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Aspects théoriques de la classification à base de treillis
125-135Views:193La classification est une notion cruciale dans les systémes orientés objets et se fait de plus en plus présente en représentation de connaissances. Elle permet principalement de trouver des regularités dans un grand tableau de nombres. Dans ce sens général, il s'agit donc d'une méthode qui joue un role important dans différents domaines scientifiques oú les connaissances sont á organiser selon certaines hiérarchies (biologie, chimie, etc.). En informatique nous parlons aussi de langages de classes sans mentionner es aspects mathématiques de la classification. Dans cet article l'auteur a pour but de proposer une introduction á la classification á travers la notion de treillis. Nous sommes persuadés que l'étude de la classification permet aux étudiants de familiariser leurs connaissances sur la modélisation et la programmation orientée objet.
The classification is a crucial notion in the object oriented systems and more and more appears in the knowledge representation. It allows us to find the regularities in a huge table of numbers. In this general sense the classification plays an important role in various domains of science, where knowledge has to be organized into hierarchy (biology, chemistry, etc.) In the computer science the languages of classes are often studied without mathematical aspects of the classification. In this paper the author has the goal to propose an introduction to the classification through the notion of lattices.We are convinced that the study of classification allows students to enlarge their knowledge on the object oriented modelling and programming. -
An improvement of the classification algorithm results
131-142Views:143One of the most important aspects of the precision of a classification is the suitable selection of a classification algorithm and a training set for a given task. Basic principles of machine learning can be used for this selection [3]. In this paper, we have focused on improving the precision of classification algorithms results. Two kinds of approaches are known: Boosting and Bagging. This paper describes the use of the first method – boosting [6] which aims at algorithms generating decision trees. A modification of the AdaBoost algorithm was implemented. Another similar method called Bagging [1] is mentioned. Results of performance tests focused on the use of the boosting method on binary decision trees are presented. The minimum number of decision trees, which enables improvement of the classification performed by a base machine learning algorithm, was found. The tests were carried out using the Reuters 21578 collection of documents and documents from an internet portal of TV Markíza. -
Development of classification module for automated question generation framework
89-102Views:188Automatic 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. -
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. -
Integrating Didactic Games in Higher Education: Benefits and Challenges
1-15Views:876In our paper, we study the reasons for the introduction of didactic games and the way of their application in higher education, especially in teaching mathematics. After describing the main characteristics and needs of Generation Z students, we outline the advantages and drawbacks of gamification and game-based learning, followed by some new aspects to their classification. The idea of device-based grouping arose because the most commonly used methods require IC tools. Gen Zs naturally accept gamified learning materials available on digital and mobile platforms, but we must not forget about traditional games either. In higher education, especially in the case of small-group teaching there should also be room for traditional, specialized didactic games, of which we focus on the benefits of card games.
Subject Classification: 97C70, 97D20, 97D40, 97U70
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Integrating elements of data science into high-school teaching: Naïve Bayes-classification algorithm and programming in Python
307-316Views:263Probability 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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Artworks as illustrations in Hungarian high school Mathematics textbooks
103-117Views:182Three different series of Hungarian Mathematics textbooks used in grade 9-12 education for the past 30 years have been analysed in this research. Our aim is to show and evaluate how the visual arts have been connected to mathematical ideas in these textbooks. We have applied the six dimensions of evaluation, which have recently been introduced in (Diego-Mantec on, Blanco, Búa Ares, & González Sequeiros, 2019) to categorise the illustrations of the three different series. We show examples for each dimension from the textbooks, and we find that even if the number of artistic illustrations in these coursebooks have significantly increased, in most cases these sporadic examples are not closely related to the mathematical context, mainly used for ornamental purposes to decorate the core text. Based on this classification we conclude that the number of artistic illustrations with underlying math concepts making students' participation more active could and should be significantly increased.
Subject Classification: 97U20
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Fehleranalyse beim Lösen von offenen Aufgaben Ergebnisse einer empirischen Studie in der Grundschule
83-113Views:206Open problems play a key role in mathematics education, also in primary school. However, children in primary school work in many relations in a different way from learner in secondary school. Therefore, the (possibly) first confrontation with an open task could be problematical. Within the framework of an international paper and pencil test it was examined how far children of primary school notice the openness of a task and which mistakes they do during working on that task. In particularly are meant by openness different interpretations of the task, which all lead to a set of numbers with more than one element as a result. For evaluation, a common classification system was adapted by slightly modification of the original system. -
What does ICT help and does not help?
33-49Views:280Year by year, ICT tools and related teaching methods are evolving a lot. Since 2016, the author of the present lines has been looking for a connection between them that supports the development of mathematical competencies and could be integrated into Transcarpathian minority Hungarian language education too. As a doctoral student at the University of Debrecen, I experienced, for example, how the interactive whiteboard revolutionized illustration in Hungarian mathematics teaching, and how it facilitated students' involvement. During my research of teaching in this regard, in some cases, the digital solution had advantageous effects versus concrete-manipulative representation of
Bruner's too.
