Search
Search Results
-
Würfel und Augensummen – ein unmögliches Paar
71-88Views:128It is well known that the values 2, 3, ..., 12 of the sum of eyes that appear when throwing two regular dice are not equally distributed. It can also be shown that no matter how the dice are falsified (or if only one of them is being manipulated) they can never reach the same probability concerning the sum of eyes ([8], 91 et seq.). This discovery can be generalized for n ≥ 2 dice. Various results of algebra and (real) calculus are used, so that a connection between two different mathematical fields can be realized. Such a connection is typical and often provides a large contribution for mathematics (because it frequently leads to a successful attempt of solving a special problem) and therefore examples of this sort should also be included in the mathematical education at schools as well as in the student teachers' university curriculum for the study of mathematics. -
Forming the concept of parameter with examples of problem solving
201-215Views:109Pupils are encountering difficulties with learning algebra. In order for them to understand algebraic concepts, particularly the concept of parameter it was decided by the teacher of mathematics and Information Technology to integrate the teaching of these two subjects. The aim of this study is to investigate whether, and to what degree, software can be useful in process of forming the concept of parameter. This longitudinal study was conducted in a junior high school (13-16 year old children) using different computer programs. -
Interdisciplinary Secondary-School Workshop: Physics and Statistics
179-194Views:135The paper describes a teaching unit of four hours with talented students aged 15-18. The workshop was designed as a problem-based sequence of tasks and was intended to deal with judging dice whether they are regular or loaded. We first introduced the students to the physics of free rotations of rigid bodies to develop the physics background of rolling dice. The highlight of this part was to recognise that cubes made from homogeneous material are the optimal form for six-sided objects leading to equal probabilities of the single faces. Experiments with all five regular bodies would lead to similar results; nevertheless, in our experiments we focused on regular cubes. This reinsures that the participants have their own experience with the context. Then, we studied rolling dice from the probabilistic point of view and – step-by-step – by extending tasks and simulations, we introduced the idea of the chi-squared test interactively with the students. The physics and the statistics part of the paper are largely independent and can be also be read separately. The success of the statistics part is best described by the fact that the students recognised that in some cases of loaded dice, it is easier to detect that property and in other cases one would need many data to make a decision with small error probabilities. A physical examination of the dice under inspection can lead to a quick and correct decision. Yet, such a physical check may fail for some reason. However, a statistical test will always lead to reasonable decision, but may require a large database. Furthermore, especially for smaller datasets, balancing the risk of different types of errors remains a key issue, which is a characteristic feature of statistical testing.
Subject Classification: F90, K90, M50, R30
-
Decomposition of triangles into isosceles triangles II: complete solution of the problem by using a computer
275-300Views:149We solve an open decomposition problem in elementary geometry using pure mathematics and a computer programme, utilizing a computer algebra system. -
Teaching puzzle-based learning: development of transferable skills
245-268Views:232While 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. -
Comparing various functions of the divisors of an integer in different residue classes
247-258Views:120The main goal of this paper is to investigate some problems related to the distribution of the divisors of a number in different residue classes. We study these questions modulo 3, and use mostly just elementary number theory. In some special cases, we demonstrate how this problem is related to other fields of maths, especially to combinatorics. Since the author is also a secondary school teacher, we use elementary methods that can be discussed in secondary school, mainly within the framework of group study sessions or in special maths classes. We do think that the investigation of these types of questions can motivate children to find their own way to create their own questions, and to get a deeper insight into problem solving by these experimentations. -
The appearance of the characteristic features of the mathematical thinking in the thinking of a chess player
201-211Views:143It is more and more important in 21st century's education that not only facts and subject knowledge should be taught but also the ways and methods of thinking should be learnt by students. Thinking is a human specificity which is significant both in mathematics and chess. The exercises aimed at beginner chess players are appropriate to demonstrate to students the mathematical thinking of 12-14 year-old students.
Playing chess is an abstract activity. During the game we use abstract concepts (e.g. sacrifice, stalemate). When solving a chess problem we use logical quantifiers frequently (e.g. in the case of any move of white, black has a move that...). Among the endgames we find many examples (e.g. exceptional draw options) that state impossibility. Affirmation of existence is frequent in a mate position with many moves. We know there is a mate but the question in these cases is how it can be delivered.
