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Psychology - an inherent part of mathematics education
1-18Views:114On the chronology of individual stations of psychology and their effect on mathematics education designed as working document for use in teacher training.
The article is structured as a literature survey which covers the numerous movements of psychology towards mathematics education. The current role of psychology in mathematics education documented by different statements and models of mathematics education should provide a basis for the subsequent investigations. A longitudinal analysis pausing at essential marks takes centre of the continuative considerations. The observed space of time in the chapter covers a wide range. It starts with the separation of psychology from philosophy as a self-contained discipline in the middle of the 19th and ends with the beginning of the 21st century. Each stop states the names of the originators and the branches of psychology they founded. These stops are accompanied by short descriptions of each single research objective on the one hand, and their contributions to mathematics education on the other hand. For this purpose, context-relevant publications in mathematics education are integrated and analysed. The evaluation of the influence of concepts of psychology on teaching technology in mathematics is addressed repeatedly and of great importance. The layout of this paper is designed for the use as a template for a unit in teacher-training courses. The conclusion of the article where the author refers to experiences when teaching elements of psychology in mathematics education courses at several universities in Austria is intended for a proof on behalf of the requested use.Subject Classification: 01A70, 01-XX, 97-03, 97D80
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Inquiry based mathematics education and the development of learning trajectories
63-89Views:409This article is based on the panel on inquiry based mathematics education and the development of learning trajectories held at the VARGA 100 Conference. After an introduction presenting the theme and organization of the panel, this article focuses on the diversity of conceptualizations of inquiry based education existing today in mathematics education and their influence on the vision and development of learning trajectories. More precisely, it considers the conceptualizations respectively associated with Realistic Mathematics Education, Genetic Constructivism, Tamás Varga’s educational approach and the Anthropological Theory of the Didactic, presented by the panellists, and also shows the efforts undertaken in European projects to reach consensusal visions.
Subject Classification: 97C30Q, 97D10, 97D20, 97D30, 97D40, 97D50
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Tamás Varga’s reform movement and the Hungarian Guided Discovery approach
11-28Views:92This paper presents Tamás Varga’s work focusing especially on the Hungarian Complex Mathematics Education reform project led by him between 1963 and 1978 and the underlying conception on mathematics education named “Guided Discovery approach”. In the first part, I describe Varga’s career. In the second part, I situate his reform project in its international and national historical context, including the international “New Math” movement and the “Guided Discovery” teaching tradition, something which is embedded in Hungarian mathematical culture. In the third part, I propose a didactic analysis of Varga’s conception on mathematics education, underlining especially certain of its characteristics which can be related to Inquiry Based Mathematics Education. Finally I briefly discuss Varga’s legacy today.
Subject Classification: 97-03, 97B20, 97D20, 97D40, 97D50
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Report of the conference "Connecting Tamás Varga’s Legacy and Current Research in Mathematics Education": November 6-8, 2019, Budapest, Hungary
5-8Views:55On the occasion of the 100th anniversary of the birth of the Hungarian mathematics educator, didactician and reform leader Tamás Varga, a conference on mathematics education has been organized in November 2019 and held at the Hungarian Academy of Science.
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Willy Servais and Tamás Varga A Belgian Hungarian perspective on teaching school mathematics
29-38Views:35Willy 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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Connections between discovery learning through the Pósa Method and the secondary school leaving examination in three Hungarian mathematics classrooms
67-85Views:166The Pósa Method is a guided discovery learning method that has been used in Hungarian education in the form of extracurricular activities for "gifted" mathematics students. A four-year experiment implemented the method in three more "average" classrooms. This article reports on the relationship between the Pósa Method and the standardized secondary school leaving mathematics exam (Matura Exam in short) in Hungary. Data consists of students' survey responses, teacher interviews, and exam results from the three Hungarian classrooms who took part in the four-year experiment. We identify aspects of the Pósa Method that can benefit and hinder exam performance. In addition, we find that learning through the Pósa Method for the four years of high school has adequately prepared students for the exam.
Subject Classification: 97D44, 97D54, 97D64
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Teaching undergraduate mathematics - a problem solving course for first year
183-206Views:64In this paper we describe a problem solving course for first year undergraduate mathematics students who would be future school teachers.
Subject Classification: 97B50, 97B70, 97D50, 97D60, 97F60, 97U30
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Capturing how students' abilities and teaching experiences affect teachers' beliefs about mathematics teaching and learning
195-212Views:79We developed an instrument to investigate the effect of students' abilities and teaching experiences on teachers' beliefs about teaching and learning of mathematics. In this pilot study, we used the instrument to measure the beliefs of 43 Indonesian math teachers and five additional teachers. Then, for further investigation, we interviewed those five additional teachers. Results from the 43 teachers' responses to the instrument show that in contrast to teachers with less than five years of teaching, teachers with more than five years elicit significantly different beliefs about mathematics teaching and learning in different contexts related to students' abilities. Teachers' reports in the further investigation indicate that teaching experiences with high and low ability students in teaching mathematics could be a possible explanation of this contrast.
