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| 11:00am - 1:00pm | Computer-Aided I/II: Computer-Aided Experiments and Explorations in the Math Classroom (working group) Session Chair: János Karsai | ||||||
| VSP 1.02 | |||||||
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Dynamic Demonstrations in the Math Classroom for Applied Sciences University of Szeged, Hungary
Most benefit with less effort; this is a common reasoning in applied sciences while learning Mathematics. Deep theories are needed but there is no time for deep study. Hence, dynamic applications help in understanding the main points, but they can hide technical details.
In our talk, we will present some examples, didactic concerns as well as our experiences of using dynamic applications in the math courses given for life sciences. We emphasize the importance of complex modeling approach.
The potential of mobile learning in elementary and secondary school mathematics. Comenius University in Bratislava, Slovak Republic
Nowadays, using mobile technologies for younger generations is becoming an everyday reality. Smartphones and tablets combined with mobile Internet are easily accessible to students and are a part of their everyday life. In the past few years, we witnessed mobile technology entering school environments too. According to the European Commission/ICT cluster, 2010 there exists an increasing discrepancy between the possibilities of using ICT at home and in school, therefore schools should support the development of modern technical environment, thus connecting their experience with these devices at home with school and prepare them for real life situations. Mobile learning is a type of education which uses mobile technologies, like the smartphone, tablet or PDA with access to the internet. Such education is very attractive to students and so it increases the attractiveness of the subject itself. For the teacher new teaching possibilities open up while using mobile learning or blended learning. Through these methods they are "granted access" to different interactive and multimedial study materials on the internet. Mobile technologies are suitable for constructivist learning and for different modern methods of teaching too. In this presentation we would like to show several methods and forms of teaching mathematics in elementary and secondary schools using mobile technologies. These methodics were created by the soon to be teachers of mathematics on the Comenius University and by the teachers themselves within the new EMATIK+ project. It has been shown that the appropriate software for mobile and blended learning are free softwares like: GeoGebra, HotPotatoes, Open-Sankore and LMS MOODLE. We will present the views of the teachers of mathematics too on m-learning in the pilot survey and research made in the field of e-testing.
Collaborative learning with GeoGebra package University of Novi Sad, Serbia
In this paper we describe the research on collaborative learning of calculus contents by using GeoGebra package. The whole process of collaborative learning was presented. The students were divided in small four member groups in the frame of Kagan (1994) principles of collaborative learning for examining the functions and drawing their graphs. Two groups of students were formed, the experimental (working with the GeoGebra) and the control one (working without it).
After the collaborative learning all students were tested and the results of experimental group were significantly better than the results of students in control group.
Computer assisted investigation in the teaching of mathematics University of South Bohemia, Faculty of Education, Czech Republic
Procedures based on experimentation and discovery play an important role in mathematical education. Computer Algebra and Dynamic Geometry Systems are suitable environments for the implementation of activities based on these procedures. The presentation brings several specific examples of various complexity that come from school practice. These examples on different topics, from financial mathematics to curves and surfaces, show how the use of CAS and DGS, jointly or separately, can facilitate the understanding of the relevant mathematical phenomena or property, or the solving of a given mathematical problem.
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| 2:15pm - 4:15pm | Workshop 1 | ||||||
| VSP 1.02 | |||||||
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Using and Developing Interactive, Creative, Mathematics Textbooks (cBooks) 1University of Southampton, United Kingdom; 2Utrecht University, the Netherlands
When we look at e-books, designed for mathematics education, we can distinguish two streams. On the one hand we see publishers of traditional Mathematics textbook come with digital versions of their products, mostly static pdf-documents that can be downloaded and used on different devices. Anticipating on new interactive possibilities, sometimes limited interactivity is build in. On the other hand we see innovative groups of designers that traditionally develop highly interactive tools and microworlds for mathematics education. Initially many of these tools were implemented as standalone applications. More and more these tools are integrated with written tasks, producing interactive worksheets, dynamic web pages and e-books for maths. The European ‘MC-squared’ project aims to start several so-called ‘Communities of Interest’ (CoI) in a number of European countries (Fischer, 2001) that work on digital, interactive, creative, mathematics textbooks, called cBooks. The cBooks are authored in the Digital Mathematics Environment in which authors can construct books with various interactive ‘widgets’.
In this 120 minutes workshop you will: * Be given a short overview of the MC-squared project and the architecture of the Digital Mathematics Environment; * Be shown two examples of cBooks on building blocks, number and fractions, as well as a myriad of widgets that could be integrated in the cBooks; * Learn how to make your own, simple, interactive cBook; * Be shown how these books can be used with students by sharing the book you’ve made with other workshop attendees; Fischer, G. (2001). Communities of interest: learning through the interaction of multiple knowledge systems. In the Proceedings of the 24th IRIS Conference S. Bjornestad, R. Moe, A. Morch, A. Opdahl (Eds.) (pp. 1-14). August 2001, Ulvik, Department of Information Science, Bergen, Norway.
