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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. -
Comment les enseignants en formation initiale utilisent les technologies informatiques dans leurs classes
187-208Views:8The research presented here deals with the way French pre-service teachers assimilate the working of technology tools and the effects on professional practice of integrating these tools into classes. We focused on the professional writings of pre-service teachers regarding the use of technology in their teaching. The results show that, besides official instructions, the motivations put forward by pre-service teachers who integrated technology in their classes are mainly their students' interest in computers and how powerful this tool is. They also show that in such an environment teachers tend to keep in the background and to leave the students to interact chiefly with the computer. We also noticed that the specificities of managing a classroom in computer environment are not taken into account unless they generate problems.
Résumé. La recherche présentée ici porte sur l'appropriation des outils informatiques par les enseignants français en formation initiale et les effets de leur intégration dans les classes sur les pratiques professionnelles. Nous avons pris comme objet d'étude des écrits professionnels, élaborés par ces professeurs stagiaires, portant sur l'utilisation des TIC dans leur enseignement. Les résultats obtenus font apparaître qu'outre les injonctions institutionnelles, les motivations invoquées par les stagiaires pour recourir à l'informatique concernent surtout l'attrait de leurs élèves pour l'ordinateur et la puissance de cet outil. Dans le cadre des usages en classe, nos résultats montrent que l'enseignant a tendance à s'effacer devant l'ordinateur, considéré comme l'interlocuteur privilégié de l'élève. Nous avons aussi pu constater que les spécificités de la gestion de la classe en environnement informatique ne sont prises en compte que lorsqu'elles se révèlent sources de problèmes. -
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. -
Blind versus wise use of CAS
407-417Views:1During my courses for mathematics major students I often use technology linked to the arising problems. In such cases I noted that some students were used to learn just some procedures, which made them able to solve (partially) some problems and when they got the result, they accepted it passively and did not relate it to the initial problem.
In this paper I outline a strategy and investigate some simple exercises about how to develop a critical attitude towards the results obtained by technology in an introductory course to CAS.
I believe that wise use of technology offers an effective method in teaching mathematics, without reducing the students' mental contribution. -
Live & Learn: When a wrong program works
195-208Views:6In this paper an interesting and surprising case study of my programming education practice is presented. This case underlines the importance of methods, standards and rules of thumb of the programming process. These elements of the programming technology can be taught well in education and they can guarantee the quality of the implemented programs. However the case described in this paper brings an anomaly when a programming standard is violated during the programming process and, although it should imply that the implemented program code works badly, the program works perfectly. This anomaly is caused by a typical implementation problem: the boundary and rules of the machine representation of numbers. This anomaly is going to be analyzed and the appropriate conclusions of our case study will be deducted. -
Shall we use one more representation? Suggestions about establishing the notion of recursion in teaching informatics in primary schools
209-229Views:8Among the most prominent developmental tasks of primary school education one finds increasing pupils' cognitive capacity with especial regard to observing, interpreting, coding and proving skills, which form an integral part of information and communication culture.
Info-technology (problem solving with the tools and methods of informatics), a subject matter within informatics, provides outstanding opportunities to reach the aims outlined above.
This study presents methodological ideas related to the subfield Algorithmization and data modelling of Info-technology. More specifically, it presents teaching methods to be applied while establishing the notion of recursion in grades 3–8 of primary education, and at the same time it also focuses on various realization possibilities of the prominent developmental tasks mentioned above. -
Synthesis of concurrent programs
301-317Views:14Students need a well defined method to be successful in the complex process of writing a concurrent program. In this paper we show a step by step method to create such programs. The method based on UML which has been thought to students during previous courses. UML provides standard and relatively simple tools to describe concurrent systems, and from the description the program can be derived.