Subject Classification: C20
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Better understanding mathematics by algorithmic thinking and computer programming
295-305Views:72Tamás Varga’s mathematics education experiment covered not just mathematics, but also other related topics. In many of his works he clearly stated that computer science can support the understanding of mathematics as much as mathematics supports informatics. On the other hand, not much later than the introduction of the new curriculum in 1978, personal computers started to spread, making it possible to teach informatics in classes and in extracurricular activities. Varga’s guided discovery approach has a didactic value for other age groups as well, not only in primary school. Its long-lasting effect can be observed even in present times. Having reviewed several educational results in the spirit of Tamás Varga, we have decided to present an extracurricular course. It is an open study group for age 12-18. Students solve problems by developing Python programs and, according to our experiences, this results in a deeper understanding of mathematical concepts.
Subject Classification: 97B10, 97B20, 97D50, 97N80, 97P20, 97P30, 97P40, 97P50, 97U70
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Visualisation in geometry education as a tool for teaching with better understanding
337-346Views:86In primary and secondary geometry education, some problems exist with pupils’ space thinking and understanding of geometric notions. Visualisation plays an important role in geometry education, and the development of pupils’ visualisation skills can support their spatial imagination. The authors present their own thoughts on the potential of including visualisation in geometry education, based on the analysis of the Hungarian National Core Curriculum and Slovak National Curriculum. Tasks for visualisation are also found in international studies, for example the Programme for International Student Assessment (PISA). Augmented reality (AR) and other information and communication technology (ICT) tools bring new possibilities to develop geometric thinking and space imagination, and they also support mathematics education with better understanding.
Subject Classification: 97U10, 97G10
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Some logical issues in discrete mathematics and algorithmic thinking
243-258Views:52The role of logic in mathematics education has been widely discussed from the seventies and eighties during the “modern maths period” till now, and remains still a rather controversial issue in the international community. Nevertheless, the relevance of discrete mathematics and algorithmic thinking for the development of heuristic and logical competences is both one of the main points of the program of Tamás Varga, and of some didactic teams in France. In this paper, we first present the semantic perspective in mathematics education and the role of logic in the Hungarian tradition. Then, we present insights on the role of research problems in the French tradition. Finely, we raise some didactical issues in algorithmic thinking at the interface of mathematics and computer science.
Subject Classification: 97E30
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Experiences in the education of mathematics during the digital curriculum from the perspective of high school students
111-128Views:101Due to the COVID-19 epidemic, Hungarian schools had to switch to a digital curriculum for an extended period between 2019 and 2021. In this article, we report on the experiences regarding the education of mathematics during the digital curriculum in the light of the reinstated on-site education, all through the eyes of high school students. Distance education brought pedagogical renewal to the lives of many groups. Students were asked about the positives and negatives of this situation.
Subject Classification: 97C90
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A retrospective look at discovery learning using the Pósa Method in three Hungarian secondary mathematics classrooms
183-202Views:148While 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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Promoting a meaningful learning of double integrals through routes of digital tasks
107-134Views:108Within a wider project aimed at innovating the teaching of mathematics for freshmen, in this study we describe the design and the implementation of two routes of digital tasks aimed at fostering students' approach to double integrals. The tasks are built on a formative assessment frame and classical works on problem solving. They provide facilitative and response-specific feedback and the possibility to request different hints. In this way, students may be guided to the development of well-connected knowledge, operative and decision-making skills. We investigated the effects of the interaction with the digital tasks on the learning of engineering freshmen, by comparing the behaviours of students who worked with the digital tasks (experimental group, N=19) and students who did not (control group, N=19). We detected that students in the experimental group showed more exibility of thinking and obtained better results in the final exam than students in the control group. The results confirmed the effectiveness of the experimental educational path and offered us interesting indications for further studies.
Subject Classification: 97D40, 97U70, 44A45
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Our digital education habits in the light of their environmental impact: the role of green computing in education
69-86Views:69With the increasing use of IT tools, the environmental impacts they generate have also increased. Education is increasingly relying on digital tools to become a major emitter of CO2 itself. Therefore, the task of education is to teach future generations how to use IT tools efficiently while being environmentally aware. In addition to some forms of green computing, we show the level and ratio of those teachers who have corresponding IT knowledge in the Hungarian education. In this study, we present the justification of the problem through a case study, which estimates the Internet traffic of a website streaming popular educational resources. In addition, we will examine the extent to which national and international educational organization and guidance documents address the development of digital environmentally aware thinking. Based on the content of this study, we suggest some considerations for content developers to decide if they really need to create the digital content.
Subject Classification: 97P99, 94-06, 94-02
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The tradition of problem-posing in Hungarian mathematics teaching
233-254Views:121Based on the literature, Pólya was influential in problem-posing research. The present paper draws attention to a book written with Pólya's collaboration, which has not yet received sufficient emphasis in the problem-posing literature. On the other hand, Pólya's impact on mathematics education in Hungary has been significant, including the problem-posing paradigm. Two works, published only in Hungarian, that rely heavily on problem-posing are highlighted. Furthermore, it is presented how problem-posing appeared in the Hungarian Complex Mathematics Teaching Experiment (1962-78) led by Tamás Varga.
