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Session Overview |
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| 8:00am - 9:00am | Registration: Registration | ||||||
| Jacob-Volhard-Hörsaal | |||||||
| 9:00am - 9:30am | Opening: Opening ceremony | ||||||
| Jacob-Volhard-Hörsaal | |||||||
| 9:30am - 10:30am | Keynote I: Keynote (Jürgen Richter-Gebert) – Lehrertag | ||||||
| Jacob-Volhard-Hörsaal | |||||||
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Mathematics on electronic media in a changing world University of Technology, Germany
Technical innovations are a ubiquitous phenomenon in our time. While new possibilities emerge, at the same time old possibilities disappear. Most probably internet based communication of mathematics in the future will observe several significant shifts in various categories: from PCs to tablets, from mouse to (multi-)touch and from Java to JavaScript.
The talk discusses implications on the design of interactive math that come along with these changes and tries to exhibit positive and negative aspects of these changes. The talk will be illustrated by demonstrations of various software projects done by the author including the interactive mathematics software Cinderella, the iOS App iOrnament and several interactive installations in Museums and public exhibitions. | ||||||
| 10:30am - 11:00am | Coffee Fr AM: Coffee break | ||||||
| Georg-Cantor-Haus | |||||||
| 11:00am - 1:00pm | CADG Tools I/II: CADG Tools, Teaching and Learning (working group) Session Chair: Matija Lokar | ||||||
| VSP 1.03 | |||||||
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Lessons Learned In Course »Computer Tools In Mathematics« University of Ljubljana, Slovenia
In first cycle professional study program Practical Mathematics at Faculty of mathematics and physics, University of Ljubljana, we have a course entitled Computer tools in mathematics. The main aim of this course is to show the students the practical usage of various tools in solving mathematical problems. There were several decisions we had to make during lectures preparation. In the talk we will discuss some of them:
• One tool covering all aspects of teaching and learning mathematics, or many "smaller" tools Some teachers claim it is best to stay in the same environment all the time. However, is that really the case? Aren’t smaller tools often more flexible and versatile? • How powerful should tools be when they are considered teaching tools Should tools capability be limited to the level of math knowledge of the students? On the other hand, should the students be exposed to “real life” tools without limitations? • Support materials development and usage We will discuss our approach producing GeoGebra Wiki (http://lokar.fmf.uni-lj.si/wikiji/GeoGebraWiki). • Relation between “mathematical answer” and result obtained from computer tool – some practical examples how students internalize the notion of “correct result” when working in different environments with different tools will be given • Openness of construction process One of the task for students is also to prepare detailed explanation of approach in solving a certain mathematical problem, we observed several problems students have. Besides the obvious ones like the difficulties in using appropriate mathematical language, there are also “technical” ones in usage of tools.
An approach to the study of systems of equations with GeoGebra: learning opportunities provided by the integration of CAS view UNSAM, Argentine Republic
Solve systems of equations in school -at least in Argentina- is usually a task for students is presented in a series of techniques that "allow” find your solution. Generally, these techniques work as an end in themselves. Finally and as a strategy of “verification" tends to give a “new resolution method”: graph.
How to overcome educational obstacles that are generated from a fragmented approach of knowledge? What can make the DGS, in particular the CAS environment? What epistemic and instrumental value (Artigue 2002) acquires the techniques work with software? Redefine work with pencil and paper? These and other questions will try to provide theoretical elements for didactic reflection on the potential guarding technologies for teaching mathematics. Specifically, this paper attempts to examine a series of dynamic problems (Bifano & Villella, 2012) to study the intersection of manipulating different parts of the same equations and reflections that have emerged from experience working with teachers to solve and analyze such problems. Search the conditions for a system of equations has solution engage students into an exploration´s work and the establishment of guesses to try and find the answer. The multiple connections that enable the integration of GeoGebra different views (Hohenwarter & Jones, 2007), it contains a teaching which enables other potential opportunities for reflection on the nature of mathematical work. Moreover, the inclusion of an artifact in class -computer - produced changes in both the types of problems that can be proposed, such as the type of classroom management that this requires (Balacheff, 1997). In this sense, the contributions of Trouche (2004) on the processes of instrumental genesis can guide the discussion about the different instrumental genesis that shaped the class.
