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Lehre der Trigonometrie anhand realistischer Aufgaben im Online-Unterricht
87-105Views:108The aim of our study was to explore the effects of the active use of realistic exercises in the field of trigonometry. We taught a group of 14 pupils, who were in grade 11. The most of them told us they did not plan mathematics-related studies in the future. We included realistic exercises into our teaching plan, which covered the fields of scalar product, as well as the sine and cosine theorems. Our teaching experiment was done within the framework of online teaching. Effects on the motivation, performance and results of the students were taken into consideration. We also attempted to examine the effects of online teaching on motivation and whether the use of realistic exercises is worthwhile in an online classroom environment. Performance of the students showed a tendency of improvement when they were dealing with the material through realistic exercises even despite the teaching happened online.
Subject Classification: 97C70, 97D40, 97G60
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Teaching geometry using computer visualizations
259-277Views:20In this work we study the development of students' creativity using computer-aided-teaching during IT classroom. Teaching geometry in Bolyai Grammar School specialized natural science classes is not an easy task. Here is introduced a new didactic means of teaching geometry which nevertheless requires the same effort to understand the material, but uses a different more active method to familiarize students with the topics. Traditional methods, and the use of compasses and rulers are not omitted either, as they develop the students' motor skills. -
The application of modelling tasks in the classroom – why and how? with reflections on an EU teacher training course
231-244Views:34The aim of the article is to present the concept of mathematical modelling in the classroom. LEMA (Learning and Education in and through Modelling and Applications) was an EU Comenius funded project in which mathematics educators from six countries worked to produce materials to support teachers' professional development. A group of voluntary Hungarian mathematics teachers were taught modelling for a year and we were and still are given feedback continously. The article leads us from the general concept of mathematical modelling to its practice in the classroom. It presents difficulties that teachers have to face when doing modelling lessons and their students' reactions are also mentioned. We present sample tasks from the material of the teacher training course as well as tasks that were created by the participants. -
Teaching XML
317-335Views:31The author has been teaching XML at the Faculty of Informatics, University of Debrecen since the end of the nineties. This paper gives an overview of XML technology from an educators viewpoint that is based on the experience that the author has gained teaching XML over the years. A detailed description of the XML course is provided. Methodological issues are also discussed. -
The Project Method and investigation in school mathematics
241-255Views:39The 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. -
Problemorientierung im Mathematikunterricht – ein Gesichtspunkt der Qualitätssteigerung
251-291Views:7The aim of this article is to give a synopsis of problem orientation in mathematics education and to stimulate the discussion of the development and research about problem-orientated mathematics teaching. At the beginning we present historical viewpoints of problem orientation and their connection with recent theories of cognition (constructivism). Secondly we give characterizations of concepts that stand in the context of problem-orientation and discuss different forms of working with open problems in mathematics teaching. Arguments for more problem orientation in mathematics education will be discussed afterwards. Since experience shows that the implementation of open problems in classroom produces barriers, we then discuss mathematical beliefs and their role in mathematical learning and teaching. A list of literature at the end is not only for references but also can be used to further research.
Zusammenfassung. Ziel des Beitrags ist es, eine Synopsis in Bezug auf Problemorientierung im Mathematikunterricht zu geben und die Diskussion bezüglich Entwicklung und Forschung eines problemorientierten Mathematikunterrichts zu stimulieren. Als Erstes werden historische Gesichtspunkte von Problemorientierung und deren Verkn üpfung mit neueren Erkenntnistheorien (Konstruktivismus) vorgestellt. Zweitens werden Erläuterungen zu Begriffen, die im Kontext von Problemorientierung stehen, gegeben und verschiedene Ausprägungen der Behandlung offener Probleme im Mathematikunterricht diskutiert. Argumente für eine stärkere Berücksichtigung von Problemorientierung im Mathematikunterricht werden danach erörtert. Auf Barrieren bei der Implementierung von offenen Problemen im Unterricht, die durch mathematische Beliefs (Vorstellungen, Überzeugungen) geprägt sind, wird zum Schluss eingegangen. Die abschließend aufgeführte Literaturliste dient nicht nur dem Beleg der Zitate, sondern kann auch zu weiterer Vertiefung genutzt werden. -
Teaching graph algorithms with Visage
35-50Views:27Combinatorial 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. -
Group Work at High School According to the Method of Tamás Varga
167-176Views:78The aim of our research is to develop students’ logical thinking. For this reason, Hungarian mathematics teachers need to be encouraged to try new methods which induce greater student involvement. Research all over the world prove that self-instruction or self-verbalizing has high effect on the learning process. This was one of the key elements of Tamás Varga’s experiment in high school. In our classroom experiments we are using a special cooperative method from Kagan among 14-18 years old students, called Sage and Scribe structure. We are looking for the answers to the following question: Does this method make mathematics lessons more enjoyable and more comfortable for students? Furthermore, we assume this structure could open the gate toward other collaborative and cooperative teaching technics.
