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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 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. -
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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On the nine-point conic of hyperbolic triangles
195-211Views:118In the Cayley–Klein model, we review some basic results concerning the geometry of hyperbolic triangles. We introduce a new definition of the circumcircle of a hyperbolic triangle, guaranteed to exist in every case, and describe its main properties. Our central theorem establishes, by means of purely elementary projective geometric arguments, that a hyperbolic triangle has a nine-point conic if and only if it is a right triangle.
Subject Classification: 51M09
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An examination of descriptive statistical knowledge of 12th-grade secondary school students - comparing and analysing their answers to closed and open questions
63-81Views:214In this article, we examine the conceptual knowledge of 12th-grade students in the field of descriptive statistics (hereafter statistics), how their knowledge is aligned with the output requirements, and how they can apply their conceptual knowledge in terms of means, graphs, and dispersion indicators. What is the proportion and the result of their answers to (semi-)open questions for which they have the necessary conceptual knowledge, but which they encounter less frequently (or not at all) in the classroom and during questioning? In spring 2020, before the outbreak of the pandemic in Hungary, a traditional-classroom, “paper-based” survey was conducted with 159 graduating students and their teachers from 3 secondary schools. According to the results of the survey, the majority of students have no difficulties in solving the type of tasks included in the final exam. Solving more complex, open-ended tasks with longer texts is more challenging, despite having all the tools to solve them, based on their conceptual knowledge and comprehension skills. A valuable supplement to the analysis and interpretation of the results is the student attitudes test, also included in the questionnaire.
Subject Classification: 97K40, 97-11, 97D60
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Freudenthal fantasy on the bus, an American adaptation
133-142Views:180In the 1960’s two mathematicians, Hans Freudenthal in the Netherlands and Tamás Varga in Hungary, had argued that people learn mathematics by being actively involved and investigating realistic mathematical problems. Their method lives on in today’s teaching and learning through the various components of cooperative and active learning, by taking ownership in learning, and learning through student dialogue. The goal is to create a welcoming classroom atmosphere in which play takes the front seat. One such scenario is visiting various (animal) stations at the zoo by bus (illustrated by pictures). Passengers are getting on and off the bus at each station (illustrated by arrows), which is modeled on the open number line. This adapted and modified action research was carried out with 5-yearl-old children in public schools of Staten Island, NY in 2019.
Subject Classification: 97D40, 97F20, 97F30
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The time spent on board games pays off: links between board game playing and competency motivation
119-131Views:348The impact playing has on the development of thinking is an important topic of psychology of learning, brain research and mathematics didactics.
Our research is also connected to the aforementioned topic. We investigated the effects of playing board games on competence motivation and the development of mathematical competencies.
In this paper, we present the results of an experiment carried out in a secondary school class.
The experimental group spent one of three weekly mathematics lessons playing board games.
Apart from the several advantages of playing games in general, we can conclude that, based on the results of the national competence measurement, the mathematical competence of the students developed properly.
The readiness and the progress of the pupils were compared on the basis of input and output tests and an initial knowledge measurement and, at the same time, we compared their level of mathematical competence with the results of the national competence
measurement.Subject Classification: 97C70, 97D40
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Sage and scribe – asymmetrical pair work that can easily fit into any mathematics lesson, yet still have cooperative benefits
133-164Views:739This article uses a case study experiment to learn the characteristics of a pair work, called the sage and scribe method (Kagan, 2008). We also wished to explore the positive and negative effects of the systematic application of this single cooperative element without any other structural changes during the lessons. In the case study experiment, we asked two teachers, accustomed to traditional frontal teaching methods, to substitute individual work tasks in their standard lesson plans with the sage and scribe method. Our experiments indicate that this method wastes insignificant time, requires little extra effort on the part of the teacher, yet has many of the positive effects of cooperative methods: in our experiments, students received immediate feedback, corrected each other’s mistakes, learned from each other in meaningful discussions and engaged in collaborative reasoning to address emerging problems.