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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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| 11:00am - 1:00pm | Future Trends I/II: Future Trends in Interactive Geometry (working group) Session Chair: Masataka Kaneko | ||||||
| VSP 1.04 | |||||||
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Using Dynamic Geometry as a Robotics Interface 1PH Ludwigsburg, Germany; 2University of Halle-Wittenberg, Germany
Lego Mindstorms is an easy to use construction kit for robotics that perfectly fits to educational purposes. There are several programming environments available that have different advantages and disadvantages. Graphical programming environments like Lego NXT-G are easy to use but often limited in the possibilitys of implemented projects. On the other hand, librarys for common programming languages like BricxCC or LeJOS provide a wide range of functions for flexible progamms, but deeper programming skills are required before one gets started.
The interactive geometry software Cinderella comes with an easy to learn but powerfull functional scripting interface CindyScript to control geometric constructions and perform mathematical calculations. But using an internal timer integrated in Cinderella's scripting environment even sophisticated interactive animations ans simulations can be implemented with only a little programming effort. We use a Cinderella plugin that wraps LeJOS commands to CindyScript functions to controll the Mindstorms NXT motors and sensors. Connecting the framework with geometric construction elements, one gets a new kind of interactive robot remote control. Vivid visualization of sensor results are possible. In this talk we present some applications developed during a course for teacher students in computer science. The examples arise from the boarderline of mathematics, physics and computer sience and cover aspects of mathematical areas like functional thinking or analytical geometry. Sketchometry - Dynamic Mathematics on Mobile Devices University of Bayreuth, Germany
The internet goes mobile and mobile devices like tablet computers and smartphones are entering the classrooms. How do these developments influence learning? What opportunities are opened for mathematics education?
These questions are the introduction of the talk about sketchometry, a new kind of dynamic mathematics software, especially designed for mobile devices. Lines, circles or triangles are simply sketched with the finger on the screen. The software transforms them into geometric objects. The fingers become compass or ruler. In contrast to PCs, software on mobile devices is ready-to-use right after power-up. Creating constructions with sketchometry in mathematics lessons becomes as easy as the usage of pocket calculators. The mobile devices can be used point by point by the students. It is not necessary to go to a computer lab for the whole lesson. For a gainful integration of these advantages in mathematics education, it is necessary to rethink didactical and methodical concepts. Traditional media, as text books or printed worksheets, can be used together with sketchometry on the mobile device. The usage of computers does no longer happen isolated but integrated.
Cross-browser graphic user interface for interactive geometry University of East Sarajevo, Bosnia and Herzegovina
This presentation will introduce a new cross-browser dynamic geometry system with graphic user interface, based on JSXGraph. JSXGraph is library for interactive geometry completely implemented in JavaScript and it does not rely on any other library. Developed cross-browser interactive geometry system depends only on JavaScript and it does not require any additional plugins. It has a very small footprint and works on all devices, including multi-touch devices running iOS, Android, firefoxOS and Windows 8. This system is an excellent solution for the development platform-independent interactive web elements for geometry constructions.
Geometric Algebra – A foundation for the combination of Dynamic Geometry Systems with Computer Algebra Systems? TU Darmstadt, Germany
Geometric Algebra is a very general mathematical system including many other systems such as linear algebra, complex numbers, Plücker coordinates, projective geometry or quaternions.
The specific compass ruler algebra, for instance, is very well suited to compute similar to working with compass and ruler. Geometric objects such as circles and lines as well as geometric operations with them can be handled very easily inside of the algebra. A circle, for instance, can be described based on the outer product of three points of the circle. The compass ruler algebra is a 4D algebra describing the 2D plane with two additional basis vectors representing the origin and infinity. You are able to directly compute with infinity, for instance, when expressing the center point of a circle as the inversion of infinity in the circle. Gaalop is an easy to handle tool in order to compute and visualize with compass ruler algebra. While in the background a computer algebra system is responsible for the symbolic computations, its visualizing component offers basic DGS functionality. Based on this combination of geometry and algebra Gaalop is also very well suited for proving of geometric relations.