Subject Classification: 97D40
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Forming the concept of congruence II.
1-12Views:31This paper is a continuation of the article Forming the concept of congruence I., where I gave theoretical background to the topic, description of the traditional method of representing the isometries of the plane with its effect on the evolution of congruence concept.
In this paper I describe a new method of representing the isometries of the plane. This method is closer to the abstract idea of 3-dimensional motion. The planar isometries are considered as restrictions of 3-dimensional motions and these are represented with free translocations given by flags.
About the terminology: I use some important concepts connected to teaching of congruence, which have to be distinguished. My goal is to analyse different teaching methods of the 2-dimensional congruencies. I use the term 3-dimensional motion for the orientation preserving (direct) 3-dimensional isometry (which is also called rigid motion or rigid body move). When referring the concrete manipulative representation of the planar congruencies I will use the term translocation. -
Mechanisms for teaching introductory programming using active learning
407-421Views:27One of the requirements of teaching introductory programming to students whose branch of learning is engineering or science is bridging the gap between in-class lectures and real-world applications. Traditional passive approaches to lecturing often focus on the syntax of a language with little or no discussion of the process involved in using the language to design algorithms to solve real-world problems. One way of overcoming the limitations of traditional lecturing is by tailoring lectures towards becoming more student-oriented, a pedagogical methodology known as active learning. This paper explores mechanisms for implementing active learning in introductory programming courses in computer science. -
Forming the concept of congruence I.
181-192Views:9Teaching isometries of the plane plays a major role in the formation of the congruence-concept in the Hungarian curricula.
In the present paper I investigate the way the isometries of the plane are traditionally introduced in most of the textbooks, especially the influence of the representations on the congruence concept, created in the teaching process.
I am going to publish a second part on this topic about a non-traditional approach (Forming the concept of congruence II). The main idea is to introduce the isometries of the two dimensional plane with the help of concrete, enactive experiences in the three dimensional space, using transparent paper as a legitimate enactive tool for building the concept of geometric motion. I will show that this is both in strict analogy with the axioms of 3-dimensional motion and at the same time close to the children's intuitive concept of congruence. -
Research studies in didactics of mathematics supported by the Operant Motive Test
153-173Views:33The present paper reports a case-study which took place within an EUsupported international program organized for research and development of multi-grade schools (NEMED, [16] [26]). One of the main goals of the research was to develop the connection between disadvantageous social situations and the efficiency (success or failure) in learning mathematics especially from the point of view of average and above-average (talented) students: Why does the talent of children with socially disadvantageous background remain undiscovered? How can we make school mathematics more aware of hidden talents?
The author was looking for a didactical solution that compensated for social disadvantages without restricting the development of "average" students by using sociological, educational, psychological and mathematical (experimental and theoretical) studies in interaction with a series of experimental (hypothesis testing and exploratory) investigations.
We constructed tools and methods for exploration and experimental teaching, adapted to Hungarian conditions (Curriculum Development, teacher training, materials, interviews, Kuhl's motivation test, Malara's "researchers and practicing teachers in cooperation" method, etc., see [18], [20]).
