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Teaching centroids in theory and in practice
67-88Views:36The main aim of this paper is to present an inquiry-based professional development activity about the teaching of centroids and to highlight some common misconceptions related to centroids. The second aim is to emphasize a major hindering factor in planning inquiry based teaching/learning activities connected with abstract mathematical notions. Our basic problem was to determine the centroid of simple systems such as: systems of collinear points, arbitrary system of points, polygons, polygonal shapes. The only inconvenience was that we needed practical activities where students could validate their findings and calculations with simple tools. At this point we faced the following situation: we have an abstract definition for the centroid of a finite system of points, while in practice we don't even have such systems. The same is valid for geometric objects like triangles, polygons. In practice we have triangular objects, polygonal shapes (domains) and not triangles, polygons. Thus in practice for validating the centroid of a system formed by 4,5,... points we also need the centroid of a polygonal shape, formed by an infinite number of points. We could use, of course, basic definitions, but our intention was to organize inquiry based learning activities, where students can understand fundamental concepts and properties before defining them. -
Central axonometry in engineer training and engineering practice
17-28Views:22This 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 formation of area concept with the help of manipulative activities
121-139Views:33Examining the performance of Hungarian students of Grades 4-12 in connection with area measurement, we found many deficiencies and thinking failures. In the light of this background, it seems reasonable to review the educational practice and to identify those teaching movements that trigger the explored problems and to design a teaching experiment that tries to avoid and exclude them. Based on result we make recommendations for the broad teaching practice. In our study we report on one part of a multi-stage teaching experiment in which we dealt with the comparison of the areas of figures, the decomposition of figures and the special role of the rectangle in the process of area concept formation. The conclusion of the post-test is that manipulative activities are important and necessary in Grades 5 and 6, more types of equidecomposition activities are needed and the number of measuring tasks with grid as a tool should also be increased. -
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. -
Learning and teaching combinatorics with Sage
389-398Views:44Learning Mathematics is not an easy task, since this subject works with especially abstract concepts and sophisticated deductions. Many students lose their interest in the subject due to lack of success. Computer algebra systems (CAS) provide new ways of learning and teaching Mathematics. Numerous teachers use them to demonstrate concepts, deductions and algorithms and to make learning process more interesting especially in higher education. It is an even more efficient way to improve the learning process, if students can use the system themselves, which helps them to practice the curriculum.
Sage is a free, open-source math software system that supports research and teaching algebra, analysis, geometry, number theory, cryptography, numerical computation, and related areas. I have been using it for several years to aid the instruction of Discrete Mathematics at Óbuda University. In this article I show some examples how representations provided by this system can help in teaching combinatorics. -
Experiences using CAS and multimedia int teaching vectorcalculus
363-382Views:31The 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. -
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. -
The development of geometrical concepts in lower primary mathematics teaching: the square and the rectangle
153-171Views:41Our 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. -
Differentiated instruction not only for Mathematics teachers
163-182Views:168The aim of differentiated development in a heterogeneous group of learners (DDHG) is to reduce school leaving without education, using an adaptive and innovative teaching-learning environment and using the most effective strategies, methods and techniques. Furthermore, this strategy helps in developing skills for learners and building cooperation between learners in heterogeneous classes through the use of the special, status-management educational procedure, and finally its strength is to sort the status ranking among learners, and to change the social structure of the class. Our goal is to figure out how to share best practices with teachers. One of the effective ways to renew teaching practice is through further training for teachers. As a trainer of the Logic-based subprogram of the Complex Basic Program (CBP) the author of the paper has experienced how well logic-based and decision-making strategies work in other subjects as well as in mathematics.
Subject Classification: 97D40
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Teaching multiparadigm programming based on object-oriented experiences
171-182Views:25Multiparadigm programming is an emerging practice in computer technology. Co-existence of object-oriented, generic and functional techniques can better handle variability of projects. The present paper gives an overview of teaching multiparadigm programming approach through typical language concepts, tools in higher education. Students learning multiparadigm-oriented subjects would gain considerable expertise, which is highly needed by the industrial side in large-scale application development. -
Teaching model-based testing
1-17Views:1205Different testing methodologies should play an important role in the education of informatics. In the model-based testing (MBT) approach, the specification of the system is described with a formal model. This model can be used to revise the correctness of the specification and as a starting point for automatic test generation. The main problem with MBT is however, that there is a huge gap between theory and practice and that this approach has a high learning curve. To cope with these problems, current paper shows, how the MBT approach can be introduced to students through a small scale example.