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| 1:00pm - 2:15pm | Lunch I: Lunch (incl. Coffee break II) | ||||||
| Georg-Cantor-Haus | |||||||
| 2:15pm - 4:15pm | AKMUI I: Vorträge – Lehrertag | ||||||
| VSP 1.04 | |||||||
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Historische astronomische Daten und moderne CAS-Rechner MLU Halle, Germany
Die Modellierung realer Zusammenhänge ist oft mit einer typischen Aufgabe verbunden: Aus Messdaten soll ein analytischer Ausdruck abgeleitet werden, der den Daten „gut angepasst“ ist. Der Vortrag greift dies für ein historisches Astronomie-Problem auf: Die Frage der Funktionsanpassung für originale Messwerte wird für den Kometen von 1618 betrachtet. Die reale Datensituation wird genutzt, um für Schüler Funktionsapproximation mit einem CAS-Rechner erleb- und nachvollziehbar werden zu lassen.Die Modellierung realer Zusammenhänge ist oft mit einer typischen Aufgabe verbunden: Aus Messdaten soll ein analytischer Ausdruck abgeleitet werden, der den Daten „gut angepasst“ ist. Der Vortrag greift dies für ein historisches Astronomie-Problem auf: Die Frage der Funktionsanpassung für originale Messwerte wird für den Kometen von 1618 betrachtet. Die reale Datensituation wird genutzt, um für Schüler Funktionsapproximation mit einem CAS-Rechner erleb- und nachvollziehbar werden zu lassen.
Unterrichtsmaterial vor dem Hintergrund verschiedener Werkzeuge – einige Beispiele der Bildverarbeitung
„Wem, wie, wann, wo und warum nutzen Werkzeuge?“ Unter anderem auf diese Fragen der Tagungseinladung möchte ich – am Beispiel der für den Mathematikunterricht reduzierten grundlegenden Konzepte der Bildverarbeitung – einige Antworten anbieten und diese zur Diskussion stellen.
Ich habe mich damit auseinandergesetzt, welche Möglichkeiten die Bildverarbeitung bietet, alternative Zugänge zu vielen Teilbereichen der Schulmathematik zu schaffen. Die dabei benutzten Werkzeuge sind Maple 18, MaplePlayer und Excel. Mit meinem Vortrag will ich Ihnen – möglichst unterrichtsnah – einige Beispiele meiner Arbeit vorstellen und Material zur Verfügung stellen, das – mit bzw. ohne den Einsatz der oben genannten Werkzeuge – zum Unterrichtseinsatz genutzt werden kann. Digitale Werkzeugkomponenten
Über welche Kompetenzen sollen Schülerinnen und Schüler zum Abitur bzw. nach Abschluss der Sekundarstufe I beim Umgang mit digitalen Werkzeugen verfügen? Inwiefern geht es dabei um etwas anderes bzw. um mehr als um die Bedienung von Software und Hardware? Die Bildungsstandards der KMK lassen da einiges offen. Eine gemeinsame Arbeitsgruppe von MNU und T3 beschäftigt sich seit 2013 mit der Fragestellung, was unter ›Digitalen Werkzeugkompetenzen‹ zu verstehen ist. Erste Ergebnisse werden im Vortrag vorgestellt und anhand von Aufgabenbeispielen zur Sekundarstufe I und II konkretisiert. Damit verbunden wird der Frage nachgegangen, wie Lernende ihren Einsatz von digitalen Werkzeugen im Arbeitsprozess und schriftlichen Überprüfungen dokumentieren sollten.
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| 2:15pm - 4:15pm | CADG Tools II/II: CADG Tools, Teaching and Learning + Learning (general topic) Session Chair: Matija Lokar | ||||||
| VSP 1.03 | |||||||
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GeoGebra Support of Discrete Mathematics Course for Future Math Teachers G.S.Scovorodu Kharkiv national pedagogical university, Ukraine
A course of discrete mathematics calls for different accentuation when taught for different future specialists. A lot of ingenuity is needed to create a successful course for would-be teachers of mathematics. The basic topics of our course includes set theory, mathematical logic, combinatorics, and graph theory in teaching which we make a good use of GeoGebra tools. A wide range of instruments (such as Spreadsheet and Algebra view, some categories of commands and tools, etc) allow a teacher to expand the range of tasks, increase their complexity, foster a research component, create dynamic demonstrations to explore mathematical objects with multiple representations.
We offer the collection of interactive dynamic demonstrations designed in GeoGebra in support of discrete mathematics course for future teachers of mathematics. The collection focuses on four types of demonstrations: teaching and learning basic concepts, stages of theorem proving, implementation of well-known algorithms from graph theory, the real-life problem-solving situations. One of the course assignments includes a student's project on creating a dynamic demonstration. The best projects formed the supplement of the collection. The result of the experimental implementation (2010-2014) of the collection of interactive dynamic demonstrations will be presented.