At the same time, ICT "canned" learning materials (videos, presentations, ...) allow for a shift towards repetitive learning instead of simultaneous active participation, which can be compensated for by the "retrieval-enhanced" learning method.
I have conducted and intend to conduct several research projects in a Transcarpathian Hungarian primary school. In the research so far, I examined whether, in addition to the financial and infrastructural features of the Transcarpathian Hungarian school, the increased "ICT-supported" and the "retrieval-enhanced" learning method could be integrated into institutional mathematics education. I examined the use of two types of ICT devices: one was the interactive whiteboard, and the other was providing one computer per student.
In this article, I describe my experiences, gained during one semester, in the class taught with the interactive whiteboard on the one hand, and in the class taught according to the "retrieval-enhanced" learning method on the other hand.
I compare the effectiveness of the classes to their previous achievements, to each other, and to a class in Hungary.Subject Classification: 97U70
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Gamification in Higher Education
87-106Views:876The way of thinking and the way of life of the today's children and teenagers have changed radically. Some of the well-established pedagogical methods that were used for decades have become obsolete. Therefore, we need to look for a new method to approach Generations Z and Alpha. Gamification, which has been known since 2010 and means the use of game elements in other areas of life, offers an opportunity to do so.
In addition to a brief description of gamification, my article shows some possibilities for using it at the university. Furthermore, I investigate the impact of gamification on the student in "Algorithms and Data Structures" university course.Subject Classification: 97P30
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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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Manipulatives and semiotic tools of Game of Go as playful and creative activity to learn mathematics in early grades in France
197-206Views:205This research develops resources to teach mathematics in French primary school by using the game of Go. A group of searchers, teachers and go players meet at university to produce teaching resources. These resources are implemented in the classroom. Then the group evaluate this implementation and improve the resources. The aim of this classroom research is to study the opportunities of the game of Go to learn mathematics and to propose a teacher training course to implement the game of Go in French primary schools in accordance with the French syllabus. Game of Go appears as a manipulative and semiotic tool to learn mathematics at primary school.
Subject Classification: 97D50, 97U60
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Square root in secondary school
59-72Views:290Although in Hungary, for decades, the calculation method of the square root of a real number is not in the mathematics curriculum, many of the taught concepts and procedures can be carried out using different square root finding methods. These provide an opportunity for students in secondary school to practice and deepen understand the compulsory curriculum. This article presents seven square-root- nding methods, currently teachable in secondary schools.
Subject Classification: A33, A34, F53, F54
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Solving word problems - a crucial step in lower secondary school education
47-68Views:315Algebra is considered one of the most important parts of Mathematics teaching and learning, because it lays the foundations of abstract thinking as well as reasoning abilities among the lower secondary school pupils who have just transited from the world of numbers and computations to the area of equalities, signs, symbols and letters. The present article focuses on the fact that how the transition from arithmetic to algebra can be made more smooth. We have concentrated our experiments towards the approach of algebraic reasoning and its utilities in filling the gap between arithmetic and beginning algebra in lower secondary school education.We also underline the importance of another approach in overcoming the challenges in the transition from arithmetic to algebra, to enhance and make algebraic learning more effective, with special considerations to word problem-solving processes. In our opinion, we have to go through three phases in the introducing of algebra in Grade 7 Mathematics education: Regula Falsi method (based only on numerical calculations); functional approach to algebra (which combines the numerical computation with letter-symbolic manipulation); and writing equations to word problems. The conclusions of the present article would be helpful to Mathematics teachers for applying themselves to develop the pupils’ interest in word problem-solving processes during algebra teaching classroom activities.
Subject Classification: 97B10, 97C30, 97C50, 97D10, 97D40
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Consequences of a virtual encounter with George Pólya
173-182Views:219The consequences of a virtual encounter with George Pólya as a teacher are recorded. An instance of his influence on my mathematical thinking is recounted through work on one of the problems in one of his books.
Subject Classification: 01A99, 11A05, 97-03, 97D50
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Willy Servais and Tamás Varga A Belgian Hungarian perspective on teaching school mathematics
29-38Views:207Willy Servais and Tamás Varga had a major influence on the development of mathematics education during the 1960s and 1970s, both in their home countries and internationally. In 1971 they jointly published Teaching School Mathematics–A Unesco Source Book, a review of curriculum reforms that were under way in different parts of the world. The book, presenting several modern syllabuses as well as examples of classroom techniques and segments of teacher-student dialogues, provided an often consulted guide to the field of mathematics education. We re-read this book and in this way acquire a unique insight into the modernization efforts of school mathematics during the 1960s and early 1970s. We take this opportunity to discuss the sometimes partly divergent views of Servais and Varga on modern mathematics education as reflected in this book.
