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Cognitive conflict as a tool of overcoming obstacles in understanding infinity
279-295Views:130This paper, based on current research, is devoted to obstacles that appear in the process of understanding the concept of infinity. In its introductory part the fundamental types of epistemology obstacles are formulated. The cognitive conflict and its role in overcoming these obstacles are analysed in the following section of this research. Finally, the authors focus on research methodology and the results of the three-year research project. The problems are illustrated by means of real experimental interviews. -
Apollonea.com project: integrating geometry and collaboration in education
183-194Views:96This article presents the Apollonea.com project, which aims to make the solutions to Apollonius’ problems accessible to students and teachers through modern technology. The web platform contains more than 150 interactive constructions created by students using GeoGebra, allowing for dynamic manipulation and visualization of solutions to various variants of Apollonius’ problems. The project combines classical geometric problems with an interdisciplinary approach, teamwork, and the use of modern technology. The article describes the process of developing the Apollonea.com website, the use of GeoGebra in the project, the structure and functions of the website, and its educational benefits in enhancing students’ geometric skills. The project demonstrates how traditional mathematics education can be connected with modern ICT tools.
Subject Classification: 97U50, 97G40, 51M04, 68U05
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E-learning management systems in Hungarian higher education
357-383Views:91Computers, informatics, and information technology have an ever-increasing role in the establishment and spread of new educational forms and methods. The role of e-learning as a new educational model is increasing in the world of computer networks, because of a widespread access to the net and a growing demand for learning beside work.
Technological elements of e-learning can be separated as Learning Management System, authoring system, course material and a browser. Learning Management System is the software package that creates the structure of the whole educational process: course organisation, course material presentation, tracking student work, recording results, and the completion of the program.
This publication shows examples of Learning Management Systems used in Hungarian higher education. Summarizing and systematizing expectations and demands expressed in connection with learning management systems, the present work tries to help the reader orientate on an ever-expanding market. -
A case study of the integration of Algorithm Visualizations in Hungarian programming education
51-66Views:285In this study, I will introduce how Algorithm Visualizations (AV) can help programming education or, in this case, the acquisition of basic programming theorems. I used two di erent methods to test this: in the first round, I examined in a larger group how much the students' ability to solve specific tasks changes after being introduced to a visualization tool, and then, what was their motivation and experience during this process. In the second round, I looked for the components that could be important when choosing a tool with the help of an in-depth interview with a smaller number of individuals. In both cases, I describe the research, experience, and results of the study, and then summarize them at the end.
Subject Classification: 97P10
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Different approaches of interplay between experimentation and theoretical consideration in dynamic geometry exploration: An example from exploring Simson line
63-81Views:166Dynamic geometry environment (DGE) is a powerful tool for exploration and discovering geometric properties because it allows users to (virtually) manipulate geometric objects. There are two possible components in the process of exploration in DGE, viz. experimentation and theoretical consideration. In most cases, there is interplay between these two components. Different people may use DGE differently. Depending on the specific mathematical tasks and the background of individual users, some approaches of interplay are more experimental whereas some other approaches of interplay are more theoretical. In this paper, different approaches of exploring a geometric task using Sketchpad (a DGE) by three individual participants will be discussed. They represent three different approaches of interplay between experimentation and theoretical consid- eration. An understanding of these approaches may contribute to an understanding on the mechanism of exploration in DGE. -
Un point d'heuristique important et mal connu: la particularisation
235-245Views:155Cet article est consacré á la présentation d'un point d'heuristique d'une grande importance et sur lequel on insiste trés peu dans notre enseignement. C'est donc une cause fréquente d'échec pour de nombreux éléves. Il s'agit du procédé consistant á particulariser lorsqu'on dispose d'une hypothése dont l'énoncé commence par "quel que soit...". Plusieurs exemples dans divers domaines des mathématiques sont proposés.
This article is devoted to the presentation of a point of heuristics of a great importance, and on which we do not lay much emphasis in our teaching. Then, it is a frequent cause of failure for many pupils. It concerns the followings process: to particularize when we dispose of an hypothesis that begins "For any...". Several examples in various domains of mathematics are proposed. -
Programming Theorems and Their Applications
213-241Views:291One of the effective methodological approaches in programming that supports the design and development of reliable software is analogy-based programming. Within this framework, the method of problem reduction plays a key role. Reducing a given problem to another one whose solving algorithm is already known can be made more efficient by the application of programming theorems. These represent proven, abstract solutions – in a general form – to some of the most common problems in programming. In this article, we present six fundamental programming theorems as well as pose five sample problems. In solving these problems, all six programming theorems will be applied. In the process of reduction, we will employ a concise specification language. Programming theorems and solutions to the problems will be given using the structogram form. However, we will use pseudocodes as descriptions of algorithms resembling their actual implementation in Python. A functional style solution to one of the problems will also be presented, which is to illustrate that for the implementation in Python, it is sufficient to give the specification of the problem for the design of the solution. The content of the article essentially corresponds to that of the introductory lectures of a course we offered to students enrolled in the Applied Mathematics specialization.