What software to use in the teaching of the mathematical subjects? Technical University of Kosice, Faculty of Electrical Engineering and Informatics, Slovak Republic
We can consider two basic views, when using mathematical software in the teaching of the mathematical subjects.
First: How to learn to use specific software for the specific tasks, e. g., software Statistica for the subjects of Applied statistics, probability and mathematical statistics, or financial mathematics. Second: How to learn to use the software that is available to us to solve specific mathematical problems from different areas of mathematics and applied mathematics. My article describes the practical use of the software in the teaching of the mathematical subjects and my experience with its use by the students and the pedagogues. In our computer laboratory our students use the following software: MS Office (MS Excel), LibreOffice (LO Calc), MATLAB 2010b, Octave, wxMaxima, and LaTeX. We use this software in the following mathematical subjects: Operational Analysis, Linear and Quadratic Programming, Numerical Mathematics, Applied Statistics, Queuing Theory and Fundamentals of the LaTeX. Eight years of experience with the usage of these programs have shown us, how we can improve the teaching process of the mathematical subjects at the technical universities with freely available software.
Discussions induced by unexpected answers from a computer algebra system University of Tartu, Estonia
It is usually assumed that a computer algebra system (CAS) offers the answer that is expected by the student or the teacher. However, some answers can be somewhat unexpected (but not necessarily wrong). Such answers could induce rich mathematical discussion. Then again, the unexpected answers could only be obstacles to learning without any productive discussion.
The CADGME paper analyzes pair discussions of first-year students who participated in a course in elementary mathematics. The pairs had worksheets with trigonometric equations and questions. In the beginning, the students solved the equation (correctly or not) on paper without the help of a CAS. Then they solved the same equation with a particular CAS (that gave a different answer from the answer expected by the students). The given questions guided them to analyze the differences, equivalence and correctness of their own answers and CAS answers. Their discussions were audio-recorded and the paper is based both on the audio-records and on the students’ worksheets. In most cases the students engaged in active discussions about the tasks. The discussions varied in dynamics, productivity, mathematical depth, etc. This paper highlights and discusses some of the patterns that emerged from the discussions.
Dynamic Algebra in EpsilonWriter 1Aristod, France; 2Université de Lyon 1, France
Dynamic Algebra has been implemented in EpsilonWriter as a way for doing calculations by direct manipulation of algebraic objects mouse, especially step-by-step calculations with explanations. A first group of functionalities is currently available (http://epsilonwriter.com). It contains equivalent drag&drop, e.g., a drag&drop of 3x onto 2x in 2x-4=3x provides –x-4=0, a drag&drop of 3x inside the parentheses in 3x(2x+3) provides 3x²+9x. Factoring out from a sum, simplifying fractions, applying substitutions to expressions, multiplying an equation by an expression, side-by-side adding equations can also be performed by drag&drop. It is particularly easy to solve simultaneous equations. Each calculation step is explained in a way chosen by the user.
These functionalities also contain schemas for solving quadratic equations with the discriminant and cubic equations with Cardan’s formulas, see demos at: http://www.epsilonwriter.com/en/top10demos/ New functionalities have been recently implemented and will be available soon. They appear as proposals displayed in a balloon when the mouse flies over an operator or when an expression is selected. This is the case for “Expand and simplify”, “Complete the square”, and “Factor with A²-B²”. A rich set of functionalities concerns functions with: definition conditions, calculations of limits, calculations of derivatives, and drawing tables of signs of derivatives and of variations of functions. Last, a lot of approximate values of several functions, from values of the variable expressed as a sequence, can be obtain in a table for helping understanding the behavior of functions. These tools are currently used to design learning units in the MCSquared project (http://mc2-project.eu/). In the first phase, the domain concerns functions. In the second phase, it will be enlarged in particular to algebra.
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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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| 4:15pm - 4:45pm | Coffee Fr PM: Coffee break and Poster presentation | ||||||
| Georg-Cantor-Haus | |||||||
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Visual introduction to modeling systems with delay 1University of Szeged, Hungary; 2University of Szeged, Hungary
Delays can appear in many phenomena in the Nature, and hence delay systems appear in many fields of Sciences. Their mathematical theory is quite new. Since understanding needs deep mathematics, hence only advanced courses deal with delay systems in mathematical curricula. On the other hand, undergraduate math and even science students should have a first impression of delay systems.