Subject Classification: 97-03
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The shift of contents in prototypical tasks used in education reforms
203-219Views:179The paper discusses the shift of contents in prototypical tasks provoked by the current educational reform in Austria. The paper starts with the educational backboard of the process of changes in particular with the out tting of the students' abilities in different taxonomies and its implementation in the competence models of Mathematics. A methodological didactical point of view on the process is given additionally. Examples out of a specific collection of math problems which arise from the educational reform are integrated and analysed in the context of educational principles and methods. The discussion ends with a short evaluation of the role of traditional approaches to tasks in the ongoing reform. A bundle of tasks as proof that they are still alive is presented finally.
Subject Classification: 97B50, 97D40, 97D50
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Bemerkungen zur Prototypentheorie – Begriffs - und Konzeptbildung
365-389Views:126Psychological theories of prototypes are put forward by mathematical modelling. Some didactical consequences are discussed on the background of this analysis. By the help of an example (classification of convex quadrangles) hints are given for didactical interpretations of actual models of cognitive psychology dealing with problems of constructing prototypes. -
Better understanding mathematics by algorithmic thinking and computer programming
295-305Views:304Tamás Varga’s mathematics education experiment covered not just mathematics, but also other related topics. In many of his works he clearly stated that computer science can support the understanding of mathematics as much as mathematics supports informatics. On the other hand, not much later than the introduction of the new curriculum in 1978, personal computers started to spread, making it possible to teach informatics in classes and in extracurricular activities. Varga’s guided discovery approach has a didactic value for other age groups as well, not only in primary school. Its long-lasting effect can be observed even in present times. Having reviewed several educational results in the spirit of Tamás Varga, we have decided to present an extracurricular course. It is an open study group for age 12-18. Students solve problems by developing Python programs and, according to our experiences, this results in a deeper understanding of mathematical concepts.
Subject Classification: 97B10, 97B20, 97D50, 97N80, 97P20, 97P30, 97P40, 97P50, 97U70
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Capturing how students' abilities and teaching experiences affect teachers' beliefs about mathematics teaching and learning
195-212Views:254We developed an instrument to investigate the effect of students' abilities and teaching experiences on teachers' beliefs about teaching and learning of mathematics. In this pilot study, we used the instrument to measure the beliefs of 43 Indonesian math teachers and five additional teachers. Then, for further investigation, we interviewed those five additional teachers. Results from the 43 teachers' responses to the instrument show that in contrast to teachers with less than five years of teaching, teachers with more than five years elicit significantly different beliefs about mathematics teaching and learning in different contexts related to students' abilities. Teachers' reports in the further investigation indicate that teaching experiences with high and low ability students in teaching mathematics could be a possible explanation of this contrast.
Subject Classification: C20
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Development and assessment of non-cognitive skills among engineering students: a comparison across two universities
161-182Views:130Non-cognitive skills, such as logical thinking and problem solving, are crucial for success in engineering fields. To assess these skills in undergraduate engineering students, we designed a targeted test comprising four different types of tasks. The study was conducted among students at the Faculty of Engineering at the University of Debrecen, and the Faculty of Mechanical Engineering and Informatics at the University of Miskolc. The aim of this paper is to analyze the test results, gather students’ feedback, and examine the strength of the relationships between deductive reasoning, diagrammatic reasoning, and algebraic thinking.
Subject Classification: 97C20
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Mobile devices in Hungarian university statistical education
19-48Views:200The methodological renewal of university statistics education has been continuous for the last 30 years. During this time, the involvement of technology tools in learning statistics played an important role. In the Introduction, we emphasize the importance of using technological tools in learning statistics, also referring to international research. After that, we firstly examine the methodological development of university statistical education over the past three decades. To do this, we analyze the writings of statistics teachers teaching at various universities in the country. To assess the use of innovative tools, in the second half of the study, we briefly present an online questionnaire survey of students in tertiary economics and an interview survey conducted with statistics teachers.
Subject Classification: 97-01, 97U70, 87K80
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Mapping students’ motivation in a problem oriented mathematics classroom
111-121Views:231This research focuses on mapping students’ motivation by implementing problem-solving activities, namely how the problem-oriented approach affects the students’ commitment, motivation, and attitude to learning. As a practicing teacher, the author faced difficulties with motivation and sought to improve her practice in the form of action research as described in this paper. Based on the literature, the author describes sources of motivation as task interest, social environment, opportunity to discover, knowing why, using objects, and helping others. The author discusses the effect of problem-oriented teaching on the motivation of 7th-grade students. In this paper, the results of two lessons are presented.
Subject Classification: 97C20, 97D40, 97D50, 97D60
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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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Concept systematization with concept maps in data modelling
149-166Views:154An important goal of concept learning is that students can allocate concepts in the hierarchical system of concepts. In the data modelling course, first, we supported concept systematization with worksheets in which the students had to fill in the blank hierarchical figures of classification of the concepts or blank Venn diagrams describing the relationships between concepts. The hierarchical systems, however, are somewhat restricted to the description of connections. The filling in Venn diagrams did not deliver the expected result, so our attention turned to concept maps. In this paper we introduce the concept maps we drew. Then we evaluate the results of concept mapping survey conducted among students. The survey was done in three courses. We compare the results of our survey with the result of an earlier concept systematising survey.
