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Aspects théoriques de la classification à base de treillis
125-135Views:192La 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:142One 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:187Automatic 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. -
Integrating Didactic Games in Higher Education: Benefits and Challenges
1-15Views:873In 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: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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Artworks as illustrations in Hungarian high school Mathematics textbooks
103-117Views:181Three 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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Nice tiling, nice geometry!?!
269-280Views:146The 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. -
The role of representations constructed by students in learning how to solve the transportation problem
129-148Views:317The purpose of the research presented in this paper was to study the role of concrete and table representations created by students in learning how to solve an optimization problem called the transportation problem. This topic was learned in collaborative groups using table representations suggested by teachers in 2021. In 2022, the researchers decided to enrich the students’ learning environment with concrete objects and urged the students to use them to present the problem to be solved. The students did it successfully and, to be able to record it in their notebooks, they constructed a table representation by themselves without any help from their teacher. After that, they managed to solve the problem by manipulating the objects. At the same time, each step in the solution was presented with changes in the table. The students were assessed before (pre-test) and after collaborative learning (test) in both academic years. The pre-test results were similar, but the test results were better in 2022. Therefore, it can be concluded that using concrete and table representations constructed by students in learning how to solve transportation problems makes collaborative learning more constructivist and more effective than when they use only table representations suggested by their teachers.
Subject Classification: 97M10, 97M40
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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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The transition problem in Hungary: curricular approach
1-16Views:296The curricular background of the transition problem from highschool to universty is analysed in Hungary. While students finish their mathematical studies successfully at highschool, pass their final exams, this knowledge seems to disappear at their first year at university. We investigate the mathematical knowledge expected by the Hungarian universities and compare it to expectations of the National Core Curriculum. Based on the levelling tests of four universities we created a seven problem test for highschool students containing very basic problems required both by the universities and the National Core Curriculum. We analyse the results of the test.
Subject Classification: D34, D35
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Didactical remarks on the changes in the requirements of the matriculation exam in Mathematics in Hungary
95-110Views:201Students within the Hungarian education system typically take a matriculation exam to obtain a secondary education certificate, which also serves as a prerequisite for university admission. Public education is regulated at different levels. One of its most fundamental elements is the National Core Curriculum, the current version of which came into force in September 2020. It is crucial to adapt the requirements of the matriculation exam in mathematics to this and ensure transparent communication about the changes. Regarding this, there exists a sample paper that contains tasks that one can reasonably expect in the actual exam in the spring. Since I have been working as a private math tutor for almost a decade and have been preparing students for the matriculation exam since then, I intend to highlight the most outstanding features from a didactic point of view based on the analysis of this sample paper.
Subject Classification: 97A30, 97B10, 97B70, 97D60, 97U40
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Lehre der Trigonometrie anhand realistischer Aufgaben im Online-Unterricht
87-105Views:214The aim of our study was to explore the effects of the active use of realistic exercises in the field of trigonometry. We taught a group of 14 pupils, who were in grade 11. The most of them told us they did not plan mathematics-related studies in the future. We included realistic exercises into our teaching plan, which covered the fields of scalar product, as well as the sine and cosine theorems. Our teaching experiment was done within the framework of online teaching. Effects on the motivation, performance and results of the students were taken into consideration. We also attempted to examine the effects of online teaching on motivation and whether the use of realistic exercises is worthwhile in an online classroom environment. Performance of the students showed a tendency of improvement when they were dealing with the material through realistic exercises even despite the teaching happened online.
Subject Classification: 97C70, 97D40, 97G60
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Solution of an open reality based word-problem in two secondary schools
143-156Views:258This survey through an open reality based word problem is intended to assess - in two secondary schools in Komárom (Hungary) and in Komarno (Slovakia, Hungarian name: Révkomárom) in grade 10 - the ability of students to realize openness of a task. The comparison is justified by the fact that the language of teaching is Hungarian in both secondary schools, but with different curricula. This survey is related to the Content Pedagogy Research Program by the Hungarian Academy of Sciences. It is preceded by several surveys with a word problem (Pocket Money) of the third author and led by her between 2012 and 2015, and within that project in 2017 within a large sample test, among about 1500 students and university students in Hungary (?, ?) (?, ?). In our research we wanted first to assess how openly work students in two schools of the two cities mentioned in solving the same task. The answer to this question was similar to the large sample test results, so most of the students worked in a closed way, when solving this word problem. So we went on and tried to explore how students thought about their own solution given to this task, through mixed-type interviews.
Subject Classification: 97D70, 97F90, 97D50, 97M10
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Online tests in Comprehensive Exams – during and after the pandemic
77-93Views:316The 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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Virtual manipulatives in inquiry-based approach of 3D problems by French 5th graders
229-240Views:219The aim of this research is to study the appropriation of a 3D environment by learners in an a-didactical situation of problem solving. We try to evaluate the relevance of the virtual 3D environment in the development of students' cognitive and metacognitive abilities. We implanted a problem-solving activity related to a 3D cube situation with an empty part in the cube in different French primary school areas in May 2019. In the experimental group each learner works individually with a PC-computer where the virtual environment ANIPPO is implemented. In the control group the pupils work in a traditional class environment. We present the results of this pre-experimentation.
