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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:169Dynamic 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. -
Many paths lead to statistical inference: Should teaching it focus on elementary approaches or reflect this multiplicity?
259-293Views:243For statistics education, a key question is how to design learning paths to statistical inference that are elementary enough that the learners can understand the concepts and that are rich enough to develop the full complexity of statistical inference later on. There are two ways to approach this problem: One is to restrict the complexity. Informal Inference considers a reduced situation and refers to resampling methods, which may be completely outsourced to computing power. The other is to find informal ways to explore situations of statistical inference, also supported with the graphing and simulating facilities of computers. The latter orientates towards the full complexity of statistical inference though it tries to reduce it for the early learning encoun-ters. We argue for the informal-ways approach as it connects to Bayesian methods of inference and allows for a full concept of probability in comparison to the Informal Inference, which reduces probability to a mere frequentist concept and – based on this – restricts inference to a few special cases. We also develop a didactic framework for our analysis, which includes the approach of Tamás Varga.
Subject Classification: 97K10, 97K70, 97K50, 97D20
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Development of spatial perception in high school with GeoGebra
211-230Views:157In everyday life, on numerous occasions we need to project 3D space onto a plane in order to activate our spatial perception. While our ability in this area can be improved, and considering several national and international research results, the development is even necessary on all levels of education. GeoGebra, as a supplement to previously used tools, has proven to be very useful respective to the development. We have many possibilities to display spatial elements in GeoGebra and to apply such kind of worksheets among 15-18 year old students. I show the results of the 2011/2012 school years connected to the development of spatial perception and the results of an input case survey, which also justifies the need for development. -
Report on the "English Language Section of Varga Tamás Days 2009"
169-175Views:147The 9th English Language Section as a part of the Varga Tamás Days was organised by the Department of Mathematics Education at the Teacher Training Institute of the Eötvös Loránd University. We report on the talks and the following discussions in this section. -
A retrospective look at discovery learning using the Pósa Method in three Hungarian secondary mathematics classrooms
183-202Views:361While the Pósa Method was originally created for mathematical talent management through extracurricular activities, three "average" public secondary school classrooms in Hungary have taken part in a four-year experiment to implement the Pósa Method, which is based on guided discovery learning of mathematics. In this paper, we examine the students' and teachers' reflections on the Pósa Method, and how student perspectives have changed between their first and last year of high school. Overall, teachers and students had a positive experience with the Pósa Method. Furthermore, our research indicated that this implementation has met several objectives of the Pósa Method, including enjoyment of mathematics and autonomous thinking.
Subject Classification: 97D40
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Veranschaulichung der Lehrstoffstruktur durch Galois-Graphen
217-229Views:181In 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. -
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. -
Report of meeting Researches in Didactics of Mathematics and Computer Sciences: January 30 - February 1, 2009, Debrecen, Hungary
165-186Views:149The meeting Researches in Didactics of Mathematics and Computer Sciences was held in Debrecen, Hungary from January 30 to February 1, 2009. The 49 Hungarian participants – including 15 PhD students – came from 18 cities and represented 29 institutions of higher education. The abstracts of the talks and the posters and also the list of participants are presented in this report. -
Understanding the spatiotemporal sample: a practical view for teaching geologist students
89-99Views:143One of the most fundamental concept of statistics is the (random) sample. Our experience – acquired during the years of undergraduate education – showed that prior to industrial practice, the students in geology (and, most probably, in many other non-mathematics oriented disciplines as well) are often confused by the possible multiple interpretation of the sample. The confusion increases even further, when samples from stationary temporal, spatial or spatio-temporal phenomena are considered. Our goal in the present paper is to give a viable alternative to this overly mathematical approach, which is proven to be far too demanding for geologist students.
Using the results of an environmental pollution analysis we tried to show the notion of the spatiotemporal sample and some of its basic characteristics. On the basis of these considerations we give the definition of the spatiotemporal sample in order to be satisfactory from both the theoretical and the practical points of view. -
Report of Meeting Researches in Didactics of Mathematics and Computer Sciences: 31 March – 2 April, 2023 Oradea, Romania
83-107Views:370The meeting Researches in Didactics of Mathematics and Computer Sciences was held in Oradea, Romania, at Partium Christian University, from 31 March to 2 April, 2023. It was organized by the Doctoral School of Mathematical and Computational Sciences of the University of Debrecen and Partium Christian University. The 85 participants – including 18 PhD students – came from 9 countries and represented 30 institutions of higher and secondary education. There were 4 plenary and 53 session talks in the program.
