EMATIK : Innovation Trends In Education Of Future Teachers and in Further Education of Mathematics Teachers

L ILLA KOREŇOVÁ , ŠTEFAN SOLČAN, MONIKA DILLINGEROVÁ , HELENA BEREKOVÁ ,

VIERA UHERČÍKOVÁ , IVAN TRENČANSKÝ , PAVEL CHALMOVIANSKÝ , MARIANNA

POLEDNOVÁ , EDUARD BOĎA, M ILOŠ BOŽEK , SOŇA KUDLI ČKOVÁ , ISTVÁN LÉNÁRT *

Comenius University, Bratislava, Slovakia, (* also ELTE University Budapest) ABSTRACT. Using ICT (Information and Communication Technology ) in the education of future teachers and in the further education of highschool mathematics teachers seems to be a very important issue in European Research Area. Instrumenting one of EU priorities within the ESF initiative, an innovation project named EMATIK has run at Comenius University. The target group are highschool teachers and future teachers of mathematics virtually from all schools in Slovakia. We summarize basic facts and lessons learned from the project, which should have a strong impact on innovating teaching of mathematics at highschools and the future of education.

Keywords: mathematics education, ICT, e-learning

1. INTRODUCTION

During the period March 2006 – September 2008, there was realized a common project of European Social Fund (ESF) and the Faculty of Mathematics, Physics and Informatics (FMPI) at Comenius University, Bratislava: Innovation trends in education of the future teachers and in the additional education of the matematics teachers (via e-learning) [1]. This project was organized and guaranteed by teachers of Department of Algebra, Geometry and Didactics of Mathematics at FMPI. The aim of the project was to modernise the education of next generation teachers and provide them with an aid to system of lifelong education in teaching mathematics. We bring some facts on the project and some impressions on the results reached by the organizers. We assume that the investment into the education and motivation of highschool teachers will in return increase the quality of education of our future students. The rest of the paper is structured as follows. Part 2 recalls history and outlines background. Parts 3 and 4 characterise the project impact, parts 5 and 6 bring some details on e-learning courses and conferences. Finally, we evaluate knowledge and comparisons selected.

2. HISTORY AND BACKGROUND

We start with a brief history on education in mathematics and teaching of mathematics in Slovakia. 1.10.1940 – foundation of the Faculty of Sciences of Slovak University (Prírodovedecká fakulta Slovenskej Univerzity (PF SU) )

26. 8. 1944 – foundation of Matematical Institute of Slovak University (since 1954 SU renamed to Comenius University) 1. 10. 1960 – creation of Department of Geometry of the Faculty of Sciences of the Comenius University During 70th and 80th – didactics of mathematics as a part of Department of Geometry, or later founded Department of Algebra. 1987 – creation of Department of Didactics of Mathematics 2005 – growth of Department of Algebra, Geometry and Didactics of Mathematics The background of the project is rooted in the following considerations and observations. Everybody has found his sympathy to mathematics via teachers or parents (or both), rarely in another way. Thus the fact that someone becomes a good mathematician (not teacher) is a consequence of a positive activity of teachers (on basic school, high school or university). A teacher, insufficiently prepared or not enough motivated, can discourage a good student thus we want to bring in the education of the future teachers high level of skills and the inner motivation. The social and financial acknowledgement we cannot garantee but we are trying to encourage teachers tired by years of teaching, offer to teachers new knowledge, resources, and ways to deepen and widen their knowledge base. Last, but not least, we intend to give them a feeling of understanding their own ambitions.

3. EMATIK PROJECT

Project with the official name „Innovation trends in education of the future teachers and in the additional education of the matematics teachers (via e-learning)“ started in the spring 2006 by creating of Ematik-team (technical, administrative

and lecturers parts), headed by coordinator Lilla Koreňová. Project manager was Stefan Solčan, assistant Monika Dillingerová, and garant Ivan Trenčanský. The activities of the project were grouped into the following areas: 1. Creation of distance education center 2. Preparing of lecturers for e-learning 3. Development of the courses 4. Education in e-learning courses via internet 5. Conferences EMATIK 2007 and 2008 6. Monitoring and publicity

Figure 2.1 The homepage of the Ematik project

4. E-LEARNING COURSES

The main activity was education in 12 e-learning courses. It started by development of courses by each of the lecturers, implementing them into the chosen e-learning system Moodle and realizing the courses after introductory seminar for the participants of our courses. Each of the 10 weeks courses had its kernel in individual home work of the participants in 10 lessons. The courses were realized in 4 terms: September–December 2006, February–May 2007, September–December 2007 and February–May 2008. Each successful participant was awarded by a certificate. The vivid interest and a strong response from our target groups can be seen in these facts: Within the first term of courses we taught 12 courses in 18 groups for 326 participants, including students and 234 teachers from basic, secondary schools, and universities or PhD students. The best 36 concluding works of participants were selected for final seminar. The second term consisted from 12 courses for 316 participants, including 233 teachers, and 23 best concluding works were presented. Within the third and fourth term, we opened 6 of 12 prepared courses for 199 participants, including 140 teachers, respectively 222 participants, including 129 teachers. In total, 1063 participants with 736 teachers shared the novel courses and spreaded the know-how to hundreds of schools. The best achievements were presented at two conferences.