In our talk, we consider the didactic problems of teaching delay systems to students without or partly having the required knowledge. We present a short easy-to-understand visual way of introducing delay systems with the help of series of dynamic demonstrations developed in Mathematica. The basic concepts, properties, the difference between systems without and with delay are treated via elementary examples. We also give applications appearing in engineering and sciences. The interactive demonstrations will be available on our website www.model.u-szeged.hu.
COGNITIVE-VISUAL APPROACH TO THE TEACHING TOPIC "DERIVATIVE OF A FUNCTION 1University of Novi Sad, Serbia; 2Gimnazija Pirot
This paper presents the problems based on the examining functions and their properties with the use of derivative. The main idea is to analyze function and its derivative without using their analytical expressions. The package Geogebra is used for presenting visualized problems.
Key words: graph, function, derivative of function, visualized problems, cognitive-visual approach
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| 4:45pm - 5:45pm | Keynote II: Keynote (Ralph-Johan Back) – Lehrertag | ||||||
| Jacob-Volhard-Hörsaal | |||||||
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Structured derivations in practice: experiences from the E-math project Abo Akademi University, Finland
Structured derivations is a new method for presenting mathematical arguments. It is a further development of Dijsktra's calculational style reasoning, and can be used for all kinds of mathematics: proofs, calculations, geometric constructions, etc. It combines the three main proof paradigms, equational reasoning, forward reasoning (Hilbert style) and backward reasoning (Gentzen style), in one single proof format. The use of structured derivations in high school mathematics education has recently been piloted in the E-math project (an EU project 2011 - 2013). The project has created new mathematics textbooks based on the systematic use of structured derivations. These textbooks cover the whole national mathematics curriculum for first year in high school, in Finland, Sweden, and Estonia. The textbooks have been implemented on a new software platform for interactive e-books created in the project. This platform has special support for displaying and writing mathematics on a computer. The new math e-books have been piloted in 15 high schools in 2012 - 13, with some 1 000 students participating in the pilots. The talk will present the main findings from the E-math project, and discuss conclusions that can be drawn from the pilots.
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| 6:00pm - 8:00pm | Welcome: Welcome reception | ||||||
| Café Einstein Von-Seckendorff-Platz 1 | |||||||
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| 9:30am - 10:30am | Keynote III: Keynote (Tomás Recio) | |||
| Jacob-Volhard-Hörsaal | ||||
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Dynamic Geometry and Mathematics: few trains on a two-way track Universidad de Cantabria, Spain
Dynamic geometry is designed as a helpful tool for mathematics comprehension. This can be thought as a single direction: from dynamic geometry to mathematics. But, conversely, some non elementary mathematics seem to be required to understand (and to improve) dynamic geometry performance.
Noticing this mutual interaction is neither very popular, nor strictly new (e.g. consider the paradigmatic case of Cinderella). On the other hand, we would argue in the talk how to trace back the origins of such interaction in order to include some fancy names such as, say, Babbage or Watt, ending up with Nash.... Yet, we think it is perhaps convenient to insist now and again on the importance of traveling back and forth along this two way track, for the benefit of mathematics education. This would be the main idea in my lecture, exemplified by some situations I have been recently dealing with, in order to improve dynamic geometry features for locus computation and automatic theorem proving.
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| 10:30am - 11:00am | Coffee Sa AM: Coffee break and Poster presentation | |||
| Georg-Cantor-Haus | ||||
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Several Aspects of Using Computer Supports for Mathematics Learning of Foreign Students from the CIS (Countries of Independent States) G. S. Skovoroda Kharkiv National Pedagogical University, Kharkiv, Ukraine
The article is devoted to peculiarities of foreign students training in a modern computer environment. Special attention is given to the problems which are relating in studying of mathematical disciplines. The problems of graphical solution of mathematical and computer tasks with using modern computer technology and visual facilities are discussed.
From real world to derivative – How to effectively include mathematical modeling and GeoGebra in mathematics education 1University of Novi Sad, Serbia; 2Technical Colegge of Applied Sciences, Serbia
Direct application of innovative teaching method based on principles of mathematical modeling is illustrated by example of the teaching unit related to introducing the concept of the first derivative processed by using mathematical modeling as teaching method and GeoGebra as software tool. Step by step procedure of making mathematical model is explained – from starting preparation, to implementation of the mathematical model and drawing conclusions from it.