We present the chess problem on beginners' level although these exercises appear in the game of advanced players and chess masters too, in a more complex form. We chose the mathematical tasks from arithmetic, number theory, geometry and the topic of equations. Students encounter these in classes, admission exams and student circles. Revealing the common features of mathematical and chess thinking shows how we can help the development of students' mathematical skills with the education of chess. -
The role of computer in the process of solving of mathematical problems (results of research)
67-80Views:119We would like to present results of an almost two years investigations about the role computer in the process of solving of mathematical problems. In these investigations took part 35 students of the secondary school (generalists) in the age 17–19 years. Each of these students solved following problem:
Find all values of the parameter m so that the function
f(x) = |mx + 1| − |2x − m| is:
a) bounded,
b) bounded only from the bottom,
c) bounded only from above,
first without a computer and next with a special computer program. We would like to show results of these researches. -
Integral part problems derived from a solution of an in mum problem
43-53Views:89In this paper, we solve the following two integral part problems:
Find all r ϵ R satisfying r^2 = [r]*([r]+1), resp. r^2≤[r]*([r]+1).
These problems have been mainly motivated by a solution of an infimum problem of Z. Boros and Á. Száz. -
Darstellungen und Vorstellungen und ihre Bedeutung für eine wirksame Metakognition beim Problemlösen und Begründen
195-220Views:117Metacognition has one of the highest effect sizes concerning successful learning. However metacognitive activities during task solving and problem solving are not directly obvious. But they can appear by writing someone's thoughts down. The following analysis, which focusses on the level of argumentation as well as on the way of derivation, shows that the quality of representation is an essential condition for the possibility of metacognition. -
A KöMaL problem in a new view
191-201Views:46The object of this paper is finding the general solution f : R^3 → R of the system of functional equations (1) valid for all x, y, z, t ϵ R. First f is expressed by a function of one variable which satisfies a system of two functional equations.This system is resolved by using an algebraic reformulation of the problem in terms of orbits and transversals. Finally the general solution of (1) is obtained. -
Combinatorics – competition – Excel
427-435Views:115In 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. -
Rational errors in learning fractions among 5th grade students
347-358Views:148Our 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
-
CS unplugged in higher education
1-23Views:136Nowadays, there is a significant lack of workforce in the IT industry, even though it is one of the most lucrative professions. According to researchers' forecasts, the existing shortage is growing, so the wages offered will be higher, yet it seems that young people are not attracted to the profession. This problem draws attention to the need to change the curriculum so that it can attract students more. One possible solution is to supplement the curriculum with CS Unplugged activities, which makes it easier to understand and deepen difficult concepts and make IT lessons more colorful. In my article, besides presenting the already known CS Unplugged activities, I will deal with how this can be applied in Hungarian higher education as well. -
Cultivating algorithmic thinking: an important issue for both technical and HUMAN sciences
107-116Views:119Algorithmic thinking is a valuable skill that all people should master. In this paper we propose a one-semester, algorithm-oriented computer science course for human science students. According to our experience such an initiative could succeed only if the next recipe is followed: interesting and practical content + exciting didactical methods + minimal programming. More explicitly, we suggest: (1) A special, simple, minimal, pseudo-code like imperative programming language that integrates a graphic library. (2) Interesting, practical and problem-oriented content with philosophical implications. (3) Exciting, human science related didactical methods including art-based, inter-cultural elements. -
Live & Learn: When a wrong program works
195-208Views:96In this paper an interesting and surprising case study of my programming education practice is presented. This case underlines the importance of methods, standards and rules of thumb of the programming process. These elements of the programming technology can be taught well in education and they can guarantee the quality of the implemented programs. However the case described in this paper brings an anomaly when a programming standard is violated during the programming process and, although it should imply that the implemented program code works badly, the program works perfectly. This anomaly is caused by a typical implementation problem: the boundary and rules of the machine representation of numbers. This anomaly is going to be analyzed and the appropriate conclusions of our case study will be deducted. -
Blind versus wise use of CAS
407-417Views:147During my courses for mathematics major students I often use technology linked to the arising problems. In such cases I noted that some students were used to learn just some procedures, which made them able to solve (partially) some problems and when they got the result, they accepted it passively and did not relate it to the initial problem.
In this paper I outline a strategy and investigate some simple exercises about how to develop a critical attitude towards the results obtained by technology in an introductory course to CAS.
I believe that wise use of technology offers an effective method in teaching mathematics, without reducing the students' mental contribution. -
Self-regulated learning in mathematics lessons at secondary level
139-160Views:17Self-regulation is a prerequisite to be able to set goals and to find suitable ways to reach them. Furthermore, it is an important ability which affects different areas of every day’s life. In educational context, self-regulation is often linked to self-regulated learning. The concept of self-regulated learning as well as key terms related to this topic such as problem-solving and modelling tasks will be discussed, while an emphasis lays on the role of the teacher. In this paper, a study on the attitudes of mathematics teachers towards self-regulated learning is presented. It focuses on teachers’ assessment of the possibility and limitations of self-regulated learning in mathematics lessons. It can be observed that most of the surveyed teachers try to incorporate self-regulatory processes in their teaching, but encounter difficulties related to various factors, such as their students, framework conditions, and the time required for such learning processes.