Subject Classification: 97D50, 97U60, 97U70
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Simple Variations on The Tower of Hanoi: A Study of Recurrences and Proofs by Induction
131-158Views:376The Tower of Hanoi problem was formulated in 1883 by mathematician Edouard Lucas. For over a century, this problem has become familiar to many of us in disciplines such as computer programming, algorithms, and discrete mathematics. Several variations to Lucas' original problem exist today, and interestingly some remain unsolved and continue to ignite research questions. Nevertheless, simple variations can still lead to interesting recurrences, which in turn are associated with exemplary proofs by induction. We explore this richness of the Tower of Hanoi beyond its classical setting to compliment the study of recurrences and proofs by induction, and clarify their pitfalls. Both topics are essential components of any typical introduction to algorithms or discrete mathematics.
Subject Classification: A20, C30, D40, D50, E50, M10, N70, P20, Q30, R20
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Strategies used in solving proportion problems among seventh-grade students
101-127Views:93In 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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Realizing the problem-solving phases of Pólya in classroom practice
219-232Views:345When teaching mathematical problem-solving is mentioned, the name of Pólya György inevitably comes to mind. Many problem-solving lessons are planned using Pólya's steps and helping questions, and teachers often rely on his heuristics even if their application happens unconsciously. In this article, we would like to examine how the two phases, Making a plan and Looking back, can be realized in a secondary school mathematics lesson. A case study was designed to observe and analyse a lesson delivered using cooperative work.
Subject Classification: 97B10, 97C70, 97D40, 97D50
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Inquiry based mathematics education and the development of learning trajectories
63-89Views:1587This article is based on the panel on inquiry based mathematics education and the development of learning trajectories held at the VARGA 100 Conference. After an introduction presenting the theme and organization of the panel, this article focuses on the diversity of conceptualizations of inquiry based education existing today in mathematics education and their influence on the vision and development of learning trajectories. More precisely, it considers the conceptualizations respectively associated with Realistic Mathematics Education, Genetic Constructivism, Tamás Varga’s educational approach and the Anthropological Theory of the Didactic, presented by the panellists, and also shows the efforts undertaken in European projects to reach consensusal visions.
Subject Classification: 97C30Q, 97D10, 97D20, 97D30, 97D40, 97D50
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Teaching performance testing
17-33Views:243Performance testing plays a vital role in the verification of large scale software systems. It is used for testing the speed, responsiveness, capacity and stability of the investigated system. However, despite the significance of this topic, the effort invested in teaching performance testing in Computer Science is insufficient. The current paper shows, how the fundamentals of performance testing can be demonstrated to students both from a theoretical and a practical viewpoint through step-by-step practical examples that are used in the industry. It is also discussed how a basic toolchain can be set up for performance tests using only free tools. With the presented examples, the reader will be able to take first steps in the performance testing area.
Subject Classification: 68M15
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Decision based examination of object-oriented programming and Design Patterns
83-109Views:152On 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. -
Straight line or line segment? Students’ concepts and their thought processes
327-336Views:265The article focuses on students’ understanding of the concept of a straight line. Attention is paid to whether students of various ages work with only part of a straight line shown or if they are aware that it can be extended. The presented results were obtained by a qualitative analysis of tests given to nearly 1,500 Czech students. The paper introduces the statistics of students’ solutions, and discusses the students’ thought processes. The results show that most of the tested students, even after completing upper secondary school, are not aware that a straight line can be extended. Finally, we present some recommendations for fostering the appropriate concept of a straight line in mathematics teaching.
Subject Classification: 97C30, 97D70, 97G40
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Introductory Computer Programming Courses in Mathematics Curriculum
19-30Views:262We present the results of surveys and curricular research on introductory computer programming courses that are required or recommended for mathematics degrees at U.S. colleges and universities. Our target schools were those with populations between 5,000 and 20,000 undergraduate students. A key result is a synopsis of programming languages in use in these introductory courses with Java, Python and C + + holding the top three spots. We found that 85% of the 340 schools in our pool require or recommend an introductory programming course as a component of a mathematics degree. Furthermore, most of these introductory programming courses are taught by faculty outside of the mathematics department. These results indicate that mathematics faculty value computer programming and should be actively involved in setting learning outcomes, incorporating skills and concepts learned in introductory programming courses into subsequent mathematics courses, and determining programming languages in use.
Subject Classification: 97D30, 97P20, 97P40
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Experiences in the education of mathematics during the digital curriculum from the perspective of high school students
111-128Views:308Due to the COVID-19 epidemic, Hungarian schools had to switch to a digital curriculum for an extended period between 2019 and 2021. In this article, we report on the experiences regarding the education of mathematics during the digital curriculum in the light of the reinstated on-site education, all through the eyes of high school students. Distance education brought pedagogical renewal to the lives of many groups. Students were asked about the positives and negatives of this situation.
Subject Classification: 97C90
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"How to be well-connected?" An example for instructional process planning with Problem Graphs
145-155Views:222Teachers’ 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