First we give a brief introduction to the concurrent systems. This is followed by the description of the method, and finally we demonstrate the method on a small problem. -
Forming the concept of parameter with examples of problem solving
201-215Views:12Pupils are encountering difficulties with learning algebra. In order for them to understand algebraic concepts, particularly the concept of parameter it was decided by the teacher of mathematics and Information Technology to integrate the teaching of these two subjects. The aim of this study is to investigate whether, and to what degree, software can be useful in process of forming the concept of parameter. This longitudinal study was conducted in a junior high school (13-16 year old children) using different computer programs. -
Some aspects of teaching the technology of designing and planning information systems in health care
131-144Views:8In this article, we use the well-known ideas of technology in designing of new information systems in health care. We explain the principle that "making a health care application" "is more than writing a program", "it requires a strong co-operation and continuous contact" between the system analysts and users. The concept of the information system must contain the work of the whole system, which means that the planning and designing process should focus on the services, which really support the customer's functions. It has to be compatible with the earlier information systems based on several decade's experience. In this paper we use the most important elements of system theory. First of all we explain why it is important to take into account the behaviour of those, who operate the information system, and also their habits and way of thinking when planning then information system. We emphasise that it is importance to overview the whole information system and its functionality because it is a major aspect of the system planning.
This paper can be used in university courses especially in teaching SDM, SSADM, Martin, etc. technologies for information system analysts, program designers and programmers. -
Is it possible to develop some elements of metacognition in a Mathematics classroom environment?
123-132Views:78In an earlier exploratory survey, we investigated the metacognitive activities of 9th grade students, and found that they have only limited experience in the “looking back” phase of the problem solving process. This paper presents the results of a teaching experiment focusing on ninth-grade students’ metacognitive activities in the process of solving several open-ended geometry problems. We conclude that promoting students’ metacognitive abilities makes their problem solving process more effective.
Subject Classification: 97D50, 97G40
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Application of computer algebra systems in automatic assessment of math skills
395-408Views:9Mathematics is one of those areas of education, where the student's progress is measured almost solely by testing his or her ability of problem solving. It has been two years now that the authors develop and use Web-based math courses where the assessment of student's progress is fully automatic. More than 150 types of problems in linear algebra and calculus have been implemented in the form of Java-driven tests. Those tests that involve symbolic computations are linked with Mathematica computational kernel through the Jlink mechanism. An individual test features random generation of an unlimited number of problems of a given type with difficulty level being controlled flat design time. Each test incorporates the evaluation of the student's solution. Various methods of grading can be set at design time, depending on the particular purpose that a test is used for (self-assessment or administrative exam). Each test is equipped with the correct solution presentation on demand. In those problems that involve a considerable amount of computational effort (e.g. Gauss elimination), additional special tools are offered in a test window so that the student can concentrate on the method of solution rather than on arithmetic computations. (Another obvious benefit is that the student is thus protected from the risk of frustrating computational errors). Individual tests can be combined into comprehensive exams whose parameters can be set up at design time (e.g., number of problems, difficulty level, grading system, time allowed for solution). The results of an exam can be automatically stored in a database with all authentication and security requirements satisfied. -
Teaching XML
317-335Views:9The author has been teaching XML at the Faculty of Informatics, University of Debrecen since the end of the nineties. This paper gives an overview of XML technology from an educators viewpoint that is based on the experience that the author has gained teaching XML over the years. A detailed description of the XML course is provided. Methodological issues are also discussed. -
Compositions of dilations and isometries in calculator-based dynamic geometry
257-266Views:10In 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. -
Practice based course of Information Technology Service Management for BSc students
229-246Views:5Currently IT systems are primarily observed as sets of services, for example finance services can be accessed via the Internet. The IT background of an enterprise has strategic importance in its activity, but it is also important, that IT helps the enterprise supporting course of business really from the background. To implement this kind of conception, IT Infrastructure Library (ITIL) provides practical advices.