Subject Classification: 97D50
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Straight line or line segment? Students’ concepts and their thought processes
327-336Views:65The article focuses on students’ understanding of the concept of a straight line. Attention is paid to whether students of various ages work with only part of a straight line shown or if they are aware that it can be extended. The presented results were obtained by a qualitative analysis of tests given to nearly 1,500 Czech students. The paper introduces the statistics of students’ solutions, and discusses the students’ thought processes. The results show that most of the tested students, even after completing upper secondary school, are not aware that a straight line can be extended. Finally, we present some recommendations for fostering the appropriate concept of a straight line in mathematics teaching.
Subject Classification: 97C30, 97D70, 97G40
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Mobile devices in Hungarian university statistical education
19-48Views:49The 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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Integrating Didactic Games in Higher Education: Benefits and Challenges
1-15Views:214In 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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Supporting the education of engineering mathematics using the immediate feedback method
49-61Views:45In the literature, several methods are suggested to deal with problems regarding the efficiency of mathematics education including techniques that help integrate new knowledge into long-term memory. We examined how effective the application of the immediate feedback method is in teaching engineering mathematics. The article presents the method used and the results obtained during the study.
Subject Classification: 97D40, 97D60
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Mathematical Laboratory: Semiotic mediation and cultural artefacts in the mathematics classroom
183-195Views:52Aim 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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Zur Veränderung des Stellenwertesvon Beweisen im Mathematikunterricht - eine Analyse von ungarischen Abiturprüfungenzwischen 1981 und 2020
35-55Views:42Proofs are not just an essential, crucial part of mathematics as a science, they also have a long tradition in Hungarian mathematics classrooms. However, the school in general and, mathematics education in particular, have been changed in the last few decades enormously, including the final secondary school examinations in mathematics. The current paper's main goal is to answer the question, how has been changed the weight and the content of reasoning and especially proving tasks in the relevant examinations.
Subject Classification: 97E54, 97D64, 97U44
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What does ICT help and does not help?
33-49Views:74Year by year, ICT tools and related teaching methods are evolving a lot. Since 2016, the author of the present lines has been looking for a connection between them that supports the development of mathematical competencies and could be integrated into Transcarpathian minority Hungarian language education too. As a doctoral student at the University of Debrecen, I experienced, for example, how the interactive whiteboard revolutionized illustration in Hungarian mathematics teaching, and how it facilitated students' involvement. During my research of teaching in this regard, in some cases, the digital solution had advantageous effects versus concrete-manipulative representation of
Bruner's too.
At the same time, ICT "canned" learning materials (videos, presentations, ...) allow for a shift towards repetitive learning instead of simultaneous active participation, which can be compensated for by the "retrieval-enhanced" learning method.
I have conducted and intend to conduct several research projects in a Transcarpathian Hungarian primary school. In the research so far, I examined whether, in addition to the financial and infrastructural features of the Transcarpathian Hungarian school, the increased "ICT-supported" and the "retrieval-enhanced" learning method could be integrated into institutional mathematics education. I examined the use of two types of ICT devices: one was the interactive whiteboard, and the other was providing one computer per student.
In this article, I describe my experiences, gained during one semester, in the class taught with the interactive whiteboard on the one hand, and in the class taught according to the "retrieval-enhanced" learning method on the other hand.
I compare the effectiveness of the classes to their previous achievements, to each other, and to a class in Hungary.Subject Classification: 97U70
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Guided Discovery in Hungarian Education Using Problem Threads: The Pósa Method in Secondary Mathematics Classrooms
51-67Views:72In Hungary, ‘guided discovery’ refers to instruction in which students learn mathematical concepts through task sequences that foster mathematical thinking. A prominent figure of guided discovery is Lajos Pósa, who developed his method to teach gifted students. Rather than teaching mathematics through thematic blocks, the Pósa Method employs webs of interconnected problem threads in which problems are built on each other, and different threads are presented simultaneously, so that students work on problems from multiple threads at the same time. It was found that this method has been successful as extracurricular training for gifted students since the 1980s; however since 2017, as part of an ongoing research, the method has been applied to mainstream curriculum in two public secondary school classrooms. The present paper examines the design and implementation processes of problem threads in this public secondary school context.
Subject Classification: 97D40
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Artworks as illustrations in Hungarian high school Mathematics textbooks
103-117Views:32Three different series of Hungarian Mathematics textbooks used in grade 9-12 education for the past 30 years have been analysed in this research. Our aim is to show and evaluate how the visual arts have been connected to mathematical ideas in these textbooks. We have applied the six dimensions of evaluation, which have recently been introduced in (Diego-Mantec on, Blanco, Búa Ares, & González Sequeiros, 2019) to categorise the illustrations of the three different series. We show examples for each dimension from the textbooks, and we find that even if the number of artistic illustrations in these coursebooks have significantly increased, in most cases these sporadic examples are not closely related to the mathematical context, mainly used for ornamental purposes to decorate the core text. Based on this classification we conclude that the number of artistic illustrations with underlying math concepts making students' participation more active could and should be significantly increased.
Subject Classification: 97U20