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Analysis of the affective factors of learning mathematics among teacher trainees
225-254Views:156The Hungarian National Core Curricula gives primacy to the development of abilities and the practical application of knowledge. The task of the training programme is primarily to prepare teacher trainees for the teaching and educating profession. As teachers, they are going to plan, organize, help, guide, control and evaluate the learning of mathematics of individuals and groups of students from the age of 6 to 10 (12), and cultivate their mathematical skills, thinking and positive attitude towards any mathematical activities. In order to train educators who are able to meet the above requirements on high standard, it is necessary to update the teacher training programme based on the trainees' preliminary knowledge and motivation level.
The key to learn about the child's mind and achieve conscious development is the systematization of factual knowledge and methodological awareness. The modern, flexible approach to subject pedagogy, based on pedagogy, psychology and epistemology, qualifies trainees to educate learners who understand and like mathematics. Therefore, it is essential to develop the trainees' positive approach to mathematics and arouse their demand for continuous professional improvement. (Programme of the four-year primary school teacher training, 1995.)
In our research we are looking for ways of ascertaining the starting parameters which have influence on the planning of the studies of mathematics and subject pedagogy. In this article we introduce a questionnaire by the means of which we collected information on the trainees' attitude and its changing towards mathematics. With the help of the analysis of the answers we paint a picture of the ELTE TÓFK (Eötvös Loránd University, Faculty of Elementary and Nursery School Teacher's Training) third year students' attitude to the subject, and we compare it to the tendencies noticed in the mass education. The energy invested in learning is influenced by the assumption of the relevance and importance of the subjects. Therefore we considered it also our task to reveal. Besides the students' attitude toward mathematics and their assumption about their own competence we have collected data also on their performance in the subject. Summarising the research results we show the advantages of the questionnaire, and summarise the observations which would indicate need for methodological changes in the mathematics teacher training. -
An interactive animation for learning sorting algorithms: How students reduced the number of comparisons in a sorting algorithm by playing a didactic game
45-62Views:204Learning programming and understanding algorithms is one of the hardest tasks for novice computer science students. One of the basic algorithms they learn during the introductory programming and algorithms courses are the sorting algorithms. Students like learning these and other algorithms by animations and didactic games, however, these animations are not educationally useful in every case. In this article, we present our educational sorting game, which can be used to introduce the topic of sorting algorithms. The didactic game can be used later too, as a demonstrative tool for explaining the more efficient, quicksort algorithm. We conducted a pedagogical experiment, in which we examined the process of development of sorting algorithms by students while they used the mentioned didactic game. The results showed that students were able to create an algorithm to solve the sorting problem, and they improved its effectiveness by reducing the number of comparisons in the algorithm. They were also able to understand the importance of the efficiency of algorithms when we demonstrated them the quicksort algorithm using the same tool after the experiment. -
Zbigniew Michalewicz - Matthew Michalewicz: Puzzle Based Learning: An introduction to critical thinking, mathematics, and problem solving. Hybrid Publishers Melbourne 2008 (Book review)
415-420Views:265Based on their experiences with engineering, mathematics, computer science, business students concerning the puzzle based learning in different countries the authors summarize their main problem solving teaching ideas. With help of interesting, motivating, nice problems they analyze the main mathematical principles and problem types. The review gives an overview about the main ideas, results of an interesting book. -
Integrating Didactic Games in Higher Education: Benefits and Challenges
1-15Views:876In our paper, we study the reasons for the introduction of didactic games and the way of their application in higher education, especially in teaching mathematics. After describing the main characteristics and needs of Generation Z students, we outline the advantages and drawbacks of gamification and game-based learning, followed by some new aspects to their classification. The idea of device-based grouping arose because the most commonly used methods require IC tools. Gen Zs naturally accept gamified learning materials available on digital and mobile platforms, but we must not forget about traditional games either. In higher education, especially in the case of small-group teaching there should also be room for traditional, specialized didactic games, of which we focus on the benefits of card games.
