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Promoting a meaningful learning of double integrals through routes of digital tasks
107-134Views:168Within a wider project aimed at innovating the teaching of mathematics for freshmen, in this study we describe the design and the implementation of two routes of digital tasks aimed at fostering students' approach to double integrals. The tasks are built on a formative assessment frame and classical works on problem solving. They provide facilitative and response-specific feedback and the possibility to request different hints. In this way, students may be guided to the development of well-connected knowledge, operative and decision-making skills. We investigated the effects of the interaction with the digital tasks on the learning of engineering freshmen, by comparing the behaviours of students who worked with the digital tasks (experimental group, N=19) and students who did not (control group, N=19). We detected that students in the experimental group showed more exibility of thinking and obtained better results in the final exam than students in the control group. The results confirmed the effectiveness of the experimental educational path and offered us interesting indications for further studies.
Subject Classification: 97D40, 97U70, 44A45
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The appearance of the characteristic features of the mathematical thinking in the thinking of a chess player
201-211Views:31It 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. -
On an international training of mathematically talented students: assets of the 20 years of the “Nagy Károly Mathematical Student-meetings”
77-89Views:25The focus of this paper is to present the gems of the "Nagy Károly Mathematical Student-meetings" in Rév-Komárom (Slovakia) from 1991 to 2010. During these 20 years there was done a lot of work to train mathematically talented students with Hungarian mother tongue and to develop their mathematical thinking, and to teach them problem solving and heuristic strategies for successful acting on the competitions. We collected the most interesting problems and methods presented by the trainer teachers. -
Hyperbolische 5-Rechtecke
111-123Views:21The main topic of this paper is the investigation of 5-pentagons whose interior angles are all right angles within the hyperbolic geometry (so-called 5-rectangles). Some knowledge of elementary hyperbolic geometry is required.
At first the existence of such a polygon is shown by construction within the Kleinmodel. Then two formulas due to D. M. Y. Sommerville [3] are proved. This means to juggle with trigonometric formulas of hyperbolic geometry.
In the last years a big number of papers concerning hyperbolic geometry was published. This proves that the interest in this nice discipline is growing again. -
Teaching probability using graph representations
103-122Views:27The main objective of this paper is to present an elementary approach to classical probability theory, based on a Van Hiele type framework, using graph representation and counting techniques, highly suitable for teaching in lower and upper secondary schools. The main advantage of this approach is that it is not based on set theoretical, or combinatorial knowledge, hence it is more suitable for beginners and facilitates the transitions from level 0 to level 3. We also mention a few teaching experiences on different levels (lower secondary school, upper secondary school, teacher training, professional development, university students) based on this approach. -
Applications of methods of descriptive geometry in solving ordinary geometric problems
103-115Views:24The importance of descriptive geometry is well-known in two fields. Spatial objects can be mapped bijectively onto a plane and then we can make constructions concerning the spatial objects. The other significance of descriptive geometry is that mathematical visual perception of objects in three-dimensional space can be improved by the aid of it. The topic of this paper is an unusual application of descriptive geometry. We may come across many geometric problems in mathematical competitions, in entrance examinations and in exercise books whose solution is expected in a classical way, however, the solution can be found more easily and many times more general than it is by the standard manner. We demonstrate some of these problems to encourage to use this geometric method. Understanding the solution requires very little knowledge of descriptive geometry, however, finding a solution needs to have some idea of descriptive geometry. -
MRP tasks, critical thinking and intrinsic motivation to proving
149-168Views:23The lack of students' need for proof is often discussed. This is an important topic, on which quite a few others have written ([26], [27], [28], [17], [8]). Nevertheless, there is limited research knowledge about how teacher can participate in process of raising of students' intrinsic motivation to proving. In this article, we discuss relationships between intrinsic motivation to proving, critical thinking and special activity – engaging with so-called MRP tasks. We present here results of a research carried out by author in two elementary schools (21 classes, grade 5-9) in Ruzomberok, Slovakia. We identified the interesting relationship between students' dealing with MRP tasks and increasing of their intrinsic motivation to proving. -
"How to be well-connected?" An example for instructional process planning with Problem Graphs
145-155Views:90Teachers’ design capacity at work is in the focus of didactical research worldwide, and fostering this capacity is unarguably a possible turning point in the conveyance of mathematical knowledge. In Hungary, the tradition hallmarked by Tamás Varga is particularly demanding towards teachers as they are supposed to be able to plan their long-term processes very carefully. In this contribution, an extensive teaching material designed in the spirit of this tradition will be presented from the field of Geometry. For exposing its inner structure, a representational tool, the Problem Graph is introduced. The paper aims to demonstrate that this tool has potential for analyzing existing resources, helping teachers to reflect on their own preparatory and classroom work, and supporting the creation of new designs.