Subject Classification: 97D40
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A case study of the integration of Algorithm Visualizations in Hungarian programming education
51-66Views:285In this study, I will introduce how Algorithm Visualizations (AV) can help programming education or, in this case, the acquisition of basic programming theorems. I used two di erent methods to test this: in the first round, I examined in a larger group how much the students' ability to solve specific tasks changes after being introduced to a visualization tool, and then, what was their motivation and experience during this process. In the second round, I looked for the components that could be important when choosing a tool with the help of an in-depth interview with a smaller number of individuals. In both cases, I describe the research, experience, and results of the study, and then summarize them at the end.
Subject Classification: 97P10
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Task variations for backtrack
107-120Views:197This article has been written for informatics teachers who want to issue back-track based tasks on their lessons or as homework or on competitions. We present a few methods to generate a more complicated problem from a simpler task, which will be more complex, and its solution needs a good idea or trick. Starting from an example, we lead the reader through increasingly di cult task variations.
Subject Classification: 97P50
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Computer cooking vs. problem solving
35-58Views:307Computer cooking is a task-related phenomenon where students (end-users) must blindly follow a long list of orders without any connection to the content of the problem, if there is any. Despite its low efficacy, this method is widely used and accepted in informatics both in the learning-teaching process and testing. The National Base Curriculum 2020 in Hungary is in complete accordance with the ‘Informatics Reference Framework for Schools’, but the course books hardly use the latest results of computer education research. The present paper provides examples of how the results of computer education research can be integrated into teaching-learning materials and classroom practices and discusses the effectiveness and consequences of the different solutions, where tool-centred approaches are compared to problem-focused solutions.
Subject Classification: 94-01
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The tools for developing a spatial geometric approach
207-216Views:192Tamás Varga writes about the use of tools: "The rational use of tools - the colored bars, the Dienes set, the logical set, the geoboard, and some other tools - is an element of our experiment that is important for all students, but especially for disadvantaged learners." (Varga T. 1977) The range of tools that can be used well in teaching has grown significantly over the years. This paper compares spatial geometric modeling kits. Tamás Varga uses the possibilities of the Babylon building set available in Hungary in the 1970s, collects space and flat geometry problems for this (Varga T. 1973). Similarly, structured kits with significantly more options have been developed later, e.g. ZomeTool and 4D Frame. These tools are regularly used in the programs of the International Experience Workshop (http://www.elmenymuhely.-hu/?lang=en). Teachers, schools that have become familiar with the versatile possibilities of these sets, use them often in the optional and regular classes. We recorded a lesson on video where secondary students worked with the 4D Frame kit. We make some comments and offer some thoughts on this lesson.
Subject Classification: 97G40, 97D40
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Trigonometric identities via combinatorics
73-91Views:240In this paper we consider the combinatorial approach of the multi-angle formulas sin nΘ and cos nΘ. We describe a simple "drawing rule" for deriving the formulas immediately. We recall some theoretical background, historical remarks, and show some topics that is connected to this problem, as Chebyshev polynomials, matching polynomials, Lucas polynomial sequences.
Subject Classification: 05A19
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Metacognition – necessities and possibilities in teaching and learning mathematics
69-87Views:226This article focuses on the design of mathematics lessons as well as on the research in mathematics didactics from the perspective that metacognition is necessary and possible.
Humans are able to self-reflect on their thoughts and actions. They are able to make themselves the subject of their thoughts and reflections. In particular, it is possible to become aware of one’s own cognition, which means the way in which one thinks about something, and thus regulate and control it. This is what the term metacognition, thinking about one’s own thinking, stands for.
Human thinking tends to biases and faults. Both are often caused by fast thinking. Certain biases occur in mathematical thinking. Overall, this makes it necessary to think slow and to reflect on one’s own thinking in a targeted manner.
The cognitive processes of thinking, learning and understanding in mathematics become more effective and successful when they are supplemented and extended by metacognitive processes. However, it depends on a specific design of the mathematics lessons and the corresponding tasks in mathematics.Subject Classification: 97C30, 97C70, 97D40, 97D50, 97D70
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Teaching undergraduate mathematics - a problem solving course for first year
183-206Views:235In this paper we describe a problem solving course for first year undergraduate mathematics students who would be future school teachers.
Subject Classification: 97B50, 97B70, 97D50, 97D60, 97F60, 97U30
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What can we learn from Tamás Varga’s work regarding the arithmetic-algebra transition?
