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Analyse von Lösungswegen und Erweiterungsmöglichkeiten eines Problems für die Klassen 7–11
231-249Views:31Making several solutions for a problem i.e. the generalization, or the extension of a problem is common in the Hungarian mathematics education.
But the analysis of a problem is unusual where the connection between the mathematical content of the task and of its different formulations is examined, solutions from different fields of mathematics are presented regarding the knowledge of different age groups, the problem is generalized in different directions, and several tools (traditional and electronic) for solutions and generalizations are presented.
This kind of problem analysis makes it viable that during the solution/elaboration several kinds of mathematical knowledge and activities are recalled and connected, facilitating their use inside and outside of mathematics.
However, an analysis like this is not unfamiliar to the traditions of the Hungarian problem solving education – because it also aims at elaborating a problem – but from several points of view.
In this study, a geometric task is analysed in such a way. -
Conventions of mathematical problems and their solutions in Hungarian secondary school leaving exams
137-146Views:8Collecting and analyzing the conventions indispensable for interpreting mathematical problems and their solutions correctly assist successful education and objective evaluation. Many professional and didactic questions arose while collecting and analyzing these conventions, which needed clarification, therefore the materials involved concisely in the conventions enrich both the theory and practice of mathematics teaching. In our research we concentrated mainly on the problems and solutions of the Hungarian school leaving examinations at secondary level in mathematics. -
Powers which commute or associate as solutions of ODEs
241-254Views:14This paper is dedicated to the two classical transcendental functions: The locus of points for which powers commute, and the locus of points for which powers associate. These classical functions however are considered in a new perspective: as holomorphic solutions of ODEs passing over the points of singularity of these ODEs.
Generally, solution functions which are holomorphic at singular points of the phase space of ODEs were studied in [2,3], and it was shown in [3], that certain holomorphic functions may satisfy only singular rational ODEs. This is the frame in which the function of commuting or associating powers are considered in this paper.
First we obtain several types of ODEs satisfied by these functions. The obtained ODEs happen to have singular points, yet the solutions are proved to be holomorphic at these points, and their Taylor expansions are obtained. However it is not yet known whether these two transcendental functions can satisfy a regular rational ODE at the respective special points. The article also poses an open question about remarkable inequalities related to the commuting powers. -
Correction to Gofen (2013): "Powers which commute or associate as solutions of ODEs?", Teaching Mathematics and Computer Science 11 (2013), 241-254.
245Views:66In the article "Powers which commute or associate as solutions of ODEs?" by Alexander Gofen (Teaching Mathematics and Computer Science, 2013, 11(2), 241–254. https://doi.org/10.5485/TMCS.2013.0347), there was an error in Conjecture 1 (p. 250), and consequently, in the References (p. 254).
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Analysis of a problem in plane geometry discussed in an 11th grade group study session
181-193Views:27The 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. -
Computer cooking vs. problem solving
35-58Views:66Computer 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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Diophantine equations concerning various means of binomial coefficients
71-79Views:30The main goal of this paper is to show by elementary methods, that there are infinitely many different pairs of binomial coefficients of the form (n C 2) such that also their arithmetic, geometric and harmonic means, resp. have the same form. We give all solutions for the arithmetic mean. We also give infinitely many non-trivial solutions for the arithmetic mean of three binomial coefficients satisfying some special conditions. The proofs require the solution of some other interesting Diophantine equations, too. Since the author is also a secondary school teacher, we use elementary methods that mostly can be discussed in secondary school, mainly within the framework of group study sessions. This explains why the means are generally analysed for two terms and for binomial coefficients with "lower" value 2, since further generalizations require substantially deeper mathematical methods which are beyond the frames of this paper. -
On some problems on composition of arithmetic functions
161-181Views:17The main goal of this paper is to investigate some problems related to the commutativity of the composition of arithmetic functions. The concept of commutativity arises many times in high school maths, so it is natural to study the composition of functions, namely the equation f(g(n)) = g(f(n)), where f and g are such well known arithmetic functions as d(n), φ(n), σ(n), ω(n), or Ω(n). We study various aspects of solvability: can we exhibit infinitely many solutions; can we determine every solution; can we find suitable values in the range of both functions f and g for which the equation is, or is not solvable, respectively. We need just the basic facts about the above functions,and we use only elementary methods in the proofs. We present some interesting questions, their solutions, and raise some unsolved problems. We found that this topic can be discussed well in secondary school, mainly within the framework of group study sessions as we had some classes with a group of kids in 9th grade. We summarize the experiences of this experiment in the last section. -
Programming Theorems and Their Applications
213-241Views:117One 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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Two centuries of the equations of commutativity and associativity of exponentiation
219-233Views:24In this survey article we guide the reader through the solution of the commutative equation of exponentiation x^y = y^x and that of the associative equation of exponentiation x^(y^z) = (x^y)^z. Various characterizations of the integer, rational, real and complex solutions are discussed together with some new results and open directions. The article is supplemented by a detailed and commented bibliography on the history of these equations. -
Examining relation between talent and competence through an experiment among 11th grade students
17-34Views:32The areas of competencies that are formable, that are to be formed and developed by teaching mathematics are well-usable in recognizing talent. We can examine the competencies of a student, we can examine the competencies required to solve a certain exercise, or what competencies an exercise improves.
