Teaching Mathematics and Computer Science (DE-journals)
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Tétel Szabadon hozzáférhető Why is the gamma function so as it is?(2003-06-01) Gronau, DetlefThis is a historical note on the gamma function Γ. The question is, why is Γ(n) for naturals n equal to (n−1)! and not equal to n! (the factorial function n! = 1·2 · · · n) ? Was A. M. Legendre responsible for this transformation, or was it L. Euler? And, who was the first who gave a representation of the so called Euler gamma function?Tétel Szabadon hozzáférhető unity of mathematics(2003-06-01) Hilton, Peter; Pedersen, JeanWe give a sustained example, drawn largely from earlier publications, of how we may freely pursue a line of mathematical enquiry if we are not constrained, unnaturally, to confine ourselves to a single mathematical subdiscipline; and we draw conclusions from the study of this example which are relevant at many levels of mathematical instruction. We also include the statement and proof of a new result (Theorem 4.1) in linear algebra which is obviously fundamental to the geometrical investigation which constitutes the leit-motif of the paper.Tétel Szabadon hozzáférhető Gaussian iteration of mean values and the existence of 2^(1/2)(2003-06-01) Daróczy, ZoltánWe propose a method for proving the existence of √2 and finding its approximate value in secondary education.Tétel Szabadon hozzáférhető Aspects théoriques de la classification à base de treillis(2003-06-01) , KatalinLa 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.Tétel Szabadon hozzáférhető first course in computer science(2003-06-01) Smolarski, Dennis C.The College Board Advanced Placement exam in computer science will use the language Java starting in fall 2003. The language chosen for this exam is based on the language commonly taught in introductory computer science courses at the university level. This article reviews the purpose of an introductory course and the various suggestions for the curriculum of introductory courses published by the Association for Computing Machinery. It then proposes that such a course stress foundational concepts over specific language syntax, and then provides a list of such foundational concepts and related topics. Based on this fundamental curriculum, the article recommends C++ as the most appropriate language. An appendix provides a sample syllabus.Tétel Szabadon hozzáférhető Mathematical gems of Debrecen old mathematical textbooks from the 16-18th centuries(2003-06-01) Kántor-Varga, TündeIn the Great Library of the Debrecen Reformed College (Hungary) we find a lot of old mathematical textbooks. We present: Arithmetic of Debrecen (1577), Maróthi's Arithmetic (1743), Hatvani's introductio (1757), Karacs's Figurae Geometricae (1788), Segner's Anfangsgründe (1764) and Mayer's Mathematischer Atlas (1745). These old mathematical textbooks let us know facts about real life of the 16-18th centuries, the contemporary level of sciences, learning and teaching methods. They are rich sources of motivation in the teaching of mathematics.Tétel Szabadon hozzáférhető Teaching student teachers(2003-06-01) Humenberger, Hans; Reichel, Hans-ChristianIn this paper we summarize various aspects of teacher training and teaching student teachers (mainly concerning teachers of upper secondary school and High school). We stress several hints and recommendations to better achieve the obviously important aim: they should learn doing, understanding and teaching mathematics! Of course, our view is particularly influenced by European traditions, but we think most of them equally apply to teacher training and teaching student teachers elsewhere. Neither is the paper meant to give an all sided overview about the problem field of teacher education as a whole, nor does it contain provocative, completely new ideas. We just want to describe our view of some aspects, based primarily on our personal experience in the mentioned field.Tétel Szabadon hozzáférhető Hyperbolische 5-Rechtecke(2003-06-01) , H.The main topic of this paper is the investigation of 5-pentagons whose interior angles are all right angles within the hyperbolic geometry (so-called 5-rectangles). Some knowledge of elementary hyperbolic geometry is required. At first the existence of such a polygon is shown by construction within the Kleinmodel. Then two formulas due to D. M. Y. Sommerville [3] are proved. This means to juggle with trigonometric formulas of hyperbolic geometry. In the last years a big number of papers concerning hyperbolic geometry was published. This proves that the interest in this nice discipline is growing again.Tétel Szabadon hozzáférhető Proof without words(2003-06-01) Alsina, C.; Monreal, A.We present a visual proof of the parallelogram law and using it we can describe a visual proof of a classical theorem on convex quadrilaterals relating sides and diagonals.Tétel Szabadon hozzáférhető Tétel Szabadon hozzáférhető Illustrated analysis of Rule of Four using Maple(2003-12-01) Maróti, GyörgyRule of Four, as a basic didactic principle, was formulated among the NCTM 2000 standards (see [14]) and since then it is quoted by numerous books and publications (see [4], [9], [12]). Practically we can say it is accepted by the community of didactic experts. The usage of the Rule of Four, however, has been realized mainly in the field of calculus, in fact certain authors restrict the wording of the principle to the calculus itself (e.g. [3]). Calculus is a pleasant field, indeed. A sequence of values of a function provides us with example for numeric representation, while the formula and the graph of the function illustrate symbolic and graphical representations, respectively. In the end by wording the basic features of the function on natural language we gain textual representation. This idyllic scene, however, becomes more complex when we leave the frame of calculus. In this paper we investigate the consequences of the usage of Rule of Four outside calculus. We discuss the different types of representations and show several examples which make the multiple features of representation evident. The examples are from different fields of mathematics and are created by the computer algebra system Maple, which turns out to be an excellent tool for illustration and visualization of the maim features of mathematical objects. Next we introduce the concept of basic representation and rational