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Tamás Varga’s reform movement and the Hungarian Guided Discovery approach
11-28Views:382This 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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A mathematical and didactical analysis of the concept of orientation
111-130Views:253The development of spatial ability, in particular the development of spatial orientation is one of the aims of mathematics education.
In my work, I examine the concept of orientation, especially concepts of between, left, right, below, above, front, back, clockwise and anticlockwise. I analyze answers given for a simple orientation task prepared for elementary school pupils. I would like to call attention to the difficulties pupils have even in case of solving simple orientation problems.
We have different ways to know more about the crucial points of a concept, especially of the concept of orientation. In this study I bring out one of them. I analyze and make some didactical conclusions about the origin and the axiomatic structure of orientation. -
Learning and teaching combinatorics with Sage
389-398Views:127Learning 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. -
Visualisation in geometry education as a tool for teaching with better understanding
337-346Views:305In 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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Impact of teacher communication skills on students’ classroom engagement in mathematics learning
1-27Views:456The study investigated teachers’ communication skills in relation to students’ classroom engagement in mathematics learning. The study area is Makurdi Local Government Area in Benue State, Nigeria. This study adopted a cross-sectional research design. A sample of 34 teachers and 204 students were drawn from twenty schools. Two researcher-structured instruments were used for data collection: Mathematics Teacher’s Communication Skills Questionnaire (MTCSQ) and Students’ Engagement in Mathematics Questionnaire (SEMQ). Descriptive statistics, analysis of variance, and independent t-tests were used to address the research questions and test the hypotheses. It was found that there is significant difference among the mean ratings on behavioural, and emotional engagements of students in mathematics classes taught by teachers with poor, fair, and good communication skills. There is no significant difference among the mean ratings on combined and cognitive engagements of students in mathematics classes taught by teachers with poor, fair, and good communication skills. Equally found was that the differences between male and female students’ mean engagement in mathematics for poor, fair, and good communication skill classes were not statistically significant. It was then recommended that teacher communication skills should be fashioned in ways to accommodate and strengthen each component of students’ engagement.
Subject Classification: 97C70
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Über die Verwendung von Maple für die Simulation von Mechanismen
21-39Views:87Maple is used to do numerical computation, plot graphs and do exact symbolic manipulations and word processing. This paper demonstrates how Maple can be used for the simulation of mechanisms. This offers the possibility for students to become familiar with this particular section of mathematical modelling. The mechanism under consideration is a so-called F-mechanisms, i.e., a planar parallel 3-RRR robot with three synchronously driven cranks. It turns out that at this example it is not possible to find the poses of the moving triangle exactly by graphical methods with traditional instruments only. Hence, numerical methods are essential for the analysis of motions which can be performed by an F-mechanism. -
Illustrated analysis of Rule of Four using Maple
383-404Views:98Rule of Four, as a basic didactic principle, was formulated among the NCTM 2000 standards (see [14]) and since then it is quoted by numerous books and publications (see [4], [9], [12]). Practically we can say it is accepted by the community of didactic experts. The usage of the Rule of Four, however, has been realized mainly in the field of calculus, in fact certain authors restrict the wording of the principle to the calculus itself (e.g. [3]).
Calculus is a pleasant field, indeed. A sequence of values of a function provides us with example for numeric representation, while the formula and the graph of the function illustrate symbolic and graphical representations, respectively. In the end by wording the basic features of the function on natural language we gain textual representation.
This idyllic scene, however, becomes more complex when we leave the frame of calculus. In this paper we investigate the consequences of the usage of Rule of Four outside calculus. We discuss the different types of representations and show several examples which make the multiple features of representation evident. The examples are from different fields of mathematics and are created by the computer algebra system Maple, which turns out to be an excellent tool for illustration and visualization of the maim features of mathematical objects.
Next we introduce the concept of basic representation and rational representation, which is considered as the mathematical notion of "didactic usable" or "didactic rational" representation. In the end we generalize the notion of numeric representation, which leads us a more widely usable didactic principle which can be considered as a generalization of Rule of Four. -
Proof step analysis for proof tutoring - a learning approach to granularity
325-343Views:137We 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. -
Word problems in different textbooks at the early stage of teaching mathematics comparative analysis
31-49Views:259In a previous research, Csíkos and Szitányi (2019) studied teachers’ views and pedagogical content knowledge on the teaching of mathematical word problems. While doing so, they reviewed and compared Eastern European textbooks of Romania, Russia, Slovakia, Croatia, and Hungary to see how world problem-solving strategies are presented in commonly used textbooks. Their results suggested that teachers, in general, agreed with the approach of the textbooks regarding the explicit solution strategies and the types of word problems used for teaching problem-solving. They also revealed that the majority of the participants agreed that a word problem-solving algorithm should be introduced to the students as early as in the first school year. These results have been presented at the Varga 100 Conference in November 2019. As the findings suggested a remarkable similarity between the Eastern European textbook approaches, in the current study we decided to conduct further research involving more textbooks from China, Finland, and the United States.
Subject Classification: 97U20, 08A50
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Reappraising Learning Technologies from the Viewpoint of the Learning of Mathematics
221-246Views:126Within the context of secondary and tertiary mathematics education, most so-called learning technologies, such as virtual learning environments, bear little relation to the kinds of technologies contemporary learners use in their free time. Thus they appear alien to them and unlikely to stimulate them toward informal learning. By considering learning technologies from the perspective of the learner, through the analysis of case studies and a literature review, this article asserts that the expectation of these media might have been over-romanticised. This leads to the recommendation of five attributes for mathematical learning technologies to be more relevant to contemporary learners' needs: promoting heuristic activities derived from human history; facilitating the shift from instrumentation to instrumentalisation; facilitating learners' construction of conceptual knowledge that promotes procedural knowledge; providing appropriate scaffolding and assessment; and reappraising the curriculum. -
Understanding the spatiotemporal sample: a practical view for teaching geologist students
89-99Views:103One 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. -
On an international training of mathematically talented students: assets of the 20 years of the “Nagy Károly Mathematical Student-meetings”
77-89Views:196The focus of this paper is to present the gems of the "Nagy Károly Mathematical Student-meetings" in Rév-Komárom (Slovakia) from 1991 to 2010. During these 20 years there was done a lot of work to train mathematically talented students with Hungarian mother tongue and to develop their mathematical thinking, and to teach them problem solving and heuristic strategies for successful acting on the competitions. We collected the most interesting problems and methods presented by the trainer teachers. -
Straight line or line segment? Students’ concepts and their thought processes
327-336Views:205The 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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Didactical remarks on the changes in the requirements of the matriculation exam in Mathematics in Hungary
95-110Views:137Students within the Hungarian education system typically take a matriculation exam to obtain a secondary education certificate, which also serves as a prerequisite for university admission. Public education is regulated at different levels. One of its most fundamental elements is the National Core Curriculum, the current version of which came into force in September 2020. It is crucial to adapt the requirements of the matriculation exam in mathematics to this and ensure transparent communication about the changes. Regarding this, there exists a sample paper that contains tasks that one can reasonably expect in the actual exam in the spring. Since I have been working as a private math tutor for almost a decade and have been preparing students for the matriculation exam since then, I intend to highlight the most outstanding features from a didactic point of view based on the analysis of this sample paper.
Subject Classification: 97A30, 97B10, 97B70, 97D60, 97U40
