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Process or object? Ways of solving mathematical problems using CAS
117-132Views:91Graphing and symbol manipulating calculators are now a part of mathematics education in many countries. In Norway symbol manipulating calculators have been used at various exams in upper secondary education. An important finding in mathematics education is the duality of mathematical entities – processes and objects. Building on the theoretical development by Anna Sfard and others, the students' solutions on exam problems in upper secondary education are discussed with reference to procedural and structural knowledge. -
Report of Meeting Researches in Didactics of Mathematics and Computer Sciences: 31 March – 2 April, 2023 Oradea, Romania
83-107Views:370The meeting Researches in Didactics of Mathematics and Computer Sciences was held in Oradea, Romania, at Partium Christian University, from 31 March to 2 April, 2023. It was organized by the Doctoral School of Mathematical and Computational Sciences of the University of Debrecen and Partium Christian University. The 85 participants – including 18 PhD students – came from 9 countries and represented 30 institutions of higher and secondary education. There were 4 plenary and 53 session talks in the program.
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What can we learn from Tamás Varga’s work regarding the arithmetic-algebra transition?
39-50Views:244Tamás Varga’s Complex Mathematics Education program plays an important role in Hungarian mathematics education. In this program, attention is given to the continuous “movement” between concrete and abstract levels. In the process of transition from arithmetic to algebra, the learner moves from a concrete level to a more abstract level. In our research, we aim to track the transition process from arithmetic to algebra by studying the 5-8-grader textbooks and teacher manuals edited under Tamás Varga's supervision. For this, we use the appearance of “working backward” and “use an equation” heuristic strategies in the examined textbooks and manuals, which play a central role in the mentioned process.
Subject Classification: 97-01, 97-03, 97D50
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An idea which yields a lot of elementary inequalities
61-72Views:144The 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. -
Report on "Problem Solving in Mathematics Education": ProMath 6 Conference, 8–11 September, 2005, Debrecen, Hungary
313-319Views:198The sixth ProMath Conference was organized at the University of Debrecen (Hungary) in the year 2005. There were 12 presentations. After a short historical introduction we present the 12 abstracts written by the authors. -
Visualisation in geometry education as a tool for teaching with better understanding
337-346Views:369In 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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Teaching graph algorithms with Visage
35-50Views:194Combinatorial 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. -
Ein anderer Weg bei dem Logarithmusunterricht: Ein entwickelndes Unterrichtsexperiment
1-16Views:112In my developmental experiment I tried to fusion the expectations of the Hungarian education and the realistic mathematics education. The duration of this experiment was 33 lectures long. In this article I try to show how were introduced the definition, the rules of logarithm with real life problems and the outcome of the experiment. -
The hyperbola and Geogebra in high-school instruction
277-285Views:181In this article the results of teaching/learning hyperbola and its characteristics in high-school using computers and GeoGebra are shown. Students involved in the research attend Engineering School "Nikola Tesla" in Leposavic, Serbia. The aim of the research was to define ways and volume of computer and GeoGebra usage in mathematics instruction in order to increase significantly students' mathematical knowledge and skills. -
The development of geometrical concepts in lower primary mathematics teaching: the square and the rectangle
153-171Views:197Our 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. -
Die Methode von Prof. Tibor Szele im Unterricht begabter Schüler
143-151Views:190Prof. Tibor Szele' has attempted to develop the mathematical problemsolving, creativity include the use of investigations and host of other devices beyond the classroom, i.e. in "mathematical circles" for talented students in secondary schools. This paper of the author – who himself has taken part in Seles1s mathematical circles – quotes from these activities according his earlier notes. This description illustrates the didactic method of Prof. T. Szele. -
Heuristic arguments and rigorous proofs in secondary school education
167-184Views:184In this paper we are going to discuss some possible applications of the mechanical method, especially the lever principle, in order to formulate heuristic conjectures related to the volume of three-dimensional solids. In the secondary school educational processes the heuristic arguments are no less important than the rigorous mathematical proofs. Between the ancient Greek mathematicians Archimedes was the first who made heuristic conjectures with the methods of Mechanics and proved them with the rigorous rules of Mathematics, in a period, when the methods of integration were not known. For a present day mathematician (or a secondary school mathematics teacher) the tools of the definite integral calculus are available in order to calculate the volume of three dimensional bodies, such as paraboloids, ellipsoids, segments of a sphere or segments of an ellipsoid. But in the secondary school educational process, it is also interesting to make heuristic conjectures by the use of the Archimedean method. It can be understood easily, but it is beyond the normal secondary school curriculum, so we recommend it only to the most talented students or to the secondary schools with advanced mathematical teaching programme. -
Levels of students' understanding on infinity
317-337Views:249Here we report some results of a two-year study for grades 5-6 and 7-8 (during the academic years 2001-03). The study included a quantitative survey for approximately 150 Finnish mathematics classes out of which 10 classes were selected to a longitudinal part of the study. Additionally, 40 students from these classes participated also a qualitative study. This paper will focus on students' understanding of infinity and the development of that understanding. The results show that most of the students did not have a proper view of infinity but that the share of able students grew, as the students got older. -
Comparative geometry on plane and sphere: didactical impressions
81-101Views:66Description of experiences in teaching comparative geometry for prospective teachers of primary schools. We focus on examples that refer to changes in our students' thinking, in their mathematical knowledge and their learning and teaching attitudes. At the beginning, we expected from our students familiarity with the basics of the geographic coordinate system, such as North and South Poles, Equator, latitudes and longitudes. Spherical trigonometry was not dealt with in the whole project. -
Some thoughts on a student survey
41-59Views:144The paper analyzes a survey of college students and describes its major findings. The object of the survey, involving 154 students, was to discover and highlight the problems that arise in taking the course Economic Mathematics I. The paper, as the summary of the first phase of a research project, wishes to present these problems, ways that may lead out of them, and possible means of help that can be offered to those taking the course. -
Constructing the disk method formula for the volume obtained by revolving a curve around an axis with the help of CAS
363-376Views:164Calculus concepts should have been taught in a carefully designed learning environment, because these concepts constitute a very important base for almost all applied sciences. The integral, one of the fundamental concepts of Calculus, has a wide application area. This paper focuses on constructing the disk method formula for the volume obtained by revolving a curve around an axis with the help of a CAS.