I studied two exercises of a test taken by students of the IT specialty segment of class 11.d of Jedlik Ányos High School, a class that I teach. These exercises were parts of the thematic unit of Combinatorics and Graph Theory. I analysed what competencies a gifted student has, and what competencies I need to improve while teaching mathematics. I summarized my experience about the solutions of the students, the ways I can take care of the gifted students, and what to do to the less gifted ones. -
Apollonius' problems in grammar school
69-85Views:29In this work there are ten problems of Apollonius listed with illustrations and solution possibilities including students' solutions, too. Usually, it is rather difficult for students to grasp the essence of these problems with the use of traditional means, bows and rulers, but the use of computers offers higher accuracy. -
Teaching puzzle-based learning: development of transferable skills
245-268Views:37While computer science and engineering students are trained to recognise familiar problems with known solutions, they may not be sufficiently prepared to address novel real-world problems. A successful computer science graduate does far more than just program and we must train our students to reach the required levels of analytical and computational thinking, rather than hoping that it will just 'develop'. As a step in this direction, we have created and experimented with a new first-year level course, Puzzle-based Learning (PBL), that is aimed at getting students to think about how to frame and solve unstructured problems. The pedagogical goal is increase students' mathematical awareness and general problem solving skills by employing puzzles, which are educational, engaging, and thought provoking. In this paper we continue sharing our experiences in teaching such a course. Whereas a brief discussion on our pedagogical objectives were covered in the first paper together with the material of the first of two lectures on pattern recognition, this follow-up paper presents the material of the second of two lectures, in which additional exercises are discussed to reinforce the lesson. Along the way we provide a glimpse of some foundational ideas of computer science such as incomputability and general system development strategies such as incremental and iterative reasoning. This paper discusses the outcomes of PBL courses, which include expected improvement in the overall results achieved by students who have undertaken PBL courses, compared to those students who have not. -
Supporting the theory of math didactic using knowledge-measuring questions and analysis of the solutions
1-16Views:28New or rediscovered results presented in this paper are the results of the analysis of the problem sets used in the two-tier system secondary school final examination in mathematics, a system that was introduced in Hungary in 2005.
Many of the revealed problem arise in connection with misunderstanding the text of the problems. Causes of misinterpretation can be either that the text is lacking some important information, or that it should be interpreted not in word-to-word manner.
Theses and their argumentations presented here refer partly on the new types of problems (tests, non-standard mathematical contents), and partly on improvement of learning-teaching process in topics of equations and approximations. -
Teaching puzzle-based learning: development of basic concepts
183-204Views:17While computer science and engineering students are trained to recognise familiar problems with known solutions, they may not be sufficiently prepared to address novel real-world problems. A successful computer science graduate does far more than just program and we must train our students to reach the required levels of analytical and computational thinking, rather than hoping that it will just 'develop'. As a step in this direction, we have created and experimented with a new first-year level course, Puzzle-based Learning (PBL), that is aimed at getting students to think about how to frame and solve unstructured problems. The pedagogical goal is increase students' mathematical awareness and general problem solving skills by employing puzzles, which are educational, engaging, and thought provoking. We share our experiences in teaching such a course – apart from a brief discussion on our pedagogical objectives, we concentrate on discussing the presented material which covers (in two lectures) just one selected topic (pattern recognition). In this paper we present the ideas behind foundations for PBL and the material of the first of two lectures on pattern recognition, in which we address core concepts and provide students with sufficient exemplars to illustrate the main points. -
Teaching agile operation and leadership through linked university courses
1-32Views:100Agile software development methods, especially Scrum, are commonly used in software development companies. For this reason, our goal was that our undergraduate students gain experience as Scrum development team members and our master's students as agile leaders. To this end, we had redesigned and linked an undergraduate and a master's course, and launched the new course in the spring of 2021. The success of our approach was confirmed by a questionnaire survey of 86 undergraduate and 27 master's students. A/B testing was also performed. Our approach is a novelty compared to solutions where the Scrum Master is a course member, an instructor, or a university employee. In addition to being resource-efficient, it also offers master's students an unparalleled opportunity to develop agile leadership skills.