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| 11:00am - 1:00pm | Computer-Aided II/II: Computer-Aided Experiments and Explorations in the Math Classroom (working group) Session Chair: János Karsai | |
| VSP 1.02 | ||
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Programming in High School as a Learning Resource of Zeros of 2º degree Polynomial Function 1Pontifícia Universidade Católica de São Paulo, Brazil; 2Faculdade de Tecnologia de São José dos Campos - ETEP Cetec Educacional S.A
This paper presents a research work where has as objective to check if the proposal of an algorithm converted into a computer program can help high school students in the learning of the zeros of the 2nd degree polynomial function. The research was conducted in two stages. The first stage was with a 1st year high school student in order to verify if the activities were appropriate and, in the second part, we have selected four participants for the second stage. After the analysis of the first stage development, the activities were improved to the second one, composed of three activities, among which the software Visualg 2.0. The APOS theory by Ed Dubinsky, theoretical support of the research, presents the action levels, process, object and scheme, that allow the verification of the individual`s capacity to develop actions over an object and think about its properties. The research participants had improvements in their learning, because besides developing a computer program to determine the zeros of 2nd degree polynomial function, they have started to elaborate other functions previewing their possible solutions, presenting all the levels of the APOS theory. As research methodology we have adopted the Design Experiments. We have justified its use, for adjustments could be done during the work development. Analyzing the activities which were done we have concluded that the students have achieved a satisfactory learning level over the object of study.
Key words: 2nd degree polynomial function, APOS theory, Algorithm.
Computer-Aided Exploring the Mathematics behind Technical Problems – Examples of Classroom Practices Beuth Hochschule Berlin, Germany
The mathematics in technical problems can be discovered by computer-aided experiments. Examples are presented from four different courses in the areas of statics, elasticity, finite elements and partial differential equations. It will be reported on the implementation within the curriculum at Beuth University of Applied Sciences Berlin, the classroom experiments and the teacher’s role.
The Role of Technology in Supporting Students' Conceptual Understanding in Linear Algebra Humboldt-Universität zu Berlin, Germany
Current trends in research on the impact of technologies in mathematics education emphasize their increased role in supporting students' conceptual understanding in comparison with numerous previous studies about technology contribution in procedural understanding. This talk exemplifies the role of Dynamic Geometry Systems utilizing students' conceptual understanding of dot product of vectors in the transition between upper high school and university education. Students' conceptual understanding is identified as constituting a structured network of: concept definitions and concept images (Tall & Vinner, 1981) of dot product of vectors developed by students; three modes of description and thinking (Hillel, 2000; Sierpinska, 2000) of dot product of vectors: arithmetic, geometric and axiomatic-structural; and concept's applications in problem solving situations. Authentic video recordings and students' written works serve as two collected data sets for qualitative analysis of students' interactions in the designed Dynamic Geometry Environment, within the framework of instrumental genesis (Drijvers et al., 2010). The study is part of a larger design-based research (The Design-Based Research Collective, 2003) undergoing seven phases in a cyclic manner, ending with evaluation and dissemination of created teaching and learning materials as visual dynamic applets and worksheets.
Levels of reasoning with coherences between dynamically linked representations of functions Heidelberg University of Education, Germany
What students describe when they explore computerbased multiple representations of functions does not necessarily reflect how, or whether at all, they understand. In fact, the case of one student explaining his observations while exploring properties of a multiple representation environment suggests that such observations - even if correctly stated - could be based on a superficial perception of the properties only. A qualitative content analysis of further interviews results in a three level model of reasoning with multiple representation learning environment. The theoretical base of this analysis is formed by cognitive theories that describe the learning process in mathematics as a process of abstracting from superficial aspects of representations to structural coherencies between them.
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| 2:15pm - 4:15pm | AKMUI III: Vorträge & Arbeitsgruppen | |
| VSP 1.02 | ||
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Terme besser verstehen mit neuen Medien
Der Umgang und das Verstehen und Interpretieren von Termen sind ein wichtiger Bestandteil des Mathematikunterrichts. Der Umgang mit Termen findet im Wesentlichen auf einer formalen Ebene statt. Durch die Benutzung neuer Medien eröffnen sich für die Schülerinnen und Schüler neue Zugänge zu Termen. Dies wird im Vortrag sowohl mit Beispielen aus der Sek. I als auch aus der Sek. II gezeigt.
Vom Funktionenmikroskop zur digitalen Funktionenlupe
Das Funktionenmikroskop von A. Kirsch war ein Klassiker für die Erarbeitung eines Grundverständnisses von Steigung und Differenzierbarkeit im Sinne lokaler Glättung. Damals ein aufwändiger Foliensatz in Lehrerhand, konnte die Grundidee des ‚Hineinzoomens‘ später mit gängigen Funktionenplottern digital umgesetzt werden.
