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CAS-aided visualization in LATEX documents for mathematical education
1-18Views:9We 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. -
The requirements in statistics education – comparison of PISA mathematical tasks and tasks from the mathematical textbooks in the field of statistics
263-275Views:12This work presents the results of the analysis of both PISA items and Croatian mathematical textbooks in the field of statistics.
The analysis shows that PISA's released statistics problems have in many ways different mathematical requirements from the requirements of textbook problems in the statistics chapters, with respect to the mathematical activities, complexity and in the forms of questions. The textbook analysis shows that mathematical examples and problems often require operation and interpretation skills on a reproductive or connections level. Statistics textbook problems are given in the closed-answer form. The results also show that while PISA puts strong emphasis on the statistics field, in the current Croatian curriculum this field is barely present. These discrepancies in requirements and portion of statistics activities surely affect the results of Croatian pupils on PISA assessment in the field of mathematical literacy. -
Tamás Varga’s reform movement and the Hungarian Guided Discovery approach
11-28Views:141This 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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The effects of chess education on mathematical problem solving performance
153-168Views:11We investigate the connection between the "queen of sciences" (mathematics) and the "royal game" (chess) with respect to the development of mathematical problem solving ability in primary school education (classes 1-8, age 7-15) where facultative chess education is present. The records of the 2014 year's entrance exam in mathematics – obligatory for the enrollment to secondary grammar schools in Hungary – are compared for the whole national database and for the results of a group containing chess-player students. The problems in the tests are classified with respect to the competencies needed to solve them. For the evaluation of the results we used standard mathematical statistical methods. -
Herschel's heritage and today's technology integration: a postulated parallel
419-430Views:9During the early 20th century, advocacy of a range of mathematical technologies played a central part in movements for the reform of mathematical education which emphasised ‘practical mathematics' and the ‘mathematical laboratory'. However, as these movements faltered, few of the associated technologies were able to gain and maintain a place in school mathematics. One conspicuous exception was a technology, originally championed by the mathematician Herschel, which successfully permeated the school mathematics curriculum because of its:
• Disciplinary congruence with influential contemporary trends in mathematics.
• External currency in wider mathematical practice beyond the school.
• Adoptive facility of incorporation in classroom practice and curricular activity.
• Educational advantage of perceived benefits outweighing costs and concerns.
An analogous perspective is applied to the situation of new technologies in school mathematics in the early 21st century. At a general level, the cases of calculators and computers are contrasted. At a more specific level, the educational prospects of CAS and DGS are assessed. -
The appearance of the characteristic features of the mathematical thinking in the thinking of a chess player
201-211Views:10It 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. -
Process or object? Ways of solving mathematical problems using CAS
117-132Views:6Graphing 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. -
CAS as a didactical challenge
379-393Views:10The paper starts with the discussion of a concept of general mathematics education (mathematics education for everyone). This concept views the focus of teaching mathematics in the reduction of the demands in the field of operative knowledge and skills as well as in an increase of the demands in the fields of basic knowledge and reflection. The consequences of this concept are didactically challenging for the use of Computer Algebra Systems (CAS) in the teaching of mathematics. By reducing the operative work we reduce exactly that field in which the original potential of CAS lies. It is shown that in such maths classes the main focus of CAS is on their use as a pedagogical tool, namely as support for the development of basic knowledge and reflection as well as a model of communication with mathematical experts. -
The application of modelling tasks in the classroom – why and how? with reflections on an EU teacher training course
231-244Views:9The aim of the article is to present the concept of mathematical modelling in the classroom. LEMA (Learning and Education in and through Modelling and Applications) was an EU Comenius funded project in which mathematics educators from six countries worked to produce materials to support teachers' professional development. A group of voluntary Hungarian mathematics teachers were taught modelling for a year and we were and still are given feedback continously. The article leads us from the general concept of mathematical modelling to its practice in the classroom. It presents difficulties that teachers have to face when doing modelling lessons and their students' reactions are also mentioned. We present sample tasks from the material of the teacher training course as well as tasks that were created by the participants. -
Using the computer to visualise graph-oriented problems
15-32Views:9The 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. -
Artworks as illustrations in Hungarian high school Mathematics textbooks
103-117Views:53Three 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
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Better understanding mathematics by algorithmic thinking and computer programming
295-305Views:104Tamás 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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Report of Meeting Researches in Didactics of Mathematics and Computer Sciences: January 22-24, 2016 Bratislava, Slovakia
115-137Views:10The meeting Researches in Didactics of Mathematics and Computer Sciences was held in Bratislava, Slovakia from the 22th to the 24th of January, 2016 at Comenius University in Bratislava. It was organized by the Doctoral School of Mathematical and Computational Sciences of University of Debrecen and the Faculty of Education of Comenius University.
The 60 participants – including 47 lecturers and 15 PhD students – came from 5 countries, 23 cities and represented 32 institutions of higher and secondary education. -
Teaching reliability theory with the Computer Algebra System Maxima
45-75Views:8The use of the Computer Algebra System Maxima as a teaching aid in an MSc module in Reliability Theory is described here. Extracts from student handouts are used to show how the ideas in Reliability Theory are developed and how they are intertwined with their applications implemented in Maxima. Three themes from the lectures are used to illustrate this: (1) Normal Approximations, (2) Markov Modelling, (3) Laplace Transform Techniques.
