Articles

• Various systems in a single mathematical model

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  Gábor Geda

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  Our aim is to study differential equations and systems described by them which have great historical importance and are considered to be fundamental on different levels of education.
  Due to their simplicity these are suitable for those who deal with this topic and want to gain useful experience in this field.
  Furthermore, our aim is to give these equations a general form which facilitates the studying of the different models by computer even for an individual programmer. At the same time it facilitates the use of different mathematical auxiliary-programmes.
  By giving the equations this way we get a chance of studying the relations between the individual systems.

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• Heads or Tails gambling — what can be learned about probability?

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  Attila Máder

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  During the teaching of probability theory, a problem may appear whose solution requires the use of methods that are unfamiliar to secondary school students. In this paper, examples of methods that can resolve this difficulties are demonstrated, which could in future allow school students to tackle and solve a wide variety of problems involving probability.

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• Analyse d’obstacles lies a la notion de fonction reciproque

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  Susana Murillo-López

  André Antibi

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  L'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.

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• Different approaches of interplay between experimentation and theoretical consideration in dynamic geometry exploration: An example from exploring Simson line

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  Yip-Cheung Chan

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  Dynamic geometry environment (DGE) is a powerful tool for exploration and discovering geometric properties because it allows users to (virtually) manipulate geometric objects. There are two possible components in the process of exploration in DGE, viz. experimentation and theoretical consideration. In most cases, there is interplay between these two components. Different people may use DGE differently. Depending on the specific mathematical tasks and the background of individual users, some approaches of interplay are more experimental whereas some other approaches of interplay are more theoretical. In this paper, different approaches of exploring a geometric task using Sketchpad (a DGE) by three individual participants will be discussed. They represent three different approaches of interplay between experimentation and theoretical consid- eration. An understanding of these approaches may contribute to an understanding on the mechanism of exploration in DGE.

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• Decision based examination of object-oriented programming and Design Patterns

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  Szabolcs Márien

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  On 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.

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• The influence of computer on examining trigonometric functions

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  Djurdjica Takači

  Radivoje Stojković

  Jasminka Radovanović

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  6

  In this paper the influence of computer on examining trigonometric functions was analyzed throughout the results questionnaire. The students, as usual, had to examine two trigonometric functions, both were given with the appropriate instructions. Three groups were tested. Two of those three groups were prepared with the help of computer and the third one was taught without computer. From the analysis of the questionnaire it follows that the computer has a great influence on understanding of the connections between the graph and very complex calculations.

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• Answers offered by computer algebra systems to equations from school textbooks

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  Eno Tonisson

  Nadezhda Velikanova

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  This paper is an attempt to develop a strategy and methods for investigating and comparing the answers offered by computer algebra systems and the school answers. After primary (pilot) investigation of how well 8 computer algebra systems handle equations from school textbooks, it is possible to conclude that the systems are mostly reliable and give reasonable answers. Some remarks regarding a somewhat unexpected answer obtained can be easily explained by built-in standards and notions, which can differ from school assumptions. In other cases the differences from school could be corrected by certain commands.

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• "Frontier algorithms"

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  Zoltán Kátai

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  In this paper we present a new method to compare algorithm design strategies. As in case of frontier towns the cultures blend, the so called "frontier algorithms" are a mixture of different programming techniques like greedy, backtracking, divide and conquer, dynamic programming. In case of some of them the frontier character is hidden, so it has to be discovered. There are algorithms that combine different techniques purposively. Furthermore, determining the programming technique the algorithm is using can be a matter of point of view. The frontier algorithms represent special opportunities to highlight particular characteristics of the algorithm design strategies. According to our experience the frontier algorithms fit best to the revision classes.

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• The development of geometrical concepts in lower primary mathematics teaching: the square and the rectangle

  153-171

  Ibolya Szilágyiné Szinger

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  Our 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.

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• Fibonacci beyond binary recursion

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  Michael A. Wirth

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  The Fibonacci series is a classical algorithm taught in computer science, usually implemented in some programming language. It is hard to find a programming textbook which doesn't touch on Fibonacci, and it's most common use is in the illustration of binary recursion. There are also many ways of tailoring the basic algorithm in order to implement it. This paper discusses some novel algorithms, which help address some of the limitations of binary recursion, but also illustrate how differing algorithms can be pedagogically beneficial. We introduce a simple algorithm for accurately calculating any Fibonacci number.

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• Comment les enseignants en formation initiale utilisent les technologies informatiques dans leurs classes

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  Maha Abboud-Blanchard

  Jean-Paul Fallot

  Agnès Lenfant

  Bernard Parzysz

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  8

  The research presented here deals with the way French pre-service teachers assimilate the working of technology tools and the effects on professional practice of integrating these tools into classes. We focused on the professional writings of pre-service teachers regarding the use of technology in their teaching. The results show that, besides official instructions, the motivations put forward by pre-service teachers who integrated technology in their classes are mainly their students' interest in computers and how powerful this tool is. They also show that in such an environment teachers tend to keep in the background and to leave the students to interact chiefly with the computer. We also noticed that the specificities of managing a classroom in computer environment are not taken into account unless they generate problems.
  Résumé. La recherche présentée ici porte sur l'appropriation des outils informatiques par les enseignants français en formation initiale et les effets de leur intégration dans les classes sur les pratiques professionnelles. Nous avons pris comme objet d'étude des écrits professionnels, élaborés par ces professeurs stagiaires, portant sur l'utilisation des TIC dans leur enseignement. Les résultats obtenus font apparaître qu'outre les injonctions institutionnelles, les motivations invoquées par les stagiaires pour recourir à l'informatique concernent surtout l'attrait de leurs élèves pour l'ordinateur et la puissance de cet outil. Dans le cadre des usages en classe, nos résultats montrent que l'enseignant a tendance à s'effacer devant l'ordinateur, considéré comme l'interlocuteur privilégié de l'élève. Nous avons aussi pu constater que les spécificités de la gestion de la classe en environnement informatique ne sont prises en compte que lorsqu'elles se révèlent sources de problèmes.

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• Shall we use one more representation? Suggestions about establishing the notion of recursion in teaching informatics in primary schools

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  Cecília Sitkuné Görömbei

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  Among the most prominent developmental tasks of primary school education one finds increasing pupils' cognitive capacity with especial regard to observing, interpreting, coding and proving skills, which form an integral part of information and communication culture.
  Info-technology (problem solving with the tools and methods of informatics), a subject matter within informatics, provides outstanding opportunities to reach the aims outlined above.
  This study presents methodological ideas related to the subfield Algorithmization and data modelling of Info-technology. More specifically, it presents teaching methods to be applied while establishing the notion of recursion in grades 3–8 of primary education, and at the same time it also focuses on various realization possibilities of the prominent developmental tasks mentioned above.

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• Evaluating admission procedures for teacher education in Finland

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  Raimo Kaasila

  Markku S. Hannula

  Anu Laine

  Erkki Pehkonen

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  10

  In Finland the number of applicants for elementary teacher education is many times greater than the number of accepted persons. In this article we focus on the significance of the entrance examination procedures at three Finnish universities. Our findings imply that the differing admission procedures at the institutions yielded different student profiles. The test component "mathematics-science" used on the entrance examination in Turku was found to be a significant separating factor, but also the applicants' mathematics achievement in upper secondary school seems to be an applicable criterion for developing admission procedures.

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