In dem Vortrag wird nun eine interaktive ‚Funktionenlupe‘ mit GeoGebra vorgestellt, die mit zwei Graphikfenstern und Ortslinien einen Zugang lokal zur Steigung des Funktionsgraphen und global zur Ableitung der Funktion bietet. Sie ermöglicht einen entdeckenden, anschaulichen und (zunächst) kalkülfreien Einstieg in die Analysis und einen Aufbau von Grundvorstellungen von Steigung und Ableitungsfunktion bis hin zur Krümmung. | |
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| 11:00am - 1:00pm | Tea. & Lea. I/III: Teaching and Learning (mixed general topics) Session Chair: Ildikó Perjési-Hámori | ||
| VSP 1.02 | |||
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Nonlinear Mapping with Educational Software Holon Institute of Technology, Israel
The support by educational software of nonlinear transformations is very weak or is absent at all. CAS software been developed not for educational application and has a limited potential in math studies.
It is relatively easy to implement affine transformations of the whole space programmatically due to internal nature of computer graphics mechanism. The problem is to support nonlinear spatial transformations in a manner that is user-friendly and seamless. This lecture presents new software possibilities in studies of a wide spectrum of mathematical subjects: • Algebra, • complex analysis, • vector fields, • differential equations, • dynamical systems etc. As result, software becomes a powerful tool, which helps to discover the unity of mathematics, to visualize and dynamically explore new mathematical environments and phenomena. Explorations of Mathematical Models in Biology with MATLAB Delaware State University, United States of America
In this presentation we discuss samples of instructional materials that are designed to help students explore and discover mathematical concepts and use those concepts in building and analyzing mathematical models of life science disciplines such as biology, ecology, and environmental sciences. The main mathematical tools used are difference equations and matrices. The use of the mathematic software MATLAB is an integral part of exploring and analyzing the models. We will discuss explorations that are designed to intuitively introduce the concept of eigenvalues and eigenvalues. Then we investigate the use of eigenvalues to determine the long-term behavior of a system of linear equations. Modeling with Markov chains and an age-structured population model will be discussed.
Involving the gamification technology to provide feedback when teaching mathematics International Solomon University, Eastern-Ukrainian Branch, Ukraine
Mathematicians have been using gaming technology when teaching by means of computer for a long time. So, back in the 60-70s of the last century the mathematicians applied game situations in intelligent tutoring systems. Those training systems (programs) have ensured assistance; have provided the opportunity for the student to choose the pace of learning. The presence of gaming moments was assumed, and research assignments were offered. The possibility of providing the feedback mechanism was particularly important feature of the tutoring systems.
In earlier tutoring systems the immediate feedback was impossible, and delayed feedback coincided with the result of the completion of the learning cycle - winning the game or solving the problem. Attracting gamification, one of the most popular technologies of the 21st century, became a solution of the problem of timely feedback in the tutoring systems. The idea of gamification is to use game mechanics and elements of game design in non-game contexts in order to motivate a desired behavior. The ultimate goal of gamification is to provide the level of motivation. The feedback in the form of badges and achievements can describe students' progress, which is then used to create levels and ranks. All this inherently leads to the creation of competition among the students. This can only be achieved under ideal condition of the gamified system. In addition, we represent our own products in the context of gamification.
Model of ICT competence Assessment on oral math exam Zavod RS za šolstvo/The National Education Institute of Slovenia, Slovenia Vocational matura in Slovenia is a form of a school-leaving exam that gives students technical education and/or enables them to continue studies in vocational colleges. Mathematics is one of four subjects on vocational matura and the math exam includes written and oral part. Assessment of the oral part of the exam is presented in this talk. A candidate is presented with a task/a situation from everyday life or his professional area and derived questions. A candidate should display the competence to 'see ' mathematics in given situation and show knowledge of mathematics by using ICT. ICT in this case means a computer with adequate software (programmes for dynamic geometry, programmes for data handling, professional programmes, …) or a graphing calculator. Students get familiar with technical tools during class and learn how to use them. The criteria for assessment such exam will be presented along with some examples of exam situations.
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| 2:15pm - 3:15pm | Workshop 7 | ||
| VSP 1.02 | |||
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Famous curves studied with GeoGebra University of Miskolc, Cyprus We will study the Chapter Famous curves of the MacTutor History of Mathematics archive, see: http://www-history.mcs.st-and.ac.uk/Curves/Curves.html The GeoGebra software is suitable to represent both the set of functions, and the so called associated curves, like evolutes, or involutes, and to experience their relation. The curves are given either in explicit, implicit, parametric or polar coordinate form, and we will explore the power of the software to visualize them. The osculating circle, tangent, normals, convex boundary of family or curves or Taylor-series will be mentioned as well.