Subject Classification: D40
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Game theory for managers and mechanical manager students
73-91Views:162In 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. -
Efficient language teaching software in a multimedia context
361-374Views:161In this article I deal with the efficiency of multimedia teaching programs, analyzing possibilities for their improvement in the field of language teaching. This research has been carried out with the use of the latest technologies, language teaching software, internet based language teaching applications, digital dictionaries, online content, and the latest results from the field of computational linguistics. The goal of my research is to create a general model that serves and supports various kinds of approaches to improving efficiency; I cannot attempt to present a complete, detailed analytical review due to the complexity and size of this topic. However, my opinion is that by considering and understanding the theoretical aspects of the subject, and supported by certain important ideas, we will be able to achieve remarkable improvements in the field of learning efficiency and knowledge retention in the language teaching and learning process that might lead to outstanding results. -
Integrating elements of data science into high-school teaching: Naïve Bayes-classification algorithm and programming in Python
307-316Views:262Probability theory and mathematical statistics are traditionally one of the most difficult chapters of mathematics to teach. One of the authors, Péter Princz has experience in teaching various topics via computer programming of the problem at hand as a class activity. The proposed method is to involve programming as a didactic tool in hard-to-teach topics. The intended goal in this case is to implement a naïve Bayes-classifier algorithm in Python and demonstrate the machine-learning capabilities of it by applying it to a real-world dataset of edible or poisonous mushrooms. The students would implement the algorithm in a playful and interactive way. The proposed incremental development process aligns well with the spirit of Tamás Varga who considered computers as modern tools of experimental problem solving as early as in the 1960s.
Subject Classification: 97D40, 97D50, 97K50, 97K99, 97M60, 97P40, 97P50, 97U50
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The effect of augmented reality assisted geometry instruction on students' achiveement and attitudes
177-193Views:309In this study, geometry instruction's academic success for the students and their attitudes towards mathematics which is supported by education materials of Augmented Reality (AR) and its effect on the acceptance of AR and its usage by teachers and students have been researched. Under this research, ARGE3D software has been developed by using augmented reality technology as for the issue of geometric objects that is contained in the mathematics curriculum of 6th class of primary education. It has been provided with this software that three-dimensional static drawings can be displayed in a dynamic and interactive way. The research was conducted in two different schools by an experiment and control group. In the process of data collection, Geometry Achievement Test (GAT), Geometric Reasoning Test (GRT), Attitudes Scale for Mathematics (ASM), students' math lecture notes, semi-structured interviews with teachers and students and observation and video recordings were used. Results showed that geometry instruction with ARGE3D increased students' academic success. In addition, it was found that geometry instruction with ARGE3D became more effective on students' attitudes that had negative attitudes towards mathematics and it also provided support to reduce fear and anxiety. -
The use of e-tests in education as a tool for retrieval practice and motivation
59-76Views:273In 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
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Forming the concept of parameter with examples of problem solving
201-215Views:153Pupils are encountering difficulties with learning algebra. In order for them to understand algebraic concepts, particularly the concept of parameter it was decided by the teacher of mathematics and Information Technology to integrate the teaching of these two subjects. The aim of this study is to investigate whether, and to what degree, software can be useful in process of forming the concept of parameter. This longitudinal study was conducted in a junior high school (13-16 year old children) using different computer programs. -
Supporting the theory of math didactic using knowledge-measuring questions and analysis of the solutions
1-16Views:191New or rediscovered results presented in this paper are the results of the analysis of the problem sets used in the two-tier system secondary school final examination in mathematics, a system that was introduced in Hungary in 2005.
Many of the revealed problem arise in connection with misunderstanding the text of the problems. Causes of misinterpretation can be either that the text is lacking some important information, or that it should be interpreted not in word-to-word manner.
Theses and their argumentations presented here refer partly on the new types of problems (tests, non-standard mathematical contents), and partly on improvement of learning-teaching process in topics of equations and approximations. -
Process or object? Ways of solving mathematical problems using CAS
117-132Views:90Graphing and symbol manipulating calculators are now a part of mathematics education in many countries. In Norway symbol manipulating calculators have been used at various exams in upper secondary education. An important finding in mathematics education is the duality of mathematical entities – processes and objects. Building on the theoretical development by Anna Sfard and others, the students' solutions on exam problems in upper secondary education are discussed with reference to procedural and structural knowledge. -
Teaching Gröbner bases
57-76Views:174In 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. -
Group Work at High School According to the Method of Tamás Varga
167-176Views:257The aim of our research is to develop students’ logical thinking. For this reason, Hungarian mathematics teachers need to be encouraged to try new methods which induce greater student involvement. Research all over the world prove that self-instruction or self-verbalizing has high effect on the learning process. This was one of the key elements of Tamás Varga’s experiment in high school. In our classroom experiments we are using a special cooperative method from Kagan among 14-18 years old students, called Sage and Scribe structure. We are looking for the answers to the following question: Does this method make mathematics lessons more enjoyable and more comfortable for students? Furthermore, we assume this structure could open the gate toward other collaborative and cooperative teaching technics.
