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On a special class of generalized conics with infinitely many focal points
87-99Views:129Let a continuous, piecewise smooth curve in the Euclidean space be given. We are going to investigate the surfaces formed by the vertices of generalized cones with such a curve as the common directrix and the same area. The basic geometric idea in the background is when the curve runs through the sides of a non-void triangle ABC. Then the sum of the areas of some triangles is constant for any point of such a surface. By the help of a growth condition we prove that these are convex compact surfaces in the space provided that the points A, B and C are not collinear. The next step is to introduce the general concept of awnings spanned by a curve. As an important example awnings spanned by a circle will be considered. Estimations for the volume of the convex hull will be also given. -
Professional Competence in science education
129-137Views:77The article begins with a brief introduction aimed at sensitizing the reader to the perception of a trend in Mathematics and Computer Science Education publications towards empirical studies. Contrary to the stated trend, the characterization of Professional Competence is intended to serve as the guiding concept for the paper. The role of Professional Competence is discussed in various areas incorporating context-relevant publications in consecutive chapters. The discussion starts with the area of material development, covering Educational Standards and ends with Didactic Principles.
Subject Classification: 97xxx, 94xxx
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Kompetenzstreben und Kompetenzerwerb: Funktionale didaktische Fördermöglichkeiten durch Differenzierung und Individualisierung
1-52Views:162As a first glimpse of specific research endeavours the most important components of competence motivation are discussed in relation to didactical questions of gaining competence by inner differentiation and individualization: self-efficacy, optimal challenge, intrinsic motivation, exploration needs, internal attribution, self-determination motivation, defense of self-worth, self-concept, and achievement motivation. In this sense "competence" means ever changing standards of self-regulation of an individual interacting with the various cognitive and emotional demands of his/her environment.
In fulfilling these requirements a prototypical example of inner differentiation in mathematics instruction is given. This didactical elaboration is available as a selfinstructing unit in Hungarian and German language within the "Electronic periodical of the Department of Methodology of Mathematics" which can be reached under http://mathdid.inhun.com. -
Arithmetic progressions of higher order
225-239Views:183The aim of this article is to clarify the role of arithmetic progressions of higher order in the set of all progressions. It is important to perceive them as the pairs of progressions closely connected by simple relations of differential or cumulative progressions, i.e. by operations denoted in the text by r and s. This duality affords in a natural way the concept of an alternating arithmetic progression that deserves further studies. All these progressions can be identified with polynomials and very special, explicitly described, recursive progressions. The results mentioned here point to a very close relationship among a series of mathematical objects and to the importance of combinatorial numbers; they are presented in a form accessible to the graduates of secondary schools. -
Veranschaulichung der Lehrstoffstruktur durch Galois-Graphen
217-229Views:181In this article we compare the process diagram with the Galois-graph, the two hierarchical descriptions of the curriculum's construction from the point of didactics. We present the concrete example through the structure of convex quadrangles. As a result of the analysis it is proved that the process diagram is suitable for describing the activity of pupils, still the Galois-graph is the adequate model of the net of knowledge. The analysis also points out that in teaching of convex quadrangles the constructions of curriculum based only on property of symmetry and only on metrical property are coherent. Generalizing concept is prosperous if the pupils' existing net of knowledge lives on, at most it is amplified and completed. Teaching of convex quadrangles in Hungarian education adopts this principle. -
A Nim like game and a machine that plays it: a learning situation at the interface of mathematics and computer science
317-326Views:369The purpose of this work is to take a didactic look at a learning situation located at the interface between mathematics and computer science. This situation offers a first approach to the concept of artificial intelligence through the study of a reinforcement learning device. The learning situation, inspired by the Computer Science Unplugged approach, is based on a combinatorial game, along with a device that learns how to play this game. We studied the learning potential when the human players face the machine. After an a priori analysis using the Theory of Didactic Situations (TDS), we conducted a pre-experiment in order to strengthen our hypotheses. In this article, we will focus on the analysis of the didactic variables, the values we have chosen for these variables and their effects on students’ strategies.