5. THE COURSES, THEIR LECTURERS AND GOALS

The courses created for the Ematik project continue – they are used for students and doctorands of FMPI. 1.Didactic of mathematics Lecturer: doc. RNDr. Helena Bereková, CSc

The course is focused on actual problems in mathematics teaching on basic and secondary schools concerning the content of mathematics education, effectivity of used educating methods, evaluation of results of mathematics education as well as diagnostic analysis of student’s works. We assigne the use of history and ICT in selected parts of mathematics teaching.

Figure 5.1 Homework problem in Course 1

2. Chapters of mathematics teaching Lecturer: RNDr. Monika Dillingerová, PhD.

We deal with some methods of teaching by problem solving , with creation of problems for competitions and their analysation. We also do not forget pupils with special learning problems. There are shown activities made with pupils the last day before Christmas or on a school trip. We join mathematics with manual activity whereby embeding into geometry.

Figure 5.2 Model made by student for course 2

3. Unconventional methods in mathematics teaching Lecturer: doc. RNDr. Viera Uherčíková, CSc.

During the course the participants get knowledge about the significance of motivation in mathematics education, about the possibilities and concepts of creating motivational background, about untraditional methods, forms and motivation tools in mathematics teaching (e.g. didactic games and competitions in mathematics) and about the possibilities to reach higher efficiency of mathematics teaching.

Figure 5. 3 Divergent problems of Ing. Max Igor Bazovský used in course 3

4. Applications of descriptive geometry Lecturer: doc. RNDr. Ivan Trenčanský, CSc.

We offer the possibility to acquaint with some applications of descriptive geometry (methods of projection) in several areas of technical praxis, especially in architecture, civil engineering, topography, cartography, etc.

C,2; |C2C

,2| = |(F)(F

,)|

Figure 5.4 Reconstruction of object (made with a help of software Microstation)

4.a Didactical situations theory (DST) Lecturer: doc. RNDr. Ivan Trenčanský, CSc.

The aim of this course is to acquaint the participants with the fundamentals and the possibilities of Didactical situations theory, i.e. understanding of notions as didactic situation, didactic triangel, didactic contract, its positive but also negative impacts in education process. We also assigne the importance of using of „a priori analysis“ and „a posteriori analysis“.

Figure 5.4a Illustration to an application of DST in geometry (concluding work of participant Mgr. K.Magyarová)

5. Didactical software in teaching mathematics Lecturer: RNDr. Lilla Kore ňová, PhD.

The course offers the view about possibilities of using the internet and didactical software in teaching mathematics on primary and secondary schools. Participants of the course will be familiarized with the software of dynamic geometry(Cabri geometry, Cabri 3D), next with the plotters (graph drawers of functions: Equation grapher, Graphmatica, Autograph, Deadline, Graph, Graphic calculus), software DERIVE, MS EXCEL and graphic calculators. On template examples they will get the vision about possibilities about using this didactical software in selected spheres of mathematics on primary and elementary schools. At the end they will be familiarized with the aspects of the influence of ICT on the aims of teaching mathematics.

Figure 5. 5 Demonstration of Cabri Geometrie using 6. Fundamentals of curves and surfaces modeling Lecturer: RNDr. Pavel Chalmovianský, PhD.

This course has been set up in order to teach the students (although teachers by job) basics of Bézier curve properties and techniques for modelling. The course was oriented in a theoretical way and the practise was supposed on the side of the student either by using any computer algebra system or any other specialized modelling software such as Blender to get into touch with the objects of the course. We briefly went over the Bézier segments, B-splines, continuity of low order, rational forms of polynomial curves and splines and the end was tensor product Bézier surfaces. Since the techniques here require a lot of computing, it was advised to use a computer for elaboration of some exercises. Using curves obtained via methods learned in this course, students could produce figures such as curves with prescribed shapes to put text along it (see figure 5.6).

Figure 5. 6 Text along a curve.

7. Geometric transformations of the Euclidean space Lecturer: RNDr. Marianna Polednová, PhD.

Here we will study the basic properties of affine geometric mappings using analytic methods because

this is fundamental for imaging reality via computer. We will deal in more details with isometric and equiform transformations of Eclidean space. The theory is supported by solving problems in exercises.

Figure 5.7 The equations of mappings 8. Projective geometry Lecturer: doc. RNDr. Štefan Solčan, PhD.

In this course we explain the principles of geometry based on study of relations connected not with measurment but with position of geometric objects. Besides the most known Extended Euclidean plane we mention also the finite geometries and then structures connected with them. We also show how one can gain a new integrated view on the theory of conics in the Extended Euclidean plane.