Subject Classification: 97D10
-
The Project Method and investigation in school mathematics
241-255Views:133The Project Method (PM) is becoming more common in the teaching of mathematics. Most of the time, Project Method means solving open and relatively wide formulated problems for the application of particular mathematical topics and the solving of everyday life problems.
At present many experts in the theory of teaching mathematics advocate teaching activities as the characteristic for most mathematical work in the classroom. Thus, there is a question: whether it is possible or eventual desirable to use the PM for solving genuine mathematical problems. This paper deals with this question and discusses the connection between the PM and investigation of new mathematical knowledge for students. Our experience has shown that the PM in connection with investigations can be a useful and effective approach to teaching mathematics. -
A computational thinking problem-thread for grade 7 students and above from the Pósa method
101-110Views:226Lajos Pósa has been developing his “learning through discovery” (Győri & Juhász, 2018) method since 1988. His weekend math camps are focused on fostering problem-solving skills and high-level mathematical-thinking skills in gifted students from grades 7 to 11. One of the core aspects of the method is the structure of the problems, all problems are part of a complex, intertwined, and rich network. In this article we analyze a computational thinking problem-thread and its role in the camps’s network of problems (Gosztonyi, 2019), and show some aspects of the method. The insights gained using this method can be useful in other contexts. The possible adaptation of the method to secondary and high schools is briefly discussed as well.
Subject Classification: 97D40
-
A geometric application to the third-order recurrence relations for sequences
287-302Views:104Using a third-order linear homogeneous recurrence relation with constant coefficients, it is found a limit-point of a sequence of affixes in plane. Starting from a classic geometric problem, an application is so created and few more nice properties are found and described. -
Die Stichprobe als ein Beispiel dafür, wie im Unterricht die klassische und die bayesianische Auffassung gleichzeitig dargestellt werden kann
133-150Views:104Teaching statistics and probability in the school is a new challenge of the Hungarian didactics. It means new tasks also for the teacher- and in service-teacher training. This paper contains an example to show how can be introduced the basic notion of the inference statistics, the point- and interval-estimation by an elementary problem of the public pole. There are two concurrent theories of the inference statistics the so called classical and the Bayesian Statistics. I would like to argue the importance of the simultaneously introduction of both methods making a comparison of the methods. The mathematical tool of our elementary model is combinatorial we use some important equations to reach our goal. The most important equation is proved by two different methods in the appendix of this paper. -
Maximum and minimum problems in secondary school education
81-98Views:130The aim of this paper is to offer some possible ways of solving extreme value problems by elementary methods with which the generally available method of differential calculus can be avoided. We line up some problems which can be solved by the usage of these elementary methods in secondary school education. The importance of the extremum problems is ignored in the regular curriculum; however they are in the main stream of competition problems – therefore they are useful tools in the selection and development of talented students. The extremum problem-solving by elementary methods means the replacement of the methods of differential calculus (which are quite stereotyped) by the elementary methods collected from different fields of Mathematics, such as elementary inequalities between geometric, arithmetic and square means, the codomain of the quadratic and trigonometric functions, etc. In the first part we show some patterns that students can imitate in solving similar problems. These patterns could also provide some ideas for Hungarian teachers on how to introduce this topic in their practice. In the second part we discuss the results of a survey carried out in two secondary schools and we formulate our conclusion concerning the improvement of students' performance in solving these kind of problems. -
Solving mathematical problems by using Maple factorization algorithms
293-297Views:112Computer algebra gives methods for manipulating mathematical expression. In this paper we use the Maple software to solve some elementary problems. Computeraided approach in the instruction of mathematics helps to impart problem solving skills to students. -
Taking learning styles into consideration in e-learning based education
385-396Views:74In improving electronic teaching material processes we should take the student's learning styles or methods into consideration. The ways learners receive information may be shared into three categories (modalities): visual, auditory, kinesthetic (tactile). In this paper I present some pedagogical questions of the electronic teaching-learning environment, offer a brief survey of the different learning style theories and emphasise the importance of the modalities in encoding information. The electronic teaching material should encourage the learner to choose an appropriate form of syllabus by which his knowledge can become more efficient.
