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• How the derivative becomes visible: the case of Daniel

  81-97

  Markus Hähkiöniemi

  Views:

  34

  This paper reports how an advanced 11th-grade student (Daniel) perceived the derivative from a graph of a function at a task-based interview after a short introduction to the derivative. Daniel made very impressive observations using, for example, the steepness and the increase of a graph as well as the slope of a tangent as representations of the derivative. He followed the graphs sequentially and, for example, perceived where the derivative is increasing/decreasing. Gestures were an essential part of his thinking. Daniel's perceptions were reflected against those of a less successful student reported previously [Hähkiöniemi, NOMAD 11, no. 1 (2006)]. Unlike the student of the previous study, Daniel seemed to use the representations transparently and could see the graph as a representation of the derivative.

• Using the computer to visualise graph-oriented problems

  15-32

  Erika Gyöngyösi

  Views:

  31

  The computer, if used more effectively, could bring advances that would improve mathematical education dramatically, not least with its ability to calculate quickly and display moving graphics. There is a gap between research results of the enthusiastic innovators in the field of information technology and the current weak integration of the use of computers into mathematics teaching.
  This paper examines what exactly the real potentials of using some mathematics computer software are to support mathematics teaching and learning in graph-oriented problems, more specifically we try to estimate the value added impact of computer use in the mathematics learning process.
  While electronic computation has been used by mathematicians for five decades, it has been in the hands of teachers and learners for at most three decades but the real breakthrough of decentralised and personalised micro-computer-based computing has been widely available for less than two decades. And it is the latter facility that has brought the greatest promise for computers in mathematics education. That computational aids overall do a better job of holding students' mathematical interest and challenging them to use their intellectual power to mathematical achievement than do traditional static media is unquestionable. The real question needing investigation concerns the circumstances where each is appropriate.
  A case study enabled a specification of advantages and obstacles of using computers in graph-oriented questions. Individual students' interviews revealed two less able students' reactions, difficulties and misinterpretations while using computers in mathematics learning.
  Among research outcomes is that the mathematical achievement of the two students observed improved and this makes teaching with computers an overriding priority for each defined teaching method.
  This paper may not have been realised without the valuable help of the Hungarian Eötvös State Grant.

• Teaching probability using graph representations

  103-122

  Gabriella Zsombori

  Szilárd András

  Views:

  32

  The main objective of this paper is to present an elementary approach to classical probability theory, based on a Van Hiele type framework, using graph representation and counting techniques, highly suitable for teaching in lower and upper secondary schools. The main advantage of this approach is that it is not based on set theoretical, or combinatorial knowledge, hence it is more suitable for beginners and facilitates the transitions from level 0 to level 3. We also mention a few teaching experiences on different levels (lower secondary school, upper secondary school, teacher training, professional development, university students) based on this approach.

• Teaching graph algorithms with Visage

  35-50

  Andreas Fest

  Ulrich Kortenkamp

  Views:

  27

  Combinatorial optimization is a substantial pool for teaching authentic mathematics. Studying topics in combinatorial optimization practice different mathematical skills, and because of this have been integrated into the new Berlin curriculum for secondary schools. In addition, teachers are encouraged to use adequate teaching software. The presented software package "Visage" is a visualization tool for graph algorithms. Using the intuitive user interface of an interactive geometry system (Cinderella), graphs and networks can be drawn very easily and different textbook algorithms can be visualized on the graphs. An authoring tool for interactive worksheets and the usage of the build-in programming interface offer new ways for teaching graphs and algorithms in a classroom.

• "How to be well-connected?" An example for instructional process planning with Problem Graphs

  145-155

  Eszter Varga

  Views:

  96

  Teachers’ design capacity at work is in the focus of didactical research worldwide, and fostering this capacity is unarguably a possible turning point in the conveyance of mathematical knowledge. In Hungary, the tradition hallmarked by Tamás Varga is particularly demanding towards teachers as they are supposed to be able to plan their long-term processes very carefully. In this contribution, an extensive teaching material designed in the spirit of this tradition will be presented from the field of Geometry. For exposing its inner structure, a representational tool, the Problem Graph is introduced. The paper aims to demonstrate that this tool has potential for analyzing existing resources, helping teachers to reflect on their own preparatory and classroom work, and supporting the creation of new designs.