39-50Views:243Tamás Varga’s Complex Mathematics Education program plays an important role in Hungarian mathematics education. In this program, attention is given to the continuous “movement” between concrete and abstract levels. In the process of transition from arithmetic to algebra, the learner moves from a concrete level to a more abstract level. In our research, we aim to track the transition process from arithmetic to algebra by studying the 5-8-grader textbooks and teacher manuals edited under Tamás Varga's supervision. For this, we use the appearance of “working backward” and “use an equation” heuristic strategies in the examined textbooks and manuals, which play a central role in the mentioned process.
Subject Classification: 97-01, 97-03, 97D50
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How do secondary school students from the Kurdistan Region of Iraq understand the concept of function?
221-244Views:295The study investigates secondary school students' understanding of the concept of function. The paper focuses on three main aspects: students' ability to define the concept of function; students' ability to recognize different representations of function; and students' ability to convert between different representations. A test was developed to assess the three main constructs of the study and administered to 342 students in secondary schools in the Kurdistan Region of Iraq. According to the results, students have diffculties in recognizing different representations of function and conversion between them. Connections between different parts of the test may provide hints on educational challenges of how to appropriately teach functions.
Subject Classification: 26Bxx, 97D60
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Task reformulation as a practical tool for formation of electronic digest of tasks
1-27Views:154Creative thinking as well as thinking itself is being developed at active learning-cognitive activity of students. To make mathematic matter a subject of interest and work of students at classes, it is efficacious to submit it in a form of tasks. The tasks may be set up in a purposeful system of tasks by means of which reaching the teaching goals in the sense of quality and durability of gained knowledge may be more effective. A suitable means for presentation of tasks with their characteristics (as e.g. didactic function and cognitive level) as well as task systems themselves is an electronic digest of tasks as a database. The analysis of textbooks and digests of tasks commonly used at schools in Slovakia shows that they do not include all the types of tasks necessary for setting up complete (in the sense of didactic functions) task systems. One of the most important methods used for formation of the missing tasks is reformulation of tasks. The individual strategies of task reformulation are explained in details on examples in this article. -
Programming Theorems and Their Applications
213-241Views:291One of the effective methodological approaches in programming that supports the design and development of reliable software is analogy-based programming. Within this framework, the method of problem reduction plays a key role. Reducing a given problem to another one whose solving algorithm is already known can be made more efficient by the application of programming theorems. These represent proven, abstract solutions – in a general form – to some of the most common problems in programming. In this article, we present six fundamental programming theorems as well as pose five sample problems. In solving these problems, all six programming theorems will be applied. In the process of reduction, we will employ a concise specification language. Programming theorems and solutions to the problems will be given using the structogram form. However, we will use pseudocodes as descriptions of algorithms resembling their actual implementation in Python. A functional style solution to one of the problems will also be presented, which is to illustrate that for the implementation in Python, it is sufficient to give the specification of the problem for the design of the solution. The content of the article essentially corresponds to that of the introductory lectures of a course we offered to students enrolled in the Applied Mathematics specialization.
Subject Classification: D40
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Apollonea.com project: integrating geometry and collaboration in education
183-194Views:98This article presents the Apollonea.com project, which aims to make the solutions to Apollonius’ problems accessible to students and teachers through modern technology. The web platform contains more than 150 interactive constructions created by students using GeoGebra, allowing for dynamic manipulation and visualization of solutions to various variants of Apollonius’ problems. The project combines classical geometric problems with an interdisciplinary approach, teamwork, and the use of modern technology. The article describes the process of developing the Apollonea.com website, the use of GeoGebra in the project, the structure and functions of the website, and its educational benefits in enhancing students’ geometric skills. The project demonstrates how traditional mathematics education can be connected with modern ICT tools.
Subject Classification: 97U50, 97G40, 51M04, 68U05
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Supporting the education of engineering mathematics using the immediate feedback method
49-61Views:212In the literature, several methods are suggested to deal with problems regarding the efficiency of mathematics education including techniques that help integrate new knowledge into long-term memory. We examined how effective the application of the immediate feedback method is in teaching engineering mathematics. The article presents the method used and the results obtained during the study.
Subject Classification: 97D40, 97D60