Subject Classification: 97C70, 97D20, 97D40, 97U70
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Efficient language teaching software in a multimedia context
361-374Views:163In this article I deal with the efficiency of multimedia teaching programs, analyzing possibilities for their improvement in the field of language teaching. This research has been carried out with the use of the latest technologies, language teaching software, internet based language teaching applications, digital dictionaries, online content, and the latest results from the field of computational linguistics. The goal of my research is to create a general model that serves and supports various kinds of approaches to improving efficiency; I cannot attempt to present a complete, detailed analytical review due to the complexity and size of this topic. However, my opinion is that by considering and understanding the theoretical aspects of the subject, and supported by certain important ideas, we will be able to achieve remarkable improvements in the field of learning efficiency and knowledge retention in the language teaching and learning process that might lead to outstanding results. -
A retrospective look at discovery learning using the Pósa Method in three Hungarian secondary mathematics classrooms
183-202Views:361While the Pósa Method was originally created for mathematical talent management through extracurricular activities, three "average" public secondary school classrooms in Hungary have taken part in a four-year experiment to implement the Pósa Method, which is based on guided discovery learning of mathematics. In this paper, we examine the students' and teachers' reflections on the Pósa Method, and how student perspectives have changed between their first and last year of high school. Overall, teachers and students had a positive experience with the Pósa Method. Furthermore, our research indicated that this implementation has met several objectives of the Pósa Method, including enjoyment of mathematics and autonomous thinking.
Subject Classification: 97D40
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Virtual worlds in education – best practice, design and research considerations
309-323Views:145The article briefly describes and characterizes the virtual worlds, different kinds of virtual worlds and possibilities of their use in education. The virtual worlds beyond the passive learning opportunities offer active, constructive and collaborative learning possibilities as well. In the middle section of this article we present the most common ways of different kinds of learning methods, which are illustrated by practical examples as well. In the final sections of the article we deal with the design research approach of 3D virtual worlds. -
Engineering and Economic Mathematics for Engineering Management Students
35-50Views:177In 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. -
Recalling calculus knowledge
55-70Views:123The main purpose of educational system is not only that the students perform well at the exam, but to remember the learnt material to some degree some time after the learning. This paper investigates students' retained knowledge, focusing mainly on topics concerning derivatives and differentiation, and examines the effect of re-learning in a short period of time. Results indicate that retained knowledge should be taken into consideration in instructional design and curriculum planning for the sequencing courses. -
Proof step analysis for proof tutoring - a learning approach to granularity
325-343Views:194We present a proof step diagnosis module based on the mathematical assistant system Ωmega. The task of this module is to evaluate proof steps as typically uttered by students in tutoring sessions on mathematical proofs. In particular, we categorise the step size of proof steps performed by the student, in order to recognise if they are appropriate with respect to the student model. We propose an approach which builds on reconstructions of the proof in question via automated proof search using a cognitively motivated proof calculus. Our approach employs learning techniques and incorporates a student model, and our diagnosis module can be adjusted to different domains and users. We present a first evaluation based on empirical data. -
Using the computer to visualise graph-oriented problems
15-32Views:142The 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. -
Mobile devices in Hungarian university statistical education
19-48Views:200The methodological renewal of university statistics education has been continuous for the last 30 years. During this time, the involvement of technology tools in learning statistics played an important role. In the Introduction, we emphasize the importance of using technological tools in learning statistics, also referring to international research. After that, we firstly examine the methodological development of university statistical education over the past three decades. To do this, we analyze the writings of statistics teachers teaching at various universities in the country. To assess the use of innovative tools, in the second half of the study, we briefly present an online questionnaire survey of students in tertiary economics and an interview survey conducted with statistics teachers.