  Subject Classification: 97D40, 97D50, 97D80, 97G10, 97U30

• Linear clause generation by Tableaux and DAGs

  109-118

  Gergely Kovásznai

  Views:

  31

  Clause generation is a preliminary step in theorem proving since most of the state-of-the-art theorem proving methods act on clause sets. Several clause generating algorithms are known. Most of them rewrite a formula according to well-known logical equivalences, thus they are quite complicated and produce not very understandable information on their functioning for humans. There are other methods that can be considered as ones based on tableaux, but only in propositional logic. In this paper, we propose a new method for clause generation in first-order logic. Since it inherits rules from analytic tableaux, analytic dual tableaux, and free-variable tableaux, this method is called clause generating tableaux (CGT). All of the known clause generating algorithms are exponential, so is CGT. However, by switching to directed acyclic graphs (DAGs) from trees, we propose a linear CGT method. Another advantageous feature is the detection of valid clauses only by the closing of CGT branches. Last but not least, CGT generates a graph as output, which is visual and easy-to-understand. Thus, CGT can also be used in teaching logic and theorem proving.

• The single-source shortest paths algorithms and the dynamic programming

  25-35

  Zoltán Kátai

  Views:

  30

  In this paper we are going to present a teaching—learning method that help students look at three single-source shortest paths graph-algorithms from a so called "upperview": the algorithm based on the topological order of the nodes, the Dijkstra algorithm, the Bellman-Ford algorithm. The goal of the suggested method is, beyond the presentation of the algorithms, to offer the students a view that reveals them the basic and even the slight principal differences and similarities between the strategies. In order to succeed in this object, teachers should present the mentioned algorithms as cousin dynamic programming strategies.

• Veranschaulichung der Lehrstoffstruktur durch Galois-Graphen

  217-229

  László Fatalin

  Views:

  41

  In 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.

• Engineering and Economic Mathematics for Engineering Management Students

  35-50

  Kornélia Éva Dékány

  Views:

  34

  In this article we describe the first part of a case study, which was made with 48 Engineering Management students. The participants of the case study were MSc level students at the Szent István University, Gödöllő. We looked for methods by which we can support the most important components of competence motivation and the development of mathematical and other key competences during the mathematics lessons and individual learning. Another goal of our research was to get reliable information about students learning methods and their awareness of self-efficacy, furthermore their achievement in the subject of Engineering and Economic Mathematics. Detailed assistance was provided for the students in the e-learning portal. Knowledge tests, questionnaire and personal interviews with the students were also used. As an example we introduce one of the knowledge tests connected with the first half of the course about linear programming and graph theory. We detail its didactical background and show the results of the students.

• Brute force on 10 letters

  183-193

  Zoltán Kovács

  István Hudi

  Views:

  31

  We deal with two problems in the set of 10-character-long strings. Both problems can be solved by slightly different methods, but our approach for each is brute force. As we point out, there can be differences in effectivity even in different brute force algorithms. As an additional result, we answer an open question of Raymond Smullyan's.

• Examining relation between talent and competence through an experiment among 11th grade students

  17-34

  Veronika Árendás

  Views:

  31

  The 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.

• Über den Vergleich des mathematischen bzw. mathematikdidaktischen Vektorbegriffs durch den Galois-Graphen

  1-12

  László Fatalin

  Views:

  47

  In this article we show how to apply the method of Galois-graph – one of the means of the formal concept-analysis in order to coordinate the mathematical and didactical requirements. As an example we have chosen the concept of the "vector". As a result of the analysis it is proved that, in elaborating the right vector concept the geometric and algebraic foundations are both needed. The analysis also points out that the geometric model, based on the concept of the "directed segment" is unnecessarily overemphasized in the East-European education.

• The influence of computer on examining trigonometric functions

  111-123

  Djurdjica Takači

  Radivoje Stojković

  Jasminka Radovanović

  Views:

  23

  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.

• Game theory for managers and mechanical manager students

  73-91

  Kornélia Éva Dékány

  Views:

  25

  In this article we describe the second part of a case study, in which 48 Mechanical Management students were involved. The participants of the case study were MSc level students at Szent István University, Gödöllő.
  In the case study we looked for methods by which we can support the most important components of competence motivation and the development of mathematical and other key competences during the mathematics lessons and individual learning.
  Another goal of our research was to get reliable information about students learning methods and their awareness of self-efficiency, furthermore their achievement in the subject of Engineering and Economic Mathematics.
  Detailed assistance was provided for the students in the e-learning portal. Knowledge tests, questionnaire and personal interviews with the students were also used.
  During the semester four topics have been discussed: linear programming, graph theory, game theory and differential equations. In this article I will describe the lesson preparations, the help for examinations and the students' achievement on game theory.

• Illustrated analysis of Rule of Four using Maple

  383-404

  György Maróti

  Views:

  37

  Rule 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.