Subject Classification: 97D99, 97K99, 97P80
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On some problems on composition of arithmetic functions
161-181Views:147The 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. -
Fuzzy Datalog with background knowledge
257-281Views:155In this paper we give a possible model for handling uncertain information. The concept of fuzzy knowledge-base will be defined as a triplet of a background knowledge defined by the similarity of predicates and terms; a deduction mechanism: a fuzzy Datalog program, and a decoding set of the program, which help us to determine the uncertainty level of the results. -
Central axonometry in engineer training and engineering practice
17-28Views:147This paper is concerned with showing a unified approach for teaching central and parallel projections of the space to the plane giving special emphasis to engineer training. The basis for unification is provided by the analogies between central axonometry and parallel axonometry. Since the concept of central axonometry is not widely known in engineering practice it is necessary to introduce it during the education phase. When teaching axonometries dynamic geometry software can also be used in an interactive way. We shall provide a method to demonstrate the basic constructions of various axonometries and use these computer applications to highlight their similarities. Our paper sheds light on the advantages of a unified approach in such areas of engineering practice as making hand drawn plans and using CAD-systems. -
The development of geometrical concepts in lower primary mathematics teaching: the square and the rectangle
153-171Views:197Our research question is how lower primary geometry teaching in Hungary, particularly the concept of squares and rectangles is related to the levels formulated by van Hiele. Moreover to what extent are the concrete activities carried out at these levels effective in evolving the concepts of squares and rectangles.
In the lower primary geometry teaching (classes 1-4) the first two stages of the van Hiele levels can be put into practice. By the completion of lower primary classes level 3 cannot be reached. Although in this age the classes of concepts (rectangles, squares) are evolved, but there is not particular relationship between them. The relation of involvement is not really perceived by the children. -
Some aspects of teaching the technology of designing and planning information systems in health care
131-144Views:105In this article, we use the well-known ideas of technology in designing of new information systems in health care. We explain the principle that "making a health care application" "is more than writing a program", "it requires a strong co-operation and continuous contact" between the system analysts and users. The concept of the information system must contain the work of the whole system, which means that the planning and designing process should focus on the services, which really support the customer's functions. It has to be compatible with the earlier information systems based on several decade's experience. In this paper we use the most important elements of system theory. First of all we explain why it is important to take into account the behaviour of those, who operate the information system, and also their habits and way of thinking when planning then information system. We emphasise that it is importance to overview the whole information system and its functionality because it is a major aspect of the system planning.
This paper can be used in university courses especially in teaching SDM, SSADM, Martin, etc. technologies for information system analysts, program designers and programmers. -
Manipulative bulletin board for early categorization
1-12Views:155According to various researchers categorization is a developmentally appropriate mathematical concept for young children. Classifying objects also relates to every day activities of human life. The manipulative bulletin board (MBB) served as a kind of auxiliary means for approaching categorization by young children. In this article we investigated the kind of MBB that pre-service early childhood education teachers constructed in order to involve children in tasks of categorization, as well as, the way children manipulated these boards in order to categorize items. The MBB, as teaching aids, facilitated the engagement of the children in different categorization processes. -
Some problems of solving linear equation with fractions
339-351Views:185The aim of this paper is to offer some possible ways of solving linear equations, using manipulative tools, in which the "−" sign is found in front of an algebraic fraction which has a binomial as a numerator. It is used at 8th grade. -
Decision based examination of object-oriented programming and Design Patterns
83-109Views:153On 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. -
Comparison of teaching exponential and logarithmic functions based on mathematics textbook analysis
297-318Views:171Exponential and logarithmic functions are key mathematical concepts that play central roles in advanced mathematics. Unfortunately these are also concepts that give students serious difficulties. In this paper I would like to give an overview – based on textbook analysis – about the Hungarian, Austrian and Dutch situation of teaching exponential and logarithmic functions. This comparison could also provide some ideas for Hungarian teachers on how to embed this topic in their practice in another more "realistic" way. -
Analysis of the affective factors of learning mathematics among teacher trainees
225-254Views:156The Hungarian National Core Curricula gives primacy to the development of abilities and the practical application of knowledge. The task of the training programme is primarily to prepare teacher trainees for the teaching and educating profession. As teachers, they are going to plan, organize, help, guide, control and evaluate the learning of mathematics of individuals and groups of students from the age of 6 to 10 (12), and cultivate their mathematical skills, thinking and positive attitude towards any mathematical activities. In order to train educators who are able to meet the above requirements on high standard, it is necessary to update the teacher training programme based on the trainees' preliminary knowledge and motivation level.