In this study, a semi-structured interview was carried out. In this interview, we tried to construct the disk method formula.
The levels of constructing the disk method formula in this study are:
• Introducing the concept: evaluating the volume of an Egyptian pyramid.
• Evaluating the volume of a cone obtained by revolution (using Maple worksheet).
• Designing their own ring and evaluating its price (using Maplet).
In this study, the interview has been presented as a dialog between teacher and students. When we look at feedback from students, we see that such a teaching method effects students in a positive way and causes them to gain conceptual understanding directed towards the concepts of approximation and volume. -
Square root in secondary school
59-72Views:290Although in Hungary, for decades, the calculation method of the square root of a real number is not in the mathematics curriculum, many of the taught concepts and procedures can be carried out using different square root finding methods. These provide an opportunity for students in secondary school to practice and deepen understand the compulsory curriculum. This article presents seven square-root- nding methods, currently teachable in secondary schools.
Subject Classification: A33, A34, F53, F54
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Correction to Mneimneh (2019): "Simple variations on the Tower of Hanoi: A study of recurrences and proofs by induction” Teaching Mathematics and Computer Science 17 (2019), 131-158.
109Views:150In the article “Simple variations on the Tower of Hanoi: A study of recurrences and proofs by induction” by Saad Mneimneh (Teaching Mathematics and Computer Science, 2019, 17(2), 131–158. https://doi.org/10.5485/TMCS.2019.0459), there was an error in Table 1 (p. 155), and consequently, the first paragraph of Section 8 (p. 154) also needed correction.
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Mapping students’ motivation in a problem oriented mathematics classroom
111-121Views:231This 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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Realizing the problem-solving phases of Pólya in classroom practice
219-232Views:349When 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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CAS-aided visualization in LATEX documents for mathematical education
1-18Views:181We have been developing KETpic as a macro package of a CAS for drawing fine LATEX-pictures, and we use it efficiently in mathematical education. Printed materials for mathematics classes are prepared under several constraints, such as "without animation", "mass printings", "monochrome", and "without halftone shadings". Because of these constraints, visualization in mathematical education tends to be unsatisfactory. Taking full advantages of LATEX and CAS, KETpic enables us to provide teaching materials with figures which are effective for mathematical education. The effects are summarized as follows:
(1) The plottings of KETpic are accurate due to CAS, and enable students to deduce mathematical laws.
(2) KETpic can provide adequate pictures for students' various interest. For example, when some students who understand a matter try to modify it, KETpic can give them appropriate and experimental figures.
(3) Even though CAS can draw 3D-figures beautifully and automatically, it is expensive for mass printings and the figures are sometimes not easy to understand. Oppositely, 3D-graphics by KETpic are monochrome, but are richly expressive.
In this paper, we give various examples of LATEX-pictures which we drew by using KETpic. For instance, the picture which is used in order to explain the convergence theorem of Fourier series makes it easier for students to understand the idea that function series converge to another function. Also the picture of skeleton is endowed with clear perspective. KETpic gives us great potential for the teaching of combinatorial mathematics. Through these examples, we claim that KETpic should have great possibilities of rich mathematical expressions under the constraints above mentioned. -
Mathematical gems of Debrecen old mathematical textbooks from the 16-18th centuries
73-110Views:103In the Great Library of the Debrecen Reformed College (Hungary) we find a lot of old mathematical textbooks. We present: Arithmetic of Debrecen (1577), Maróthi's Arithmetic (1743), Hatvani's introductio (1757), Karacs's Figurae Geometricae (1788), Segner's Anfangsgründe (1764) and Mayer's Mathematischer Atlas (1745). These old mathematical textbooks let us know facts about real life of the 16-18th centuries, the contemporary level of sciences, learning and teaching methods. They are rich sources of motivation in the teaching of mathematics. -
Infimum problems derived from the proofs of some generalized Schwarz inequalities
41-57Views:223We define f(a;b)(r) = ar + b/r for all a, b, r Є R with r > 0. And, for some subsets A of R, we determine FA_+ (a; b) = inf (r Є A_+) f(a,b) (r) ; where A_+ ={r Є A : r > 0}. The above in ma are mainly motivated by the proofs of some recent generalized Schwarz inequalities established by the present authors.
Subject Classification: I35
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Development of high school students' geometric thinking with particular emphasis on mathematically talented students
93-110Views:180We carried out research using Zalman Usiskin's test (1982) and also a modified version of his test to see how the geometric approach of secondary school students (Grades 8-10) specialized in mathematics had changed. We observed two groups of students for several years. Our aim was to find a relation between the change of the mean of the van Hiele level of the students and the structure of the geometry syllabus. We also observed if there was a change in the geometric approach of the students during the summer holidays and if so, in what way it changed. -
Reappraising Learning Technologies from the Viewpoint of the Learning of Mathematics
221-246Views:170Within 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.