The paper describes the topic of an optional course which introduces BSc students to the subject of IT Service Management the way that it can be used as a practice supplement beside theory courses of IT Service Management. -
Report of Meeting Researches in Didactics of Mathematics and Computer Sciences: January 24-26, 2020 Sárospatak, Hungary
243-271Views:95The meeting Researches in Didactics of Mathematics and Computer Sciences was held in Sárospatak, Hungary, on the Comenius Campus of the Eszterházy Károly University, from the 24th to the 26th of February, 2020. It was organized by the Doctoral School of Mathematical and Computational Sciences of University of Debrecen and the Eszterházy Károly University. The 76 participants – including 15 PhD students – came from 9 countries, 23 cities and represented 33 institutions of higher and secondary education. There were 4 plenary, 48 session talks and 4 poster presentations in the program.
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Teaching multiparadigm programming based on object-oriented experiences
171-182Views:18Multiparadigm programming is an emerging practice in computer technology. Co-existence of object-oriented, generic and functional techniques can better handle variability of projects. The present paper gives an overview of teaching multiparadigm programming approach through typical language concepts, tools in higher education. Students learning multiparadigm-oriented subjects would gain considerable expertise, which is highly needed by the industrial side in large-scale application development. -
Comparing the IT skills and the programming knowledge of Hungarian students specialized in informatics with Romanian students attending a science course or a mathematics-informatics course
21-40Views:11The goal of this research is an analysis of the IT skills and programming knowledge of Hungarian and Romanian students attending a Science course or a Mathematics-Informatics course. Analysed was how effectively can students from different grades answer questions dealing with different subjects. After having evaluated the test results correctness of the original presumption emerged. Significance level was 5% through the analysis. Significant divergency in knowledge of Hungarian students and Romanian students of Humanities (Profil Uman) was found in 11th and 12th grades too. Romanian students attending a science course (Profil Real) and a Mathematics-Informatics course scored higher in programming than their Hungarian counterparts specialized in Informatics in the 11th grade. After the evaluation a final conclusion can be made: Romanian students of the Real Profile have the same or more practice in programming than Hungarian students specialized in Informatics, though the latters have the same or better IT skills. Unfortunately, Hungarian teachers concentrate on word processing and spreadsheet calculation and teach programming just for the students specialized in Informatics, although algorithm thinking would be important for every student before finishing secondary school. -
Psychology - an inherent part of mathematics education
1-18Views:134On 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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Report of conference XXXVIII. National Conference on Teaching Mathematics, Physics and Computer Sciences: August 25-27, 2014 Pécs, Hungary
281-303Views:2The XXXVIII. National Conference on Teaching Mathematics, Physics and Computer Sciences (MAFIOK) was held in Pécs, Hungary between 25 and 27 August, 2014 at the Pollack Mihály Faculty of Engineering and Information Technology. It was organized by the Engineering Mathematics Department. The 65 participants – including 4 invited lecturers and 53 lecturers – came from 2 countries and represented 14 institutions of higher education. -
Programming theorems on enumerator
89-108Views:11This paper deals with the examination of the programming patterns best known by programmers: the programming theorems. It is a significant issue that in what way these patterns can be formulated in order to solve a relatively broad spectrum of problems using a small number of patterns. In this paper, the well known programming theorems are applied to the processing of enumerators. To this end, the robustness of patterns gained this way will be presented, and it will be also pointed out how the programs thus constructed can be implemented in the modern object-oriented programming environments: in language C++, Java and C#. -
Interdisciplinary Secondary-School Workshop: Physics and Statistics
179-194Views:47The paper describes a teaching unit of four hours with talented students aged 15-18. The workshop was designed as a problem-based sequence of tasks and was intended to deal with judging dice whether they are regular or loaded. We first introduced the students to the physics of free rotations of rigid bodies to develop the physics background of rolling dice. The highlight of this part was to recognise that cubes made from homogeneous material are the optimal form for six-sided objects leading to equal probabilities of the single faces. Experiments with all five regular bodies would lead to similar results; nevertheless, in our experiments we focused on regular cubes. This reinsures that the participants have their own experience with the context. Then, we studied rolling dice from the probabilistic point of view and – step-by-step – by extending tasks and simulations, we introduced the idea of the chi-squared test interactively with the students. The physics and the statistics part of the paper are largely independent and can be also be read separately. The success of the statistics part is best described by the fact that the students recognised that in some cases of loaded dice, it is easier to detect that property and in other cases one would need many data to make a decision with small error probabilities. A physical examination of the dice under inspection can lead to a quick and correct decision. Yet, such a physical check may fail for some reason. However, a statistical test will always lead to reasonable decision, but may require a large database. Furthermore, especially for smaller datasets, balancing the risk of different types of errors remains a key issue, which is a characteristic feature of statistical testing.