The teaching materials and methodological guidelines are based on Bruner's representation theory (see [5]). The empirical research took place in 16 multi-grade schools located in different parts of the country. The author co-operated with nearly 250 students and 25 teachers for 3 years. In this paper we try to demonstrate how an Operant Motive Test can be involved in this research (see [18]). -
Realizing the problem-solving phases of Pólya in classroom practice
219-232Views:124When teaching mathematical problem-solving is mentioned, the name of Pólya György inevitably comes to mind. Many problem-solving lessons are planned using Pólya's steps and helping questions, and teachers often rely on his heuristics even if their application happens unconsciously. In this article, we would like to examine how the two phases, Making a plan and Looking back, can be realized in a secondary school mathematics lesson. A case study was designed to observe and analyse a lesson delivered using cooperative work.
Subject Classification: 97B10, 97C70, 97D40, 97D50
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Teaching Gröbner bases
57-76Views:23In this article we offer a demonstration of how the StudentGroebner package, a didactic oriented Maple package for Gröbner basis theory, could assist the teaching/learning process. Our approach is practical. Instead of expounding on deep didactic theory we simply give examples on how we imagine experimental learning in classroom. The educational goal is to prepare the introduction of two sophisticated algorithms, the division algorithm and Buchberger's algorithm, by gathering preliminary knowledge about them. -
Algorithmics of the knapsack type tasks
37-71Views:27We propose a new kind of approach of the teaching of knapsack type problems in the classroom. We will remind you the context of the general knapsack-task and we will classify it, including the two most popular task variants: the discrete and the continuous one. Once we briefly present the solving algorithm of the continuous variant, we will focus on the solving of the discrete task, and we will determine the complexity of the algorithms, looking for different optimizing possibilities. All these issues are presented in a useful way for highschool teachers, who are preparing students in order to participate in different programming contests. -
Ist eine schnelle tiefgehende (und nachhaltige) Änderung in der Vorstellung von Mathematiklehrern möglich? - Reflexion der Erfahrungen eines Fortbildungskurses im Bereich der mathematischen Modellierung
1-20Views:13Based on the material which was worked out within the project LEMA (2006-2009) pilot-teacher training courses were organized in the six partner countries, so in Hungary as well in the subject: Practice of Modelling tasks in the classroom. According to the tests which were filled out by the participants the conclusion was formulated that they achieved some changes in their pedagogical knowledge and in their estimation concerning their self-efficacy, but they didn't have shown any changes in their beliefs of mathematics and mathematics education. However according to their experience as project partners and leaders of the Hungarian course the authors have the idea that despite of the international results there are changes in this subject in the case of the Hungarian participants. This way can formulated the question:
Which changes can be observed in the case of the participants concerning belief towards mathematics and mathematics education after the course and how long-lasting these changes are?
The question is examined on the example of two teachers who were participants of the course. -
Computer cooking vs. problem solving
35-58Views:66Computer cooking is a task-related phenomenon where students (end-users) must blindly follow a long list of orders without any connection to the content of the problem, if there is any. Despite its low efficacy, this method is widely used and accepted in informatics both in the learning-teaching process and testing. The National Base Curriculum 2020 in Hungary is in complete accordance with the ‘Informatics Reference Framework for Schools’, but the course books hardly use the latest results of computer education research. The present paper provides examples of how the results of computer education research can be integrated into teaching-learning materials and classroom practices and discusses the effectiveness and consequences of the different solutions, where tool-centred approaches are compared to problem-focused solutions.
Subject Classification: 94-01
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Willy Servais and Tamás Varga A Belgian Hungarian perspective on teaching school mathematics
29-38Views:74Willy Servais and Tamás Varga had a major influence on the development of mathematics education during the 1960s and 1970s, both in their home countries and internationally. In 1971 they jointly published Teaching School Mathematics–A Unesco Source Book, a review of curriculum reforms that were under way in different parts of the world. The book, presenting several modern syllabuses as well as examples of classroom techniques and segments of teacher-student dialogues, provided an often consulted guide to the field of mathematics education. We re-read this book and in this way acquire a unique insight into the modernization efforts of school mathematics during the 1960s and early 1970s. We take this opportunity to discuss the sometimes partly divergent views of Servais and Varga on modern mathematics education as reflected in this book.