Subject Classification: 97D40
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The formation of area concept with the help of manipulative activities
121-139Views:158Examining 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. -
Computer cooking vs. problem solving
35-58Views:307Computer cooking is a task-related phenomenon where students (end-users) must blindly follow a long list of orders without any connection to the content of the problem, if there is any. Despite its low efficacy, this method is widely used and accepted in informatics both in the learning-teaching process and testing. The National Base Curriculum 2020 in Hungary is in complete accordance with the ‘Informatics Reference Framework for Schools’, but the course books hardly use the latest results of computer education research. The present paper provides examples of how the results of computer education research can be integrated into teaching-learning materials and classroom practices and discusses the effectiveness and consequences of the different solutions, where tool-centred approaches are compared to problem-focused solutions.
Subject Classification: 94-01
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A constructive and metacognitive teaching path at university level on the Principle of Mathematical Induction: focus on the students' behaviours, productions and awareness
133-161Views:322We present the main results about a teaching/learning path for engineering university students devoted to the Principle of Mathematical Induction (PMI). The path, of constructive and metacognitive type, is aimed at fostering an aware and meaningful learning of PMI and it is based on providing students with a range of explorations and conjecturing activities, after which the formulation of the statement of the PMI is devolved to the students themselves, organized in working groups. A specific focus is put on the quantification in the statement of PMI to bring students to a deep understanding and a mature view of PMI as a convincing method of proof. The results show the effectiveness of the metacognitive reflections on each phase of the path for what concerns a) students' handling of structural complexity of the PMI, b) students' conceptualization of quantification as a key element for the reification of the proving process by PMI; c) students' perception of the PMI as a convincing method of proof.
Subject Classification: 97B40, 97C70
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An interactive animation for learning sorting algorithms: How students reduced the number of comparisons in a sorting algorithm by playing a didactic game
45-62Views:201Learning programming and understanding algorithms is one of the hardest tasks for novice computer science students. One of the basic algorithms they learn during the introductory programming and algorithms courses are the sorting algorithms. Students like learning these and other algorithms by animations and didactic games, however, these animations are not educationally useful in every case. In this article, we present our educational sorting game, which can be used to introduce the topic of sorting algorithms. The didactic game can be used later too, as a demonstrative tool for explaining the more efficient, quicksort algorithm. We conducted a pedagogical experiment, in which we examined the process of development of sorting algorithms by students while they used the mentioned didactic game. The results showed that students were able to create an algorithm to solve the sorting problem, and they improved its effectiveness by reducing the number of comparisons in the algorithm. They were also able to understand the importance of the efficiency of algorithms when we demonstrated them the quicksort algorithm using the same tool after the experiment. -
Dynamic geometry systems in teaching geometry
67-80Views:111Computer 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. -
A proposed application of Monte Carlo method in teaching probability
37-42Views:154Pupils' misconception of probability often results from lack of experience. Combining the concept of probability and statistics, the proposed application is intended for the teachers of mathematics at an elementary school. By reformulating the task in the form of an adventure, pupils examine a mathematical problem, which is too difficult for them to solve by combinatorial method. By recommending the simulation of the problem, we have sought to provide pupils with valuable experience of experimenting, recording and evaluating data. -
Potential, actual and practical variations for teaching functions: cases study in China and France
157-166Views:211This 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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Heuristic arguments and rigorous proofs in secondary school education
167-184Views:182In this paper we are going to discuss some possible applications of the mechanical method, especially the lever principle, in order to formulate heuristic conjectures related to the volume of three-dimensional solids. In the secondary school educational processes the heuristic arguments are no less important than the rigorous mathematical proofs. Between the ancient Greek mathematicians Archimedes was the first who made heuristic conjectures with the methods of Mechanics and proved them with the rigorous rules of Mathematics, in a period, when the methods of integration were not known. For a present day mathematician (or a secondary school mathematics teacher) the tools of the definite integral calculus are available in order to calculate the volume of three dimensional bodies, such as paraboloids, ellipsoids, segments of a sphere or segments of an ellipsoid. But in the secondary school educational process, it is also interesting to make heuristic conjectures by the use of the Archimedean method. It can be understood easily, but it is beyond the normal secondary school curriculum, so we recommend it only to the most talented students or to the secondary schools with advanced mathematical teaching programme.