Subject Classification: 97-01, 97U70, 87K80
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Comparative survey on pupils' beliefs of mathematics teaching in Finland and Ukraine
13-33Views:99The focus of this comparative survey was the following research question: What are the differences and similarities in pupils' beliefs in mathematics between Finland and Ukraine? Data were gathered with the help of a questionnaire. The questionnaire consists of 32 structured statements about mathematics teaching for which the pupils were asked to rate their beliefs on a 5-step scale. The Finnish sample comprised 255 pupils, and the Ukrainian sample 200 pupils. Our data has been gathered with a non-probabilistic convenience sampling.
The main results of our survey are, as follows: Generally, pupils' beliefs of mathematics teaching and learning in Finland and Ukraine are rather far from similar. An investigation of the differences between pupils' answers across the two countries also showed beliefs that are characteristic for each country. For pupils in Finland, the characteristic beliefs seem to be, as follows: the value of strict discipline, working in small groups, and the idea that all understand. For pupils in Ukraine, the most characteristic might be the following beliefs: the use of learning games, the emphases of mathematical concepts, and teachers' explanations. -
A computational thinking problem-thread for grade 7 students and above from the Pósa method
101-110Views:292Lajos Pósa has been developing his “learning through discovery” (Győri & Juhász, 2018) method since 1988. His weekend math camps are focused on fostering problem-solving skills and high-level mathematical-thinking skills in gifted students from grades 7 to 11. One of the core aspects of the method is the structure of the problems, all problems are part of a complex, intertwined, and rich network. In this article we analyze a computational thinking problem-thread and its role in the camps’s network of problems (Gosztonyi, 2019), and show some aspects of the method. The insights gained using this method can be useful in other contexts. The possible adaptation of the method to secondary and high schools is briefly discussed as well.
Subject Classification: 97D40
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Challenges that a teacher-researcher faces during an action research – a case study
89-99Views:258This paper explores the dual role of the teacher-researcher in a four-year action research project focused on problem-based learning in mathematics. It highlights the challenges faced during the phases of planning, implementation, analysis, and reflection. Drawing on insights from the author’s experiences and observations based on both qualitative and quantitative data collection methods, the study identifies distinct challenges linked to the dual role, like differing design goals or subjective-objective voices. The author also proposes solutions to the identified challenges, such as collaboration with university experts and using reflective practices. Furthermore, the research underscores the beneficial impact of action research on enhancing teachers’ awareness and bridging the theory-practice gap, calling for further studies in this area.
Subject Classification: 97D99
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Teaching puzzle-based learning: development of transferable skills
245-268Views:344While computer science and engineering students are trained to recognise familiar problems with known solutions, they may not be sufficiently prepared to address novel real-world problems. A successful computer science graduate does far more than just program and we must train our students to reach the required levels of analytical and computational thinking, rather than hoping that it will just 'develop'. As a step in this direction, we have created and experimented with a new first-year level course, Puzzle-based Learning (PBL), that is aimed at getting students to think about how to frame and solve unstructured problems. The pedagogical goal is increase students' mathematical awareness and general problem solving skills by employing puzzles, which are educational, engaging, and thought provoking. In this paper we continue sharing our experiences in teaching such a course. Whereas a brief discussion on our pedagogical objectives were covered in the first paper together with the material of the first of two lectures on pattern recognition, this follow-up paper presents the material of the second of two lectures, in which additional exercises are discussed to reinforce the lesson. Along the way we provide a glimpse of some foundational ideas of computer science such as incomputability and general system development strategies such as incremental and iterative reasoning. This paper discusses the outcomes of PBL courses, which include expected improvement in the overall results achieved by students who have undertaken PBL courses, compared to those students who have not. -
Integrating elements of data science into high-school teaching: Naïve Bayes-classification algorithm and programming in Python
307-316Views:263Probability theory and mathematical statistics are traditionally one of the most difficult chapters of mathematics to teach. One of the authors, Péter Princz has experience in teaching various topics via computer programming of the problem at hand as a class activity. The proposed method is to involve programming as a didactic tool in hard-to-teach topics. The intended goal in this case is to implement a naïve Bayes-classifier algorithm in Python and demonstrate the machine-learning capabilities of it by applying it to a real-world dataset of edible or poisonous mushrooms. The students would implement the algorithm in a playful and interactive way. The proposed incremental development process aligns well with the spirit of Tamás Varga who considered computers as modern tools of experimental problem solving as early as in the 1960s.