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Strategies used in solving proportion problems among seventh-grade students
101-127Views:97In the 2023/2024 school year, 146 seventh-grade Hungarian students (aged 12-13) participated in our classroom experiment on solving proportion problems. At the beginning and the end of the teaching phase, both the experimental and the control groups solved a test. Regarding the answers of the students, in the pre- and post-test mostly consisting of word problems, we examined the success of solving the problems, as well as the solution strategies. For this, we used the strategies of proportional thinking that already exist in the literature of mathematical didactics. We intended to answer the following questions: To what extent and in which ways do the different types of problems and texts influence the solution strategies chosen by the students? How successfully do seventh-grade students solve proportion problems?
Subject Classification: 97D50, 97F80
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Forming the concept of congruence I.
181-192Views:145Teaching isometries of the plane plays a major role in the formation of the congruence-concept in the Hungarian curricula.
In the present paper I investigate the way the isometries of the plane are traditionally introduced in most of the textbooks, especially the influence of the representations on the congruence concept, created in the teaching process.
I am going to publish a second part on this topic about a non-traditional approach (Forming the concept of congruence II). The main idea is to introduce the isometries of the two dimensional plane with the help of concrete, enactive experiences in the three dimensional space, using transparent paper as a legitimate enactive tool for building the concept of geometric motion. I will show that this is both in strict analogy with the axioms of 3-dimensional motion and at the same time close to the children's intuitive concept of congruence. -
Various systems in a single mathematical model
1-13Views:118Our aim is to study differential equations and systems described by them which have great historical importance and are considered to be fundamental on different levels of education.
Due to their simplicity these are suitable for those who deal with this topic and want to gain useful experience in this field.
Furthermore, our aim is to give these equations a general form which facilitates the studying of the different models by computer even for an individual programmer. At the same time it facilitates the use of different mathematical auxiliary-programmes.
By giving the equations this way we get a chance of studying the relations between the individual systems. -
Analysis of a problem in plane geometry discussed in an 11th grade group study session
181-193Views:140The main aim of this paper is to show those strategies and proof methods we try to teach in secondary maths education through an interesting geometric problem: Find a relation for the sides of a triangle where an angle is the double of another angle. Is the converse also true? Is it possible to generalize the problem? We try to answer these questions while discussing the upcoming difficulties in detail and presenting more possible solutions. Hopefully the paper can be successfully used in study group sessions and problem solving seminars in secondary schools. -
Fostering engineering freshmen’s shifts of attention by using Matlab LiveScript for solving mathematical tasks
1-14Views:270We designed an experimental path including a summative assessment phase, where engineering freshmen are involved in solving mathematical tasks by using Matlab LiveScripts. We analyzed the students’ answers to a questionnaire about their perceived impact of the use of Matlab on their way to solve mathematical tasks. The main result is that students show shifts of attention from computations to other aspects of problem solving, moving from an operational to a structural view of mathematics.
Subject Classification: 97U70, 97H60
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Metadata formats and the new markup language from library field
85-102Views:130Using new communication technologies libraries must face continuously new questions, possibilities and expectations. This study discusses some library-related aspects of our electronic era and how computer-related data formats affect bibliographic dataprocessing to give a survey of the most important results.
Altough library systems have been supporting the creation and maintenance of MARC records for decades, the need to create other forms of metadata is relatively new. In higher education it is important that students get acquainted with these new aspects of bibliographic description and bibliographic dataprocessing.