The positive effects of mathematical modeling and GeoGebra as software tool to better understanding, the creation of advanced mathematical thinking, and the application of mathematical theory on solving real world problems are presented and illustrated.
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| 11:00am - 1:00pm | AKMUI II: Vorträge | |||
| VSP 1.03 | ||||
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app@school – App-Entwicklung als Lehr-Lern-Szenario in der Schule
Bei app@school geht es darum, dass die Schülerinnen und Schüler mithilfe analoger und digitaler Werkzeuge eine eigene mobile app konzipieren, entwickeln und publizieren. Indem der gesamte Produktionsprozess einer Applikation vollzogen wird, werden sowohl fachlich-mediale als auch soziale Kompetenzen angebahnt.
Um dieses Ziel zu verwirklichen haben wir das theoretisch fundiertes Lehr-Lern-Szenario app@school entworfen, das einerseits den pädagogischen Projektgedanken aufgreift, also auf prozessuales Lernen in (Experten)Gruppen abzielt, und sich andererseits an den Abläufen agiler Softwareentwicklung orientiert. Als Entwicklungswerkzeug verwenden wir Stencyl (http://stencyl.com), da es eine visuelle Gestaltung aller Programmbestandteile (Levels, Actors, Logik) ermöglicht. Der Einsatz dieses Tools erlaubt darüber hinaus die ideale Verzahnung der einzelnen Expertengruppen (Grafiker, Leveldesigner, Programmierer, PR-Manager, Projektmanager), die jeweils an einem Module der mobile app arbeiten. Im Rahmen eines Vortrags auf der Tagung des AK »Mathematik und Informatik« würde sowohl das didaktische Design von app@school als auch erste Ergebnisse dessen schulpraktischer Erprobung Anfang diesen Jahres zur Diskussion gestellt. Interaktives Konstruieren im länderübergreifendem bilingualen Mathematikunterricht University of Education Ludwigsburg, Germany
Für das Fach Mathematik wird ein Unterrichtsszenario aus dem INTACT-
Kontext dargestellt, in dem Schülergruppen aus zwei verschiedenen Ländern Dreieckskonstruktionen mit DGS durchführen und dazu Beschreibungen in der Fremd- bzw. der Muttersprache anfertigen. Diese Beschreibungen werden zwischen den beiden Schülergruppen der verschiedenen Länder ausgetauscht, um die Dreiecke zu rekonstruieren. Dabei sind sowohl geometrische wie auch sprachliche, interkulturelle und soziale Fertigkeiten und Kompetenzen gefordert und gefördert. Nutzbar sind die Materialien, die über eine speziell entwickelte Lernplattform zur Verfügung gestellt werden, nicht nur mit interaktiven Whiteboards sondern auch mit anderen Geräten, wie z.B. Tablet-PCs, Smartphones usw. Im multilingualen Projekt „INTACT - Interactive teaching materials across culture and technology“ der Pädagogischen Hochschule Ludwigsburg mit den Partnerländern Spanien, Irland, Ungarn, Rumänien und Portugal werden interaktive Materialien für den bilingualen Unterricht verschiedener Fächer zur Verfügung gestellt und so eine virtuelle Kooperation zwischen Schulen der ganzen Welt ermöglicht. INTACTwird für drei Jahre aus Fördermitteln der EU im Rahmen des Förderprogramms Lebenslanges Lernen - "COMENIUS Multilateral Projects" gefördert. Pen&Paper-Programmierung - Neue Chancen für digitale Medien
Der Begriff des Algorithmus gehört seit langem zu den fundamentalen
mathematischen Ideen, die im Mathematikunterricht einen spiraligen Aufbau haben. Für die Sekundarstufen bedeutet die Thematisierung für die Schüler die Chance, selbstständig Algorithmen zu entwickeln, um so einerseits Einsichten in mathematische Strukturen zu gewinnen und zu vertiefen und andererseits einen reflektierten Umgang mit den heutigen digitalen Medien zu fördern. In meinem Vortrag stelle ich die von mir entwickelte Pen&Paper-Programmiersprache Adi vor. Ursprünglich als Ausgangspunkt zum Erlernen einer Programmiersprache gedacht, nutze ich diese Sprache als Vehikel, um einer möglichen Interpretation des Begriffs "Algorithmisches Denken" nachzugehen. | |||
| 11:00am - 1:00pm | Automated Deduction: Automated Deduction in Dynamic Geometry Tools (working group) Session Chair: Predrag Janičić | |||
| VSP 1.04 | ||||
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Automatic theorem proving in Dynamic Geometry contexts: what is it good for? Two diverse points of view 1Universidad de Cantabria Spain; 2Architectural technical high school, Serbia