Subject Classification: 97U50
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Let's learn database programming in an active way
213-228Views:24This paper introduces how I applied the "learning by doing" method in the education of advanced knowledge of database systems in Software Engineering BSc program. The first goal of my method is to enable the students to use the PL/SQL and SQL as a skill, namely they get a practical competence which can be immediately used in business. In the laboratory the students independently practise the material learnt on the lecture. They get feedback for all their activities from the teacher. A software system helps administer the solutions, automatically verifies the syntax of them and helps the teacher to evaluate them. The paper summarises the results of three semesters. In the last year I compared the active learning method with the traditional method. I asked the students in a voluntary survey about the active learning method. -
Combinatorics – competition – Excel
427-435Views:31In 2001 the Informatics Points Competition of the Mathematics Journal for Secondary School Students (KÖMAL) was restarted [1]. The editors set themselves an aim to make the formerly mere programming competition a bit more varied. Therefore, every month there has been published a spreadsheet problem, a part of which was related to combinatorics. This article is intended to discuss the above mentioned problems and the solutions given to them at competitions. We will prove that traditional mathematical and programming tasks can be solved with a system developed for application purposes when applying a different way of thinking. -
Comments on the remaining velocity project with reports of school-experiments
117-133Views:14The aim of this article is to introduce different possible solutions to the exercise referring to the calculation of "remaining velocity". We explain the possible approaches to the problem with the help of either using the tools of mathematics or other subjects. During the past few years, we have made Hungarian and Slovakian secondary school students solve the exercise, choosing from both children of average and of high abilities. The experince has shown that very few students were able to solve the problem by themselves, but with the help of their teachers, the exercise and the solution has been an eye-opener experience to all of them. A lot of students were even considering to drive more carefully in the future after getting their driving licenses. -
Rational errors in learning fractions among 5th grade students
347-358Views:72Our paper focuses on empirical research in which we map out the errors in learning fractions. Errors are often logically consistent and rule-based rather than being random. When people face solving an unfamiliar problem, they usually construct rules or strategies in order to solve it (Van Lehn, 1983). These strategies tend to be systematic, often make ‘sense’ to the people who created them but often lead to incorrect solutions (Ben-Zeev, 1996). These mistakes were named rational errors by Ben-Zeev (1996). The research aims to show that when learning fractions, students produce such errors, identified in the literature, and that students who make these kinds of mistakes achieve low results in mathematics tests. The research was done among 5th-grade students.
Subject Classification: 97C10, 97C30, 97C70, 97D60, 97D70, 97F50
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Force of summation
185-199Views:25Programming theorems are important tools of programming methodology. By using analogous programming techniques, the solutions of different tasks can be created easily and fast based on programming theorems. Perhaps the summation is the simplest programming theorem that is widely-known among the programmers but once and for all the most various tasks can be solved by this theorem. The aim of the present paper is to investigate the summation programming theorem. Several different abstract levels of this theorem will be defined and the problem types that can be solved based on summation are going to be described. We will underline those points of a programming theorem that make a theorem general and that are not defined in advance, just later during its application, when the solution of a problem is derived from the theorem. -
Die aus der Studienzeit stammenden Aufzeichnungen des Johann Bolyai über die Würfelverdoppelung
307-316Views:28Hereinafter we are going to show that Bolyai Janos was preoccupied by the problem of the Duplication of the Cube, which was unknown until now by the rich Bolyai-literature.
This problem was solved using the Parabola, the Hyperbola and the Cissoide already in the ancient times. The Cissoide was created by Diocles especially for the constuction of the Duplication of the Cube without Compass and Straightedge. The hereinafter "deciphered" document of Bolyai was written during his university studies. In his study he presents the solutions discovered by then and tries to give a new one. We transcribed his notations to the present-day use and complemented it where it was necessary.
The mathematics historically background and the explication is very detailed described by Van derWaerden in Erwachende Wissenschaft [7], which is to find on English, German and Hungarian, too. That's because we dispense with this [8]. -
Straight line or line segment? Students’ concepts and their thought processes
327-336Views:100The 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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Über einen allgemeinen Übungsbegriff bei verschiedenen Unterrichtsmethoden in der Planung des Mathematikunterrichtes
175-201Views:38Practice is important in the education of mathematics but is neclected in the didactic of mathematics. One of the reasons is that practice is often defined too "narrowly" and the definitions of practice have in most cases an obscure background theory. In the article a general definition of practice is given, which – in contrast to the usual definitions – views practice from the point of the pupils (practice means activity of pupils). By utilising this definition consequences will be drawn. These consequences serve as for the more exact planning of practice in education as for the analysis of the dependency of practice from teachingsmethods.
In the second part an example will be presented for planning together practice and lesson, in two different teachingsmethods (traditionel, problemsolving). The analysis of both worksheets (one for each method, identical teachingsmaterial) was made on the basis of authors practise in lessons i.e. her own concepts and the experience with pupils at a class 5. On the basis of the expectable solutions is specified – using a criteriacatalog – what was practised.
The analysis of practice leads further to the examination of above mentioned dependency from teachingsmethods. -
Increasing the popularity and efficiency of distance education by old-new methods
211-228Views:37In our essay we aim to provide suggestions to develop distance education and we decisively focus on programmed education that is supported by e-learning environment. We both think that the shortage of programmed educational methods is causeless in Hungary's distance education. The widespread usage of info-communication devices and of the Internet makes the programmed educational methods (not as an exclusive method) possible to use in distance education together with e-learning environment. In our work we summarize the possible solutions and at the same time we also provide a case study, as an insight into our e-learning project (called Logical Programming) by Moodle.