In our department I launched the BDML development project in order to standardize bibliographic description (based on Hungarian standard 3424/1) with the help of XML. The development aims at the creation of a set of standards that provides information about the structure of bibliographic description in XML form. In the second part of the paper I give a brief presentation and summary of BDML. -
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. -
Heuristic arguments and rigorous proofs in secondary school education
167-184Views:185In 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. -
Teaching word processing – the practice
247-262Views:177I compared two surveys, which were aimed to check the word processing ability of students in high schools and universities. The surveys were carried out ten years apart from one another, in 1997 and 2006. The results clearly show that most of the students are not able to use word processors properly. In the survey of 1997 I found explanation for this underperformance in the lack of computers and teachers. However, the results of the second survey did not prove any better than the results of the first, and in 2006 neither the number of computers nor the number of teachers can be blamed. What else then? I suggest that the reason for this general ignorance, for this `modern illiteracy' is the ignorance of the teachers. Until the teachers are not prepared and the senior students of the universities leave the education system without a proper knowledge of the required subjects, there is little chance that they would be able to teach word processing at a satisfactory level. -
Teaching multiparadigm programming based on object-oriented experiences
171-182Views:168Multiparadigm programming is an emerging practice in computer technology. Co-existence of object-oriented, generic and functional techniques can better handle variability of projects. The present paper gives an overview of teaching multiparadigm programming approach through typical language concepts, tools in higher education. Students learning multiparadigm-oriented subjects would gain considerable expertise, which is highly needed by the industrial side in large-scale application development. -
Learning and teaching combinatorics with Sage
389-398Views:178Learning Mathematics is not an easy task, since this subject works with especially abstract concepts and sophisticated deductions. Many students lose their interest in the subject due to lack of success. Computer algebra systems (CAS) provide new ways of learning and teaching Mathematics. Numerous teachers use them to demonstrate concepts, deductions and algorithms and to make learning process more interesting especially in higher education. It is an even more efficient way to improve the learning process, if students can use the system themselves, which helps them to practice the curriculum.
Sage is a free, open-source math software system that supports research and teaching algebra, analysis, geometry, number theory, cryptography, numerical computation, and related areas. I have been using it for several years to aid the instruction of Discrete Mathematics at Óbuda University. In this article I show some examples how representations provided by this system can help in teaching combinatorics. -
Teaching of problem-solving strategies in mathematics in secondary schools
139-164Views:132In the Hungarian mathematics education there is no explicit teaching of problem-solving strategies. The best students can abstract the strategies from the solutions of concrete problems, but for the average students it is not enough. In our article we report about a developmental research. The topic of the research was the explicit teaching of two basic strategies (forward method, backward method). Based on our experiences we state that it is possible to increase the effectivity of students' problemsolving achievement by teaching the problem-solving strategies explicitly. -
Würfel und Augensummen – ein unmögliches Paar
71-88Views:179It is well known that the values 2, 3, ..., 12 of the sum of eyes that appear when throwing two regular dice are not equally distributed. It can also be shown that no matter how the dice are falsified (or if only one of them is being manipulated) they can never reach the same probability concerning the sum of eyes ([8], 91 et seq.). This discovery can be generalized for n ≥ 2 dice. Various results of algebra and (real) calculus are used, so that a connection between two different mathematical fields can be realized. Such a connection is typical and often provides a large contribution for mathematics (because it frequently leads to a successful attempt of solving a special problem) and therefore examples of this sort should also be included in the mathematical education at schools as well as in the student teachers' university curriculum for the study of mathematics. -
Young women's barriers to choose IT and methods to overcome them - A case study from Hungary
77-101Views:306Women's scarcity in the STEM, especially in the IT sector is pronouncedly evident. Young women are obstructed from entering and remaining in IT by a broad range of social, educational, and labor market factors. In our paper, we would like to analyze the main barriers girls face in choosing IT, while also proposing potential methods to help them overcome these obstacles. In the second part of the paper, we will present a case study to illustrate in detail how the combination of the above methods can be put into practice to address and tackle the complex set of barriers girls face. We will first introduce a Hungarian annual program, Girls' Day ("Lányok napja"), specifically aimed to promote STEM to girls, then we will present two specific events organized for the 2020 edition of the program and designed with the above principles in mind. The interactive presentation, exposing girls to female role models of the field in a gamified way, and a game development exercise, building Scratch programming skills, have attempted to provide young women both with positive perspectives and experiences in IT, which are instrumental in helping them to surmount entrenched obstacles and raise their interest in the field.
Subject Classification: 97P10, 97U30
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A role of geometry in the frame of competencies attainment
41-55Views:165We discuss aspects of the Education Reform from teaching to educational system. In this context we recognize some problems in recognition of some competencies that students need to achieve and we present how we have developed the measurement method of spatial abilities and problem solving competence. Especially, we investigate how students use spatial visualization abilities in solving various problems in other mathematical course. We have tested how students use their spatial abilities previously developed in geometry courses based on conceptual approach to solve a test based on procedural concept in Mathematical Analysis course.
