills by employing puzzles, which are educational, engaging, and thought provoking. We share our experiences in teaching such a course – apart from a brief discussion on our pedagogical objectives, we concentrate on discussing the presented material which covers (in two lectures) just one selected topic (pattern recognition). In this paper we present the ideas behind foundations for PBL and the material of the first of two lectures on pattern recognition, in which we address core concepts and provide students with sufficient exemplars to illustrate the main points. -
Motivating students with projects encompassing the whole duration of their studies
165-180Views:155Based on my ten years of teaching experience at the University of Debrecen, I can say that students majoring Software Information Technology BSc have to face a number of difficulties during their studies. I think these difficulties root from two main problems: students are unmotivated and cannot sense the coherence between the knowledge acquired in the various courses. This paper tries to give some alleviation to both of these problems by the idea of introducing some long-term projects to students, which they can work on throughout their studies, dealing with a particular aspect of the projects in each course. -
Two centuries of the equations of commutativity and associativity of exponentiation
219-233Views:78In this survey article we guide the reader through the solution of the commutative equation of exponentiation x^y = y^x and that of the associative equation of exponentiation x^(y^z) = (x^y)^z. Various characterizations of the integer, rational, real and complex solutions are discussed together with some new results and open directions. The article is supplemented by a detailed and commented bibliography on the history of these equations. -
Expressiveness of programming languages and environments: a comparative study
111-141Views:162In written and oral communication tools, the support of the understanding of our message have an important role: we can increase the expressiveness and the level of understanding of our topic by approaching it in several ways, i.e. in written methods by highlighting the important parts; in oral by changing tone and other elements of non-verbal communication. In this paper programming languages and developing environments are compared with each other in terms of their methods and their level of support to the solution of programming tasks.
There is a need to have these tools in programming and, of course, in teaching programming. What are the factors that define the distinctness and the legibility of a program? What are the basic principles which give an instrument in programmers' and students' hands in order to create a properly working program from already existing algorithms in the most efficient way? We search for the answers to these questions in this paper. -
Teaching of old historical mathematics problems with ICT tools
13-24Views:224The 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. -
Mathematische Bildung im Klagenfurter Doktorand(inn)enkolleg
67-84Views:122In 2003 we set up a programme for PhD-studies ("Doktorand(inn)enkolleg") at the University of Klagenfurt which should promote and support PhD-studies in the field of mathematics education.Within this programme it is worked on the topic "general mathematics education" from different perspectives.
In the first part of this paper intentions, the fields of work and the form of organisation are briefly demonstrated. The second and main part considers in detail the work in one of the four fields of work, and finally, the third and last part presents some experiences with regard to the contents as well as general ones. -
Analogous programming with a template class library
135-152Views:194In this paper, a template class library and its usage can be read. The classes of the library contain the C++ code of the algorithm of programming theorems. This library supports the implementation of the programs that are planned by analogous programming but the primary aim of its usage is to introduce the object-oriented programming style to show how a reusable code can be written with inheritances, overriding virtual methods, composition of objectcs and template parameters.