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| 3:15pm - 3:45pm | Tea. & Lea. II/III: Teaching and Learning (mixed general topics) Session Chair: Ildikó Perjési-Hámori | ||
| VSP 1.02 | |||
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Using of GeoGebra in Integer Linear Programming Technical University of Kosice, Faculty of Electrical Engeenering and Informatics, Slovak Republic
Linear programming (LP), is a relatively young mathematical discipline, dating from the invention of the simplex method by George. B. Dantzig in 1947. Linear programming is now regarded as one of the fundamental management methods and methods of optimization of real processes. Linear programming is taught in various disciplines particularly in mathematics, engineering and business. It is often presented in courses such as management science or operations research. Integer linear programming (ILP) is special part of linear programming with special approaches to its solution. In this work we offer the way to take advantage of GeoGebra for visualization of ILP problems that can help students to understand relationship between LP and ILP.
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| 11:00am - 1:00pm | Reasoning: Reasoning and Proving with Tool Support (working group) Session Chair: Walther Neuper | |||
| VSP 1.02 | ||||
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Playing Mathematics like a Chess Game? An Educational View on Computer Theorem Proving Graz University of Technology, Austria
We discuss a new approach to didactics of mathematics triggered by technological innovation: Computer Theorem Proving (TP) attains increasing attention by application to large proofs, for instance to the Kepler conjecture. On the other hand, respective technology is still open source and used in several prototypes. Possible kinds of interaction in such prototypes is compared with possible interaction in chess software (where the latter usually is not TP-based).
Given TP-based educational software and the context of a problem in applied mathematics, then each input of the player/learner is checked reliably by the system (a move of a certain figure to a certain field on the chessboard / a formula or method promoting a calculation within the given context); and if the player gets stuck, the system can propose a next step (a move towards winning the game / a formula or method solving the problem at hand). The result of the interaction between learner and system on the screen is expected to be close to what is written on paper during an examination on applied mathematics. The didactical analysis, which will be given for the above software-based approach, does not emphasize the fun of playing; rather, the strengths of reliability and of variability supported by TP technology will be emphasized: If an input is wrong, the system will state the incorrectness with the reliability of formal logic (while variants are handled with maximal generosity); explanations, of what is wrong in detail, can be given from TP-technology's feature of transparency. The variability of interaction follows from TP-technology’s power: if the learner is not satisfied with the progress of a calculation, she or he can go back a few steps and try another way. Or one can explore variants by going to different intermediate states and watch the system trying a solution.
Computer assisted proving from the perspective of the secondary school teacher University of South Bohemia, Faculty of Education, Czech Republic
The paper shows what a qualitative change in the effective use of proof is brought by contemporary mathematical software in the teaching of mathematics. Particular corresponding examples of school practice are presented. Such use of mathematical software, however, makes new demands on the teachers. They must for example choose suitable topics, adapt the lesson organization, change teaching methods and methods of evaluation. The paper brings the first results of the research that was done by the first author Irena Štrausová which focused on the role of the teacher when teaching mathematics using dynamic visual proofs at selected secondary schools.
Database supported automated observation of dynamic constructions University of Ljubljana, Slovenia
Proving facts in school geometry involves several processes, among others: sketching, observing, stating hypothesis, checking them, and providing deductive argumentation. Nowadays technology provides considerable support to some of these processes, in particular dynamic geometry software is a valuable aid for sketching geometric configurations and empirical checking hypotheses. Currently, considerable effort is put into developing systems for automated proving.
In the presentation we shall explore the role of automated observation, i.e. using technology to detect properties of dynamic constructions. Observation is an essential part of analysis of a construction and enables the generation of hypotheses that possibly lead to synthetic or simple algebraic proofs. Automated observation is not only a powerful ‘geometric eye’ that spots hardly perceptible properties, it also gives rise to new ‘obstacles' in geometric thinking and calls for specific demands on dynamic geometry software. In this sense we shall present some solutions that are implemented in OK Geometry software. One of them, the implicit constructions, allows that geometric objects (in a dynamic construction) are specified by required properties and not (entirely) by construction steps. Automated observation of implicit construction may bring to light properties that lead to the solution of a problem related to the studied configuration. Perhaps the most promising potential related to automated observation is the use of a database of (dynamic) geometric objects and operations. We shall present the implementation of a database related to the geometry of triangle, consisting of several thousands of characteristic points of a triangle (e.g. incentre, orthocentre), known as Kimberling centres, and a large number of lines, circles, conics, and geometric operations; many of these objects possess interesting geometric properties. In this way automated observation does not take into account only the objects of a studied construction but as well tries to relate them to the objects in the database.
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