Subject Classification: P50
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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. -
Comparison of teaching exponential and logarithmic functions based on mathematics textbook analysis
297-318Views:28Exponential 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. -
Conventions of mathematical problems and their solutions in Hungarian secondary school leaving exams
137-146Views:8Collecting and analyzing the conventions indispensable for interpreting mathematical problems and their solutions correctly assist successful education and objective evaluation. Many professional and didactic questions arose while collecting and analyzing these conventions, which needed clarification, therefore the materials involved concisely in the conventions enrich both the theory and practice of mathematics teaching. In our research we concentrated mainly on the problems and solutions of the Hungarian school leaving examinations at secondary level in mathematics. -
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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Mapping students’ motivation in a problem oriented mathematics classroom
111-121Views:64This research focuses on mapping students’ motivation by implementing problem-solving activities, namely how the problem-oriented approach affects the students’ commitment, motivation, and attitude to learning. As a practicing teacher, the author faced difficulties with motivation and sought to improve her practice in the form of action research as described in this paper. Based on the literature, the author describes sources of motivation as task interest, social environment, opportunity to discover, knowing why, using objects, and helping others. The author discusses the effect of problem-oriented teaching on the motivation of 7th-grade students. In this paper, the results of two lessons are presented.
Subject Classification: 97C20, 97D40, 97D50, 97D60
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Young women's barriers to choose IT and methods to overcome them - A case study from Hungary
77-101Views:175Women's scarcity in the STEM, especially in the IT sector is pronouncedly evident. Young women are obstructed from entering and remaining in IT by a broad range of social, educational, and labor market factors. In our paper, we would like to analyze the main barriers girls face in choosing IT, while also proposing potential methods to help them overcome these obstacles. In the second part of the paper, we will present a case study to illustrate in detail how the combination of the above methods can be put into practice to address and tackle the complex set of barriers girls face. We will first introduce a Hungarian annual program, Girls' Day ("Lányok napja"), specifically aimed to promote STEM to girls, then we will present two specific events organized for the 2020 edition of the program and designed with the above principles in mind. The interactive presentation, exposing girls to female role models of the field in a gamified way, and a game development exercise, building Scratch programming skills, have attempted to provide young women both with positive perspectives and experiences in IT, which are instrumental in helping them to surmount entrenched obstacles and raise their interest in the field.
Subject Classification: 97P10, 97U30
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Mathematical Laboratory: Semiotic mediation and cultural artefacts in the mathematics classroom
183-195Views:91Aim of this presentation is to summarize the influence of Tamas Varga on the Italian research and practice concerning didactics of mathematics since the 70s of the 20th centuries. While being in Budapest for the Conference I noticed that this influence was not known by most Hungarian mathematics educators. I guess that also in Italy, only the teacher educators of my generation know Varga’s influence on the teaching and learning of mathematics in primary school. Hence I start from a brief summary of development of mathematics curriculum in Italy (mainly in primary school) in the last decades of the 20th century. I focus some elements that may be connected with Varga’s influence and, later, some recent development of them.
Subject Classification: 97G20, 97-U6, 97A40
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Understanding the spatiotemporal sample: a practical view for teaching geologist students
89-99Views:25One of the most fundamental concept of statistics is the (random) sample. Our experience – acquired during the years of undergraduate education – showed that prior to industrial practice, the students in geology (and, most probably, in many other non-mathematics oriented disciplines as well) are often confused by the possible multiple interpretation of the sample. The confusion increases even further, when samples from stationary temporal, spatial or spatio-temporal phenomena are considered. Our goal in the present paper is to give a viable alternative to this overly mathematical approach, which is proven to be far too demanding for geologist students.
Using the results of an environmental pollution analysis we tried to show the notion of the spatiotemporal sample and some of its basic characteristics. On the basis of these considerations we give the definition of the spatiotemporal sample in order to be satisfactory from both the theoretical and the practical points of view. -
An idea which yields a lot of elementary inequalities
61-72Views:7The aim of the article is to show how studies in higher mathematics can be applied in everyday teaching practice to construct new problems for their pupils. In higher mathematics it is known that the set of real numbers with the addition and multiplication (shortly: (R,+,x)) is an ordered field. Considering a strictly monotonic increasing and continuous function σ with domain ...
By this idea, using different kinds of functions σ we show a lot of different elementary inequalities. -
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. -
Teaching word processing – the practice
247-262Views:20I compared two surveys, which were aimed to check the word processing ability of students in high schools and universities. The surveys were carried out ten years apart from one another, in 1997 and 2006. The results clearly show that most of the students are not able to use word processors properly. In the survey of 1997 I found explanation for this underperformance in the lack of computers and teachers. However, the results of the second survey did not prove any better than the results of the first, and in 2006 neither the number of computers nor the number of teachers can be blamed. What else then? I suggest that the reason for this general ignorance, for this `modern illiteracy' is the ignorance of the teachers. Until the teachers are not prepared and the senior students of the universities leave the education system without a proper knowledge of the required subjects, there is little chance that they would be able to teach word processing at a satisfactory level.