Subject Classification: 97D40, 97D50, 97K50, 97K99, 97M60, 97P40, 97P50, 97U50
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Teaching Fourier series, partial differential equations and their applications with help of computer algebra system
51-68Views:149In this paper, some examples of Fourier series and partial difference equations will be shown to demonstrate opportunities for CAS use in various circumstances. The well-known white-box – black-box teaching-learning techniques and the modularization will be used to allow the use of the same worksheet in different ways. -
Problem-solving in mathematics with the help of computers
405-422Views:99One 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. -
Cooperative learning in teaching mathematics: the case of addition and subtraction of integers
117-136Views:139In the course of teaching and learning mathematics, many of the problems are caused by the operations with integers. My paper is a presentation of an experiment by which I tried to make the acquisition of these operations easier through the use of cooperative methods and representations. The experiment was conducted in The Lower-Secondary School of Paptamási from Romania, in the school year 2009-2010. I present the results of the experiment. -
CALIBRATE and CAS/DGS resources
267-279Views:127The CALIBRATE project was initiated by the EU with the goal of expanding the use of ICT in education by increasing the amount of available learning resources via resource exchange. Although CAS/DGS can be used to easily create high quality learning resources which are also easily adaptable across national boundaries, such resources are difficult to find at CALIBRATE portals. We believe that this is due to CAS/DGS still being rather exotic to most of the people as well as with the common problem of finding existing appropriate resources. A possible solution is for CALIBRATE portals to properly equip existing and forthcoming CAS/DGS resources with suitable metadata and to provide some integration with CAS/DGS tools, enabling both beginners and power users to create and exchange CAS/DGS resources. -
Metacognition – necessities and possibilities in teaching and learning mathematics
69-87Views:227This article focuses on the design of mathematics lessons as well as on the research in mathematics didactics from the perspective that metacognition is necessary and possible.
Humans are able to self-reflect on their thoughts and actions. They are able to make themselves the subject of their thoughts and reflections. In particular, it is possible to become aware of one’s own cognition, which means the way in which one thinks about something, and thus regulate and control it. This is what the term metacognition, thinking about one’s own thinking, stands for.
Human thinking tends to biases and faults. Both are often caused by fast thinking. Certain biases occur in mathematical thinking. Overall, this makes it necessary to think slow and to reflect on one’s own thinking in a targeted manner.
The cognitive processes of thinking, learning and understanding in mathematics become more effective and successful when they are supplemented and extended by metacognitive processes. However, it depends on a specific design of the mathematics lessons and the corresponding tasks in mathematics.Subject Classification: 97C30, 97C70, 97D40, 97D50, 97D70
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Many paths lead to statistical inference: Should teaching it focus on elementary approaches or reflect this multiplicity?
259-293Views:243For statistics education, a key question is how to design learning paths to statistical inference that are elementary enough that the learners can understand the concepts and that are rich enough to develop the full complexity of statistical inference later on. There are two ways to approach this problem: One is to restrict the complexity. Informal Inference considers a reduced situation and refers to resampling methods, which may be completely outsourced to computing power. The other is to find informal ways to explore situations of statistical inference, also supported with the graphing and simulating facilities of computers. The latter orientates towards the full complexity of statistical inference though it tries to reduce it for the early learning encoun-ters. We argue for the informal-ways approach as it connects to Bayesian methods of inference and allows for a full concept of probability in comparison to the Informal Inference, which reduces probability to a mere frequentist concept and – based on this – restricts inference to a few special cases. We also develop a didactic framework for our analysis, which includes the approach of Tamás Varga.
Subject Classification: 97K10, 97K70, 97K50, 97D20
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Experiences using CAS and multimedia int teaching vectorcalculus
363-382Views:88The 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 centroids in theory and in practice
67-88Views:232The 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.