Subject Classification: 97D40, 97D50, 97D80, 97G10, 97U30
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Teaching Gröbner bases
57-76Views:21In 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. -
Online tests in Comprehensive Exams – during and after the pandemic
77-93Views:67The Covid-19 pandemic accelerated the development of electronic (e-learning) assessment methods and forced their use worldwide. Many instructors and students had to familiarize themselves with the form of distance education. During and since Covid-19 in Hungary, at the Faculty of Engineering of the University of Debrecen, the written part of the Comprehensive Exam in Mathematics is organized in a computer lab of the university using an online test. Our goal is that the results of the tests may be as reliable as possible in terms of measuring the students’ knowledge, and thus the grades given based on the test results would be realistic. In this paper, we show the analysis of a sample written exam and compare the real exam results of students who were prepared for the comprehensive exam during Covid-19 and who have participated in face-to-face education since then. The tools provided by the Moodle system necessary for comparison are also presented.
Subject Classification: 97D40, 97D70, 97U50
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Development of classification module for automated question generation framework
89-102Views:25Automatic question generation is in the focus of recent researches which includes bordering disciplines like education, text mining, knowledge-engineering. The elaborated system generates multi-choice questions from textbooks without using an external semantic database. One of the base modules of the system is the classification module defining the extracted word. This paper describes modules of the framework including a detailed analysis of the classification part. We show the operability of the elaborated system through a practical test. -
Comparative geometry on plane and sphere: didactical impressions
81-101Views:2Description of experiences in teaching comparative geometry for prospective teachers of primary schools. We focus on examples that refer to changes in our students' thinking, in their mathematical knowledge and their learning and teaching attitudes. At the beginning, we expected from our students familiarity with the basics of the geographic coordinate system, such as North and South Poles, Equator, latitudes and longitudes. Spherical trigonometry was not dealt with in the whole project. -
Our digital education habits in the light of their environmental impact: the role of green computing in education
69-86Views:113With the increasing use of IT tools, the environmental impacts they generate have also increased. Education is increasingly relying on digital tools to become a major emitter of CO2 itself. Therefore, the task of education is to teach future generations how to use IT tools efficiently while being environmentally aware. In addition to some forms of green computing, we show the level and ratio of those teachers who have corresponding IT knowledge in the Hungarian education. In this study, we present the justification of the problem through a case study, which estimates the Internet traffic of a website streaming popular educational resources. In addition, we will examine the extent to which national and international educational organization and guidance documents address the development of digital environmentally aware thinking. Based on the content of this study, we suggest some considerations for content developers to decide if they really need to create the digital content.
Subject Classification: 97P99, 94-06, 94-02
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The Project Method and investigation in school mathematics
241-255Views:27The Project Method (PM) is becoming more common in the teaching of mathematics. Most of the time, Project Method means solving open and relatively wide formulated problems for the application of particular mathematical topics and the solving of everyday life problems.
At present many experts in the theory of teaching mathematics advocate teaching activities as the characteristic for most mathematical work in the classroom. Thus, there is a question: whether it is possible or eventual desirable to use the PM for solving genuine mathematical problems. This paper deals with this question and discusses the connection between the PM and investigation of new mathematical knowledge for students. Our experience has shown that the PM in connection with investigations can be a useful and effective approach to teaching mathematics. -
Implementation opportunities of the Moodle learning management system in virtual environment the Sloodle project
275-293Views:27Using e-learning was firstly appeared in companies' sphere. It should be very useful if learning management systems were applied. Nowadays e-learning is used in different fields and gives useful informations in case of basics and its knowledge. It is essential to know the arranging technics and applicated handling methods of some supporting learning management systems of e-learning. The Moodle is the best-known learning management system.
The Second Life is one of the virtual environments which is useful in learning-teaching methods that is used in most educational institute all over the world. Sloodle is an open source project which connects the Second Life with Moodle learning management system. Sloodle is a kind of "bridge" in which different kind of activities and registering and provided in both Moodle and Second Life.
In our department, University of Debrecen Health Faculty of Nyíregyháza ILIAS learning management system has operated since February, 2008. In the interest of higher level education we decided to use and made available some courses through Moodle learning management system.
Some tools of Sloodle will be presented in our article. It will be the first study for our research in which we would use the Moodle learning management system, the virtual environment of Second Life and the project of Sloodle itself. Our article will contain the starting details and its statistical confirmation of our Sloodle project. We like to demonstrate that the results of the Sloodle-aided group are significantly better than the results of the control group in the most cases. -
The use of e-tests in education as a tool for retrieval practice and motivation
59-76Views:84In 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