Subject Classification: 97-03
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Looking back on Pólya’s teaching of problem solving
207-217Views:228This article is a personal reflection on Pólya's work on problem solving, supported by a re-reading of some of his books and viewing his film Let Us Teach Guessing. Pólya's work has had lasting impact on the goals of school mathematics, especially in establishing solving problems (including non-routine problems) as a major goal and in establishing the elements of how to teach for problem solving. His work demonstrated the importance of choosing rich problems for students to explore, equipping them with some heuristic strategies and metacognitive awareness of the problem solving process, and promoting 'looking back' as a way of learning from the problem solving experience. The ideas are all still influential. What has changed most is the nature of classrooms, with the subsequent appreciation of a supporting yet challenging classroom where students work collaboratively and play an active role in classroom discussion.
Subject Classification: 97D50, 97A30
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Comment les enseignants en formation initiale utilisent les technologies informatiques dans leurs classes
187-208Views:29The 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. -
Manipulatives and semiotic tools of Game of Go as playful and creative activity to learn mathematics in early grades in France
197-206Views:63This research develops resources to teach mathematics in French primary school by using the game of Go. A group of searchers, teachers and go players meet at university to produce teaching resources. These resources are implemented in the classroom. Then the group evaluate this implementation and improve the resources. The aim of this classroom research is to study the opportunities of the game of Go to learn mathematics and to propose a teacher training course to implement the game of Go in French primary schools in accordance with the French syllabus. Game of Go appears as a manipulative and semiotic tool to learn mathematics at primary school.
Subject Classification: 97D50, 97U60
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An examination of descriptive statistical knowledge of 12th-grade secondary school students - comparing and analysing their answers to closed and open questions
63-81Views:74In this article, we examine the conceptual knowledge of 12th-grade students in the field of descriptive statistics (hereafter statistics), how their knowledge is aligned with the output requirements, and how they can apply their conceptual knowledge in terms of means, graphs, and dispersion indicators. What is the proportion and the result of their answers to (semi-)open questions for which they have the necessary conceptual knowledge, but which they encounter less frequently (or not at all) in the classroom and during questioning? In spring 2020, before the outbreak of the pandemic in Hungary, a traditional-classroom, “paper-based” survey was conducted with 159 graduating students and their teachers from 3 secondary schools. According to the results of the survey, the majority of students have no difficulties in solving the type of tasks included in the final exam. Solving more complex, open-ended tasks with longer texts is more challenging, despite having all the tools to solve them, based on their conceptual knowledge and comprehension skills. A valuable supplement to the analysis and interpretation of the results is the student attitudes test, also included in the questionnaire.
Subject Classification: 97K40, 97-11, 97D60
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"How to be well-connected?" An example for instructional process planning with Problem Graphs
145-155Views:96Teachers’ 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
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Some Remarks on History of Mathematical Problem Solving
51-64Views:33In this contribution, it is our goal is to look on history of mathematics as a resource for a long-term study of mathematical problem solving processes and heuristics. In this way we intend to get additional information, e. g., about heuristics which proved to be extremely successful to create new mathematics. "Changing representation" and "false position" are examples of such strategies, which are illustrated by concrete examples to demonstrate the use for classroom teaching and teacher education. Our methods are based on hermeneutic principles. -
Freudenthal fantasy on the bus, an American adaptation
133-142Views:61In the 1960’s two mathematicians, Hans Freudenthal in the Netherlands and Tamás Varga in Hungary, had argued that people learn mathematics by being actively involved and investigating realistic mathematical problems. Their method lives on in today’s teaching and learning through the various components of cooperative and active learning, by taking ownership in learning, and learning through student dialogue. The goal is to create a welcoming classroom atmosphere in which play takes the front seat. One such scenario is visiting various (animal) stations at the zoo by bus (illustrated by pictures). Passengers are getting on and off the bus at each station (illustrated by arrows), which is modeled on the open number line. This adapted and modified action research was carried out with 5-yearl-old children in public schools of Staten Island, NY in 2019.
Subject Classification: 97D40, 97F20, 97F30