Figure 5. 8 Illustration from the participants Concluding Works made with Cabri Geometrie (Mgr. A.Soláriková)

9. Introduction to algebraic geometry Lecturer: Doc. RNDr. Eduard Boďa, PhD.

We start from the known results of conics theory in Eclidean plane (as a set of all poins of this plane which satisfy a quadratic equation in two variables) which allow us imaging the conics in a plane. We

aquire methods how to extend this process onto the equations of higher degree (n = 3,4,...)

Figure 5. 9 Picture of some algebraic plane curves

10. Introduction to differential geometry Lecturer: Doc. RNDr. Miloš Božek, PhD.

The participants of this course gain a possibility to acquaint themself with the basic properties of curves and surfaces which are besides the geometry very usefull for creating of objects and scenes in computer graphics, for physics (mechanics) and through it for applications in engineering (machine building, civil engineering, architecture).

Figure 5. 10 Illustration of a plane curve

11. Introduction to computing geometry Lecturer: RNDr. Soňa Kudli čková, PhD.

We concentrate on geometric background for parallel and central projections with respect to the classification of methods of projection. We describe how points in the 2-D plane or in the 3-D space can be represented by Cartesian and homogeneous coordinates. Planar and spatial transformations are introduced to construct objects from “geometric” primitives and how to manipulate existing objects.

Figure 5. 11 Vizualisation of some surfaces of revolution described by participants (of course 11) Poláčiková, Hrivová and Štefanidesová 12. Comparative geometry Lecturer: Prof. István Lénárt, Budapest

The author and owner of patent of special educational tool „Lenart Sphere“ registered in the USA, Hungary and some other countries with its help explains what have different geometries common or distinct. At the same time he shows how use this comparative geometry in mathematics teaching on basic and secondary schools.

Figure 5. 12 I.Lénárt´s workshop on Ematik 2007 Conference

6. EMATIK CONFERENCES 2007 AND 2008

The first conference held on September 10th – 12th , 2007 [2] aimed to present the views of the teachers and future teachers on distance education methods using internet (e-learning) and their skills with ICT using in teaching mathematics on different levels of schools (grammar and secondary schools and universities) and in the additional education of teachers. There were 44 registered lectures, 36 of them were presented for 111 participants. Main speakers and plenary sessions adressed extending SANET (Slovak Academic Network), Mathematics and eLTR-Technologies: A Personal Reflection (eLTR = eLearning, eTeaching &

eResearch), and Un travail experimental sur le processus de modelisations: la philosophie implicite des futures enseignants de Mathe et physique. Several lecturers discussed didactics for e-learning, usability of mathematics, using Cabri formally and methodologically, e-learning and e-examining. Prof. István Lénárt discussed possibilities of Comparative geometry in e-learning environment, and Dr. Anna Rybak: Individualization of educational process in e-learning courses. Proceedings of the Ematik Conference 2007 were printed and they are available also in e-version on http://konferencia.ematik.sk/.

Figure 6. 1 The homepage of Ematik Conference 2007 The second Ematik Conference 2008 was held in July 2008. The aims of the conference – to present know how of e-learning a using ICT in mathematics education in different types of schools and in further education of teachers. There were 27 registered lectures, 16 of them were presented. Main speakers and plenary sessions focused on graphics software and planar transformations, e-learningu for masification of university education, employing Moodle for math labs, and measuring efficiency of educational e-content. A related project European virtual lab on mathematics was presented, as well. Again, the discussion on e-learning and e-examining continued.

7. PROJECT RESULTS AND CONCLUSION

Answering the question “What one could gain?”, we have to share different points of view of our target groups. Students got knowledge and credits, possibility meet teachers in „virtual“ class in each of our e-learning courses as a new way of learning and teaching, communication with teachers in praxis. i.e. a possibility meet teachers in „virtual“ class in each of our e-learning courses. Teachers were enriched hopefully with new information, further education, certificate, knowing e-learning as a new way of learning and teaching,

new contacts with partners – teachers possibly from the opposite parts of Slovakia, the possibility meet students – future teachers in „virtual“ class in each of our e-learning courses. Project teachers created educational units in Moodle environment and gained practice with education via e-learning, new technologies and contacts. Besides the abovementioned, the equipment and financial support of ESF, we mastered the practice with paperwork, financing and monitoring of the large and possibly important ESF project.

ACKNOWLEDGEMENTS

This work was supported by a common project of ESF and Comenius University: Innovation trends in education of the future teachers and in the additional education of the matematics teachers (via e-learning) and the VEGA grant No.1/ 0763 / 09.

REFERENCES

[1] KORENOVA, L. et al. 2007. EMATIK Portal. www.ematik.sk [online] [31.01.2009]

[2] KORENOVA, L. et al. 2007. EMATIK Conference Portal. http://konferencia.ematik.sk/ [online] [31.01.2009]

[3] Proceedings of EMATIK Conference 2007 (In Slovak: Zborník príspevkov z konferencie EMATIK 2007). Komenský University, FMPI 2008. Cheaf editor: Dillingerová M.

[4] Proceedings of EMATIK Conference 2008 (In Slovak: Zborník príspevkov z konferencie EMATIK 2008).. Komenský University, FMPI 2009. Cheaf editor: Dillingerová M.