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The mathematics teacher trainee as an assistant teacher
295-306Views:192The experiment described in the article aims to answer two needs at once: that of assistant teachers in schools, and that of a more practical training of mathematics teachers. The answer suggested is a model of school experience where mathematics teacher trainees work as assistant teachers in schools. An attempt to realize this model is described, and it is evaluated positively. -
Teaching student teachers: various components of a complex task
55-72Views:111In this paper we summarize various aspects of teacher training and teaching student teachers (mainly concerning teachers of upper secondary school and High school). We stress several hints and recommendations to better achieve the obviously important aim: they should learn doing, understanding and teaching mathematics!
Of course, our view is particularly influenced by European traditions, but we think most of them equally apply to teacher training and teaching student teachers elsewhere. Neither is the paper meant to give an all sided overview about the problem field of teacher education as a whole, nor does it contain provocative, completely new ideas. We just want to describe our view of some aspects, based primarily on our personal experience in the mentioned field. -
Cultivating algorithmic thinking: an important issue for both technical and HUMAN sciences
107-116Views:223Algorithmic thinking is a valuable skill that all people should master. In this paper we propose a one-semester, algorithm-oriented computer science course for human science students. According to our experience such an initiative could succeed only if the next recipe is followed: interesting and practical content + exciting didactical methods + minimal programming. More explicitly, we suggest: (1) A special, simple, minimal, pseudo-code like imperative programming language that integrates a graphic library. (2) Interesting, practical and problem-oriented content with philosophical implications. (3) Exciting, human science related didactical methods including art-based, inter-cultural elements. -
The appearance of the characteristic features of the mathematical thinking in the thinking of a chess player
201-211Views:242It is more and more important in 21st century's education that not only facts and subject knowledge should be taught but also the ways and methods of thinking should be learnt by students. Thinking is a human specificity which is significant both in mathematics and chess. The exercises aimed at beginner chess players are appropriate to demonstrate to students the mathematical thinking of 12-14 year-old students.
Playing chess is an abstract activity. During the game we use abstract concepts (e.g. sacrifice, stalemate). When solving a chess problem we use logical quantifiers frequently (e.g. in the case of any move of white, black has a move that...). Among the endgames we find many examples (e.g. exceptional draw options) that state impossibility. Affirmation of existence is frequent in a mate position with many moves. We know there is a mate but the question in these cases is how it can be delivered.
We present the chess problem on beginners' level although these exercises appear in the game of advanced players and chess masters too, in a more complex form. We chose the mathematical tasks from arithmetic, number theory, geometry and the topic of equations. Students encounter these in classes, admission exams and student circles. Revealing the common features of mathematical and chess thinking shows how we can help the development of students' mathematical skills with the education of chess. -
Professional Competence in science education
129-137Views:139The article begins with a brief introduction aimed at sensitizing the reader to the perception of a trend in Mathematics and Computer Science Education publications towards empirical studies. Contrary to the stated trend, the characterization of Professional Competence is intended to serve as the guiding concept for the paper. The role of Professional Competence is discussed in various areas incorporating context-relevant publications in consecutive chapters. The discussion starts with the area of material development, covering Educational Standards and ends with Didactic Principles.
Subject Classification: 97xxx, 94xxx
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Maximum and minimum problems in secondary school education
81-98Views:266The aim of this paper is to offer some possible ways of solving extreme value problems by elementary methods with which the generally available method of differential calculus can be avoided. We line up some problems which can be solved by the usage of these elementary methods in secondary school education. The importance of the extremum problems is ignored in the regular curriculum; however they are in the main stream of competition problems – therefore they are useful tools in the selection and development of talented students. The extremum problem-solving by elementary methods means the replacement of the methods of differential calculus (which are quite stereotyped) by the elementary methods collected from different fields of Mathematics, such as elementary inequalities between geometric, arithmetic and square means, the codomain of the quadratic and trigonometric functions, etc. In the first part we show some patterns that students can imitate in solving similar problems. These patterns could also provide some ideas for Hungarian teachers on how to introduce this topic in their practice. In the second part we discuss the results of a survey carried out in two secondary schools and we formulate our conclusion concerning the improvement of students' performance in solving these kind of problems. -
The investigation of students' skills in the process of function concept creation
249-266Views:245Function is a basic concept of mathematics, in particular, mathematical analysis. After an analysis of the function concept development process, I propose a model of rule following and rule recognition skills development that combines features of the van Hiele levels and the levels of language about function [11]. Using this model I investigate students' rule following and rule recognition skills from the viewpoint of the preparation for the function concept of sixth grade students (12-13 years old) in the Ukrainian and Hungarian education system. -
Online tests in Comprehensive Exams – during and after the pandemic
77-93Views:395The Covid-19 pandemic accelerated the development of electronic (e-learning) assessment methods and forced their use worldwide. Many instructors and students had to familiarize themselves with the form of distance education. During and since Covid-19 in Hungary, at the Faculty of Engineering of the University of Debrecen, the written part of the Comprehensive Exam in Mathematics is organized in a computer lab of the university using an online test. Our goal is that the results of the tests may be as reliable as possible in terms of measuring the students’ knowledge, and thus the grades given based on the test results would be realistic. In this paper, we show the analysis of a sample written exam and compare the real exam results of students who were prepared for the comprehensive exam during Covid-19 and who have participated in face-to-face education since then. The tools provided by the Moodle system necessary for comparison are also presented.
Subject Classification: 97D40, 97D70, 97U50
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Interactive web portals in mathematics
347-361Views:305Many of the recent problems in higher education (less contact seminars, the heterogeneity and the increasing number of our students) call for new instructional methods. At University of Szeged we have developed a mathematical web portal which can offer a solution for such problems among the changing circumstances. This freely available, easy-to-use web-surface supports interactive mathematical problem-solving and student self assessment. Our computer program cooperates with a lot of free software (computer algebra systems, formula parsers, converters, word processors). WebMathematics Interactive has been available for the public since June 2002 on its web page http://wmi.math.u-szeged.hu. -
Reappraising Learning Technologies from the Viewpoint of the Learning of Mathematics
221-246Views:211Within 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. -
Fostering engineering freshmen’s shifts of attention by using Matlab LiveScript for solving mathematical tasks
1-14Views:326We designed an experimental path including a summative assessment phase, where engineering freshmen are involved in solving mathematical tasks by using Matlab LiveScripts. We analyzed the students’ answers to a questionnaire about their perceived impact of the use of Matlab on their way to solve mathematical tasks. The main result is that students show shifts of attention from computations to other aspects of problem solving, moving from an operational to a structural view of mathematics.
Subject Classification: 97U70, 97H60
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Understanding the spatiotemporal sample: a practical view for teaching geologist students
89-99Views:187One 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. -
Teaching of problem-solving strategies in mathematics in secondary schools
139-164Views:177In the Hungarian mathematics education there is no explicit teaching of problem-solving strategies. The best students can abstract the strategies from the solutions of concrete problems, but for the average students it is not enough. In our article we report about a developmental research. The topic of the research was the explicit teaching of two basic strategies (forward method, backward method). Based on our experiences we state that it is possible to increase the effectivity of students' problemsolving achievement by teaching the problem-solving strategies explicitly. -
The shift of contents in prototypical tasks used in education reforms
203-219Views:205The paper discusses the shift of contents in prototypical tasks provoked by the current educational reform in Austria. The paper starts with the educational backboard of the process of changes in particular with the out tting of the students' abilities in different taxonomies and its implementation in the competence models of Mathematics. A methodological didactical point of view on the process is given additionally. Examples out of a specific collection of math problems which arise from the educational reform are integrated and analysed in the context of educational principles and methods. The discussion ends with a short evaluation of the role of traditional approaches to tasks in the ongoing reform. A bundle of tasks as proof that they are still alive is presented finally.
Subject Classification: 97B50, 97D40, 97D50
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Würfel und Augensummen – ein unmögliches Paar
71-88Views:238It is well known that the values 2, 3, ..., 12 of the sum of eyes that appear when throwing two regular dice are not equally distributed. It can also be shown that no matter how the dice are falsified (or if only one of them is being manipulated) they can never reach the same probability concerning the sum of eyes ([8], 91 et seq.). This discovery can be generalized for n ≥ 2 dice. Various results of algebra and (real) calculus are used, so that a connection between two different mathematical fields can be realized. Such a connection is typical and often provides a large contribution for mathematics (because it frequently leads to a successful attempt of solving a special problem) and therefore examples of this sort should also be included in the mathematical education at schools as well as in the student teachers' university curriculum for the study of mathematics. -
Compositions of dilations and isometries in calculator-based dynamic geometry
257-266Views:157In an exploratory study pre-service elementary school teachers constructed dilations and isometries for figures drawn and transformed using dynamic geometry on calculators. Observational and self assessments of the constructed images showed that the future teachers developed high levels of confidence in their abilities to construct compositions of the geometric transformations. Scores on follow-up assessment items indicated that the prospective teachers' levels of expertise corresponded to their levels of confidence. Conclusions indicated that dynamic geometry on the calculator was an appropriate technology, but one that required careful planning, to develop these future teachers' expertise with the compositions. -
Should we draw, or should we work with numbers? Investigating proportional reasoning among 5th to 7th graders
1-28Views:79Proportional reasoning is an essential component of our everyday life and our mathematics studies. The rate of development in this area varies between age groups. In order to find out the level of students in Grades 5–7, we developed an online test. We consider it important to emphasize and support the use of visual representations in this subject, and therefore the tasks of the test on the eDia (Csapó & Molnár, 2019) interface have three types of input and output.We distinguish between ratios represented visually in the form of discrete quantities, ratios represented visually in the form of continuous quantities and ratios represented by text or numbers. Our study aimed to explore the differences between task types. Results indicate a representation-dependent developmental shift: in Grades 5–6, students perform best on tasks involving visual discrete quantities, whereas in Grade 7, performance increases markedly on text-text tasks. This suggests that visual representations function as an early scaffold, while later instruction strengthens symbolic processing.
Subject Classification: Primary: 97C30; Secondary: 97D40, 97D60
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Report of Meeting Researches in Didactics of Mathematics and Computer Sciences: 31 March – 2 April, 2023 Oradea, Romania
83-107Views:407The 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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Report on "Problem Solving in Mathematics Education": ProMath 6 Conference, 8–11 September, 2005, Debrecen, Hungary
313-319Views:261The 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. -
Heuristic arguments and rigorous proofs in secondary school education
167-184Views:250In 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. -
Dressed up problems - the danger of picking the inappropriate dress
77-94Views:282Modelling and dressed-up problems play an inevitably unavoidable role in mathematics education. In this study we would like to point out how dangerous is it to dress up mathematical problems. We go back to the principle of De Lange: The problem designer is not only dressing up the problem, but he is the solution designer, as well. We show three examples selected from Hungarian high school textbooks where the intended solution does not solve the problem, because the dressing changes the context and changes the problem itself. -
The mathematics textbook as an aid to differentiation: a first Hungarian example
35-53Views:126Differentiation is a way of teaching where each student is taught according to his/her personal needs. This technique is not widely used in Hungary yet, although this would be necessary due to the introduction of the two-level final examination and to a growing concern for equal opportunities and integrated teaching. One of the most significant aids to differentiation is an appropriate textbook, and that is why a group of professionals wrote a set of textbooks that supports this technique. The paper examines the requirements for a differentiated textbook, and the extent to which the textbook in question meets them. -
Teaching probability theory by using a web based assessment system together with computer algebra
81-95Views:195In the course of Maths Basics 2, the Faculty of Economic Science students of Kaposvár University learn the classical chapters of Probability Theory, namely random variables and the well-known probability distributions. Our teaching experiences show that students' achievement is weaker in case of problems concerning continuous random variables. From school year 2012/13 we have had an opportunity to take Maple TA, the web-based test- and assessment system, into the course of education. It is sufficient for the users of Maple TA to have a browser. Maple computer algebra system, which runs on the server, assesses students' answers in an intelligent way, and compares them with the answers that are considered correct by the teacher. In our presentation we introduce some elements of Maple TA system, the didactic considerations the test sheets were made by, as well as our research results concerning the use of Maple TA. -
Probabilistic thinking, characteristic features
13-36Views:163This paper is the first step in a series of a general research project on possible development in probability approach. Our goal is to check with quantitative methods how correct our presumptions formulated during our teaching experience were. In order to get an answer to this question, we conducted a survey among third-year students at our college about their general and scientific concepts as well as about the way they typically think. -
Erratum to the paper "The theory of functional equations in high school education" Teaching Mathematics and Computer Science 10/2 (2012), 345-360
145Views:194A correction is necessary in subsection 2.5. although this does not affect the truth of the main formula.