The talk will be dual, in the sense of addressing the same issue from two different perspectives. One, that of a secondary mathematics teacher, with experience on the introduction of ATP tools (GCLC and GeoGebra) at the classroom. Then, the point of view of a university ATP researcher and developer, with sustained involvement in math education. In both cases the main point will be to provide some answers to the key question:
What is ATP good for? The most efficient ATP systems in geometry are usually algebraic. Although these provers do not generate classical readable proofs, they give an yes/no answer and can generate non-degeneracy conditions (NDGs) that need to hold for the statement to be satisfied. Although ATPs can be applied to check the main statement of the theorem, we advocate that it is much more beneficial for pupils to apply automation on intermediate steps, and so to verify if their conclusions are correct and if they are on the right track to prove the main theorem. In such setting, pupils still need to produce global proof steps, while the ATP system takes care of details that are usually straightforward, but tedious to justify. Pupil cooperates with the machine and relies on its help, but his understanding of the geometry problem is still crucial for success. Trough this activity student develops his thinking and analytical skills, but, unlike with classical pen-and-paper proofs, in a very rigorous setting, since every conclusion must be precisely formulated and pass the automated test. Also, a careful analysis of NDGs may reveal many subtle issues about the theorem that is being proved. In this talk we will present some of our experience and ideas about using these techniques (implemented both in GCLC and GeoGebra) in high-school geometry classes.
Automated algebraic calculations of geometric figures in dynamic geometry systems University of Education Weingarten, Germany
Using Methods based on Automated Deduction in Geometry (ADG), it is possible to perform algebraic calculations on interactively constructed figures. Thus, a parameter of such a figure can be calculated algebraically as a function of other parameters of this figure. This opens up a new computer-assisted connection of synthetic elementary geometry to algebra. On the other hand, the question: "WHAT HOW depends on WHOM?" is cleared up when a figure is dynamically varied. - In this lecture there are given some selected examples in a constraint-based DGS developed for educational purposes. Their didactic relevance is explained and some resulting mathematics education problems are discussed.
Teaching loci and envelopes in GeoGebra 1University of Vigo, Spain; 2Johannes Kepler University, Austria
GeoGebra is open source mathematics education software being used in thousands of schools worldwide. Since version 4.2 (December 2012) it supports symbolic computation of locus equations as a result of joint effort of mathematicians and programmers helping the GeoGebra developer team. The joint work, based on former researches, started in 2010 and continued until present days, now enables fast locus and envelope computations even in a web browser in full HTML5 mode. In conclusion, classroom demonstrations and deeper investigations of dynamic analytical geometry is ready to use on tablets or smartphones as well.
In our talk we consider some typical grammar school topics when investigating loci is a natural way of defining mathematical objects. Such topics include definition of a parabola and other conics in different situations like synthetic definitions or points and curves associated with a triangle. In most grammar schools, however, no other than quadratic curves are discussed, but generalization of some exercises and also every day problems will introduce higher order algebraic curves. Thus our talk will mention the cubic curve ``strophoid'' as locus of heights of a triangle when one of the vertices moves on a circle. Also quartic ``cardioid'' and sextic ``nephroid'' can be of every day interest when investigating mathematics in a coffee cup. We will also focus on GeoGebra specific tips and tricks when constructing a geometric figure to be available for getting the locus equation. Among others, simplification and synthetization (via the intercept theorem) will be mentioned.