representation, which is considered as the mathematical notion of "didactic usable" or "didactic rational" representation. In the end we generalize the notion of numeric representation, which leads us a more widely usable didactic principle which can be considered as a generalization of Rule of Four.Tétel Szabadon hozzáférhető Interactive web portals in mathematics(2003-12-01) Vajda, Róbert; Kovács, ZoltánMany of the recent problems in higher education (less contact seminars, the heterogeneity and the increasing number of our students) call for new instructional methods. At University of Szeged we have developed a mathematical web portal which can offer a solution for such problems among the changing circumstances. This freely available, easy-to-use web-surface supports interactive mathematical problem-solving and student self assessment. Our computer program cooperates with a lot of free software (computer algebra systems, formula parsers, converters, word processors). WebMathematics Interactive has been available for the public since June 2002 on its web page http://wmi.math.u-szeged.hu.Tétel Szabadon hozzáférhető Experiences using CAS and multimedia int teaching vectorcalculus(2003-12-01) Perjési-Hámori, IldikóThe development of informatics brings new opportunities that need reevaluating of the teaching concepts. For this reason we have performed a comprehensive educational development for engineering students. Our main goals were to work out a new educational strategy, to develop the needed package of the subject material, to introduce the strategy in the practice, to analyze and evaluate the experiences. In the developed and adapted teaching-learning strategy the teacher is the organizer, designer and the manager of the process. In this paper we summarize the concepts, the results and experiences of the 3-years-long development.Tétel Szabadon hozzáférhető Problem-solving in mathematics with the help of computers(2003-12-01) Kovács, AndrásOne of the most important tasks of the didactics of mathematics is the describing of the process of problem-solving activity and problem-solving thinking. The psychological theories concerning the problem-solving thinking leave the special demand of school subjects out of consideration, and search for connections of universal validity. In this article we attempt to connect an abstract theory of psychology concerning problem-solving thinking and a more practical conception of the problem-solving activity of mathematics, which is based on Polya's idea. In this way we can get a structure of problem-solving, which has scientific bases and at the same time it is useful in computer aided learning. Our result was developed and tested in Hungary so this is suitable especially for the Hungarian conditions of mathematics teaching.Tétel Szabadon hozzáférhető Models of impulsive phenomena(2003-12-01) Karsai, János"Take the textbook to computer" – is said quite often. Would it be so easy? If we start such a work, we meet a lot of trouble very soon. A book stored on a CD, read on the screen of computer and containing some hyperlinks does not become automatically electronic textbook. There are difficulties also in writing merely an electronic attachment to a classical book. In this paper, we deal with some important features (actually important from our point of view) of interactive mathematics textbooks, arising mathematical, didactical and technical problems. The "principles" are illustrated with examples taken from the book-CD "Models of Impulsive Phenomena".Tétel Szabadon hozzáférhető Figures to the article "Assimilation of mathematical knowledge using Maple" on pages 321-331. (color plate)(2003-12-01) Kovács, Zoltán; Kozma, LászlóColor plateTétel Szabadon hozzáférhető On the legacy of G. Pólya(2003-12-01) Zimmermann, BerndIn this article are given some new aspects of mathematical problem solving. A framework is presented by three main resources: (1) Pólya's studies about mathematical heuristics are augmented by information drawn from a study of the history of mathematical problem solving. (2) Connections are presented between mathematical problem solving and mathematical beliefs. (3) Experience with a special program for mathematical talented students is sketched. On this background a new textbook-series has been developed and some teaching examples are taken from this context. An outlook is given on some new research on teaching of problem solving, including possible relations to modern brain research.Tétel Szabadon hozzáférhető Beweise von Sätzen mit Hilfe der Modelle der hyperbolischen Geometrie(2003-12-01) , JenőWe give simple proofs for some problems of elemental hyperbolic geometry using the Poincare's half-sphere model. Our method is that a point of a figure is transformed to a special point of the model.Tétel Szabadon hozzáférhető Assimilation of mathematical knowledge using Maple(2003-12-01) Kovács, Zoltán; Kozma, LászlóFor more than four years we have been teaching a Maple course at University of Debrecen for prospective mathematics teachers. The aim of the course is that students get some experience on mathematical visualization with Maple. At the last part of the course the student is provided with a problem of geometrical flavor. Within three or four weeks he/she must obtain a solution. In this paper we present and analyze two of student projects: rotation of the hypercube and drawing of complex functions. The concluding remark is that most of the students will profit from using Maple for such type of problems: it helps to assimilate mathematical knowledge.Tétel Szabadon hozzáférhető From iteration to one - dimensional discrete dynamical systems using CAS(2003-12-01) Sárvári, Csaba; Klincsik, MihályIn our paper we present the basic didactical framework and approaches of a course on one-dimensional discrete dynamical systems made with the help of Computer Algebra Systems (CAS) for students familiar with the fundamentals of calculus. First we review some didactical principles of teaching mathematics in general and write about the advantages of the modularization for CAS in referring to the constructivistic view of learning. Then we deal with our own development, a CAS-based collection of programs for teaching Newton's method for the calculation of roots of a real function. Included is the discussion of domains of attraction and chaotic behaviour of the iterations. We summarize our teaching experiences using CAS.