Subject Classification: F90, K90, M50, R30
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The role of computer in the process of solving of mathematical problems (results of research)
67-80Views:11We would like to present results of an almost two years investigations about the role computer in the process of solving of mathematical problems. In these investigations took part 35 students of the secondary school (generalists) in the age 17–19 years. Each of these students solved following problem:
Find all values of the parameter m so that the function
f(x) = |mx + 1| − |2x − m| is:
a) bounded,
b) bounded only from the bottom,
c) bounded only from above,
first without a computer and next with a special computer program. We would like to show results of these researches. -
Nice tiling, nice geometry!?!
269-280Views:9The squared papers in our booklets, or the squared (maybe black and white) pavements in the streets arise an amusing problem: How to deform the side segments of the square pattern, so that the side lines further remain equal (congruent) to each other? More precisely, we require that each congruent transformation of the new pattern, mapping any deformed side segment onto another one, leaves the whole (infinitely extended) pattern invariant (unchanged).
It turns out that there are exactly 14 types of such edge-transitive (or so-called isotoxal) quadrangle tilings, sometimes with two different forms (e.g. black and white) of quadrangles (see Figure 2). Such a collection of tiling can be very nice, perhaps also useful for decorative pavements in streets, in flats, etc.
I shall sketch the solution of the problem that leads to fine (and important) mathematical concepts (as barycentric triangulation of a polygonal tiling, adjacency operations, adjacency matrix, symmetry group of a tiling, D-symbol, etc). All these can be discussed in an enjoyable way, e.g. in a special mathematical circle of a secondary school, or in more elementary form as visually attractive figures in a primary school as well.
My colleague, István Prok [11] developed an attractive computer program on the Euclidean plane crystallographic groups with a nice interactive play (for free download), see our Figures 3-5.
A complete classification of such Euclidean plane tilings (not only with quadrangles) can be interesting for university students as well, hopefully also for the Reader (Audience). This is why I shall give some references, where you find also other ones.
Further problems indicate the efficiency of this theory now. All these demonstrate the usual procedure of mathematics and the (teaching) methodology as well: We start with a concrete problem, then extend it further, step-by-step by creating new manipulations, concepts and methods. So we get a theory at certain abstraction level. Then newer problems arise, etc.
This paper is an extended version of the presentation and the conference paper [7]. The author thanks the Organizers, especially their head Professor Margita Pavlekovic for the invitation, support and for the kind atmosphere of the conference. -
Report of Meeting Researches in Didactics of Mathematics and Computer Sciences, April 1-3, 2022 Baja, Hungary
135-155Views:147The meeting Researches in Didactics of Mathematics and Computer Sciences was held in Baja, Hungary, at Eötvös József College, from the 1st to the 3th of April, 2022. It was organized by the Doctoral School of Mathematical and Computational Sciences of the University of Debrecen and by Eötvös József College. The 62 participants - including 18 PhD students - came from 8 countries and represented 26 institutions of higher and secondary education. There were 3 plenary and 40 session talks in the program.
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Radio Frequency Identification from the viewpoint of students of computer science
241-250Views:1This paper aims at creating the right pedagogical attitudes in term of teaching a new technology, Radio Frequency Identification (RFID) by evaluating the social acceptance of this new method. Survey of future teachers, students of teacher master studies and students from informatics oriented secondary schools were surveyed comparing their attitudes in terms of RFID to other recent technologies. Consequences of this survey are incorporated into the curriculum of the new RFID course at our institution.