The key to learn about the child's mind and achieve conscious development is the systematization of factual knowledge and methodological awareness. The modern, flexible approach to subject pedagogy, based on pedagogy, psychology and epistemology, qualifies trainees to educate learners who understand and like mathematics. Therefore, it is essential to develop the trainees' positive approach to mathematics and arouse their demand for continuous professional improvement. (Programme of the four-year primary school teacher training, 1995.)
In our research we are looking for ways of ascertaining the starting parameters which have influence on the planning of the studies of mathematics and subject pedagogy. In this article we introduce a questionnaire by the means of which we collected information on the trainees' attitude and its changing towards mathematics. With the help of the analysis of the answers we paint a picture of the ELTE TÓFK (Eötvös Loránd University, Faculty of Elementary and Nursery School Teacher's Training) third year students' attitude to the subject, and we compare it to the tendencies noticed in the mass education. The energy invested in learning is influenced by the assumption of the relevance and importance of the subjects. Therefore we considered it also our task to reveal. Besides the students' attitude toward mathematics and their assumption about their own competence we have collected data also on their performance in the subject. Summarising the research results we show the advantages of the questionnaire, and summarise the observations which would indicate need for methodological changes in the mathematics teacher training. -
A proposed application of Monte Carlo method in teaching probability
37-42Views:156Pupils' misconception of probability often results from lack of experience. Combining the concept of probability and statistics, the proposed application is intended for the teachers of mathematics at an elementary school. By reformulating the task in the form of an adventure, pupils examine a mathematical problem, which is too difficult for them to solve by combinatorial method. By recommending the simulation of the problem, we have sought to provide pupils with valuable experience of experimenting, recording and evaluating data. -
Analyse d’obstacles lies a la notion de fonction reciproque
43-61Views:103L'article présente une réflexion sur les difficultés éventuelles des étudiants au cours de l'étude des fonctions réciproques. Nous nous intéressons à l'enseignement de cette notion dans le cadre de la transition entre le lycée et la première année d'université en France. L'article résume la façon de présenter de nos jours le concept de fonction réciproque dans le curriculum français. Nous présentons des propositions d'enseignement issues de quelques travaux de recherche anglophones. Cela nous permet de mettre en lumière l'existence d'obstacles dans l'apprentissage et de difficultés dans l'enseignement de cette notion. -
A role of geometry in the frame of competencies attainment
41-55Views:165We discuss aspects of the Education Reform from teaching to educational system. In this context we recognize some problems in recognition of some competencies that students need to achieve and we present how we have developed the measurement method of spatial abilities and problem solving competence. Especially, we investigate how students use spatial visualization abilities in solving various problems in other mathematical course. We have tested how students use their spatial abilities previously developed in geometry courses based on conceptual approach to solve a test based on procedural concept in Mathematical Analysis course. -
A new approach for explaining Rhind's Recto – and its utility in teaching
337-355Views:120The Recto is a table in the Rhind Mathematical Papyrus (RMP) of ancient Egypt containing the unit fraction decompositions of fractions 2/n (3 ≤ n ≤ 101, n odd). To the question how (and why) the decompositions were made, there exists no generally accepted answer. The fact that in some other sources of Egyptian mathematics decompositions different from those in Recto exist makes the problem more difficult.
Researchers normally try to find the answer in some formulas by which the entries of the table were calculated [see e.g. 1, 42]. We are convinced that the correct answer is not hidden in formulas but in the characteristics of Egyptian mathematics namely those of fraction and division concepts. To study them is important not only from historical point of view but also from methodological one: how to develop fraction concept and how to make division easier.
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