It is argued that Maxima is a good tool for the task, since: it is fairly easy to learn & use; it is well documented; it has extensive facilities; it is available for any operating system; and, finally, it can be freely downloaded from the Web. Maxima proves to be a useful tool even for Reliability research for certain tasks. This latter feature provides a seamless link from teaching to research – an important feature in postgraduate education. -
Interactive web portals in mathematics
347-361Views:6Many 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. -
Mathematics in Good Will Hunting II: problems from the students perspective
3-19Views:7This is the second part of a three paper long series exploring the role of mathematicians and of the mathematical content occurring in popular media. In particular we analyze the drama film Good Will Hunting. Here we investigate the mathematical content of the movie by considering the problems appearing in it. We examine how a mathematician or a mathematics student would solve these problems. Moreover, we review how these problems could be integrated into the higher education of Hungary. -
Guided Discovery in Hungarian Education Using Problem Threads: The Pósa Method in Secondary Mathematics Classrooms
51-67Views:99In 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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Problemorientierung im Mathematikunterricht – ein Gesichtspunkt der Qualitätssteigerung
251-291Views:2The aim of this article is to give a synopsis of problem orientation in mathematics education and to stimulate the discussion of the development and research about problem-orientated mathematics teaching. At the beginning we present historical viewpoints of problem orientation and their connection with recent theories of cognition (constructivism). Secondly we give characterizations of concepts that stand in the context of problem-orientation and discuss different forms of working with open problems in mathematics teaching. Arguments for more problem orientation in mathematics education will be discussed afterwards. Since experience shows that the implementation of open problems in classroom produces barriers, we then discuss mathematical beliefs and their role in mathematical learning and teaching. A list of literature at the end is not only for references but also can be used to further research.
Zusammenfassung. Ziel des Beitrags ist es, eine Synopsis in Bezug auf Problemorientierung im Mathematikunterricht zu geben und die Diskussion bezüglich Entwicklung und Forschung eines problemorientierten Mathematikunterrichts zu stimulieren. Als Erstes werden historische Gesichtspunkte von Problemorientierung und deren Verkn üpfung mit neueren Erkenntnistheorien (Konstruktivismus) vorgestellt. Zweitens werden Erläuterungen zu Begriffen, die im Kontext von Problemorientierung stehen, gegeben und verschiedene Ausprägungen der Behandlung offener Probleme im Mathematikunterricht diskutiert. Argumente für eine stärkere Berücksichtigung von Problemorientierung im Mathematikunterricht werden danach erörtert. Auf Barrieren bei der Implementierung von offenen Problemen im Unterricht, die durch mathematische Beliefs (Vorstellungen, Überzeugungen) geprägt sind, wird zum Schluss eingegangen. Die abschließend aufgeführte Literaturliste dient nicht nur dem Beleg der Zitate, sondern kann auch zu weiterer Vertiefung genutzt werden. -
Analyse von Lösungswegen und Erweiterungsmöglichkeiten eines Problems für die Klassen 7–11
231-249Views:9Making several solutions for a problem i.e. the generalization, or the extension of a problem is common in the Hungarian mathematics education.
But the analysis of a problem is unusual where the connection between the mathematical content of the task and of its different formulations is examined, solutions from different fields of mathematics are presented regarding the knowledge of different age groups, the problem is generalized in different directions, and several tools (traditional and electronic) for solutions and generalizations are presented.
This kind of problem analysis makes it viable that during the solution/elaboration several kinds of mathematical knowledge and activities are recalled and connected, facilitating their use inside and outside of mathematics.
However, an analysis like this is not unfamiliar to the traditions of the Hungarian problem solving education – because it also aims at elaborating a problem – but from several points of view.
In this study, a geometric task is analysed in such a way. -
Bemerkungen zur Prototypentheorie – Begriffs - und Konzeptbildung
365-389Views:8Psychological theories of prototypes are put forward by mathematical modelling. Some didactical consequences are discussed on the background of this analysis. By the help of an example (classification of convex quadrangles) hints are given for didactical interpretations of actual models of cognitive psychology dealing with problems of constructing prototypes. -
What does ICT help and does not help?
33-49Views:94Year 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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Analysis of the affective factors of learning mathematics among teacher trainees
225-254Views:10The 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. -
Fostering engineering freshmen’s shifts of attention by using Matlab LiveScript for solving mathematical tasks
1-14Views:77We 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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Würfel und Augensummen – ein unmögliches Paar
71-88Views:9It 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. -
Teaching meaningful mathematics with the Computer Algebra System MAXIMA using the example of inequalities
53-65Views:10The paper was originally motivated by the request to accentuate the meaningful contribution of inequalities in Mathematics Education. Additionally nationwide approved competences such as estimating come to the fore when organizing mathematical contents along some central Big Ideas. Not least the integration of computers enriches the reasonable discussion of inequalities by modern well accepted methodological principles. The freeware MAXIMA is used as Computer Algebra System (CAS) representatively.