Extending the range of computable objects in Dynamic Geometry by using Quantifier Elimination University of Vigo, Spain
Traditionally, loci in Dynamic Geometry Systems (DGS) have been displayed either tracing the locus point, either through a special command that, roughly, enhances such tracing procedure. Some DGS incorporate more sophisticated approaches being able to return algebraic knowledge about loci. Nevertheless, these enhancements deal with a restricted type of loci, those where the position of the locus points is completely determined by another point, the mover, which must lie on a linear path. If the mover point is not unique, or it is not bound to a line, standard DGS can only offer a tracing strategy, just returning a graphical answer. Consider, for instance, two circles and a point on each circle. The locus of their midpoint is a 2-dimensional region, that cannot be easily described with current DGS.
In this talk I illustrate these limitations when computing loci. Furthermore, a discussion on the application of Cylindrical Algebraic Decomposition software will be given. | |||
| 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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| 1:00pm - 2:15pm | Lunch II: Lunch (incl. Coffee break V) | |||
| Georg-Cantor-Haus | ||||
| 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. | |||
| 2:15pm - 4:15pm | Workshop 2 & 3 | |||
| VSP 1.04 | ||||
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Mapping mathematics learning resources University of Education Weingarten, Germany
In this workshop, we shall work collectively to write down a map of the sources of learning resources, be them open or not, adaptable or not, nicely made or not.
The aim is to assemble a broad panorama of the places where mathematics learning resources can be found. We shall do so in a web-based map which can be followed live during the workshop. The wish is to leverage the diversity of attendees at the conference so as to obtain some clarity in the local practices in teachers of their surroundings. The result of this workshop would support such harvesting projects as Open-Discovery-Space, i2geo.net, probably Elixier, and other initiatives. Moreover, it would help users of such repositories to get a broader overview and thus decide better what to visit at their next resourcing excursions. Controlling Lego Mindstorms robots by Cinderella's scripting interface PH Ludwigsburg, Germany
By installing the Mindstorms plugin, the interactive geometry software Cinderella can be used as a remote control for Lego Mindstorms robots.
In this workshop we will explain how to install and use the plugin. We will give a short introduction to the CindyScript programming language. Simple examples will show how to read sensor values from a Lego NXT and use them to manipulate geometric constructions. Vice versa, interactive geometric constructions are used as remote controls for motors. There will be a few Mindstorms controller on location for own explorations. | |||
| 2:15pm - 4:15pm | Workshop 4 & 5 | |||
| VSP 1.03 | ||||
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Development of Dynamic Demonstrations with Mathematica University of Szeged, Hungary
Some years ago, one of the main improvement in Mathematica was the dynamic functionality. Real-time graphics and functions for dynamic variables (Manipulate…) meant a breakthrough in the classroom usage of CAS systems. Currently, this technique has been greatly improved. In addition, the so called CDF ("Computable Document Format") has been introduced. In the workshop, the participants can get acquainted with the elements of dynamic features of Mathematica and they learn how to prepare simple CDF documents.
Colleagues are kindly encouraged to send problems, topics for discussion in the frame of the workshop. The participants should bring their laptops with Mathematica installed. A 30 days trial version can be downloaded from the website of Wolfram Research. Planned length: 3-4 hours.
Sketchometry - DGS for tablets, smartphones and interactive whiteboards University of Bayreuth, Germany
Sketchometry is a Dynamic Geometry System (DGS) for the student in classroom. In contrast to many other DGS which are primarily used by teachers as a presentation tool, sketchometry tries to be a simple tool that invites students to experiment and discover.
Sketchometry can be used on tablets, smartphones, interactive whiteboards as well as on desktop computers. The software has a touch-optimized user interface for easy sketching of geometric constructions. In the workshop we will discover sketchometry by creating various classroom examples. Please, bring your own device: tablet, smartphone or notebook.
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| 4:15pm - 4:45pm | Coffee Sa PM: Coffee break and Poster presentation | |||
| Georg-Cantor-Haus | ||||
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Visual introduction to bifurcations University of Szeged, Hungary
Investigating the dependence on parameters is essential in studying dynamical systems. In particular, the bifurcation theory is getting more and more important in most fields of engineering and sciences. Nevertheless, these theories are hardly included in standard university curricula.
We will give an intuitive introduction with the help of dynamic demonstrations developed in Mathematica. We consider elementary examples of both difference and differential equations presenting different types of bifurcation. During the whole treatment, we keep in mind the real didactic “contradiction” that the students do not or only partly have the required knowledge. The interactive demonstrations will be available on our website www.model.u-szeged.hu.
MEASURES OF GEOMETRIC OBJECTS AS THE LIMIT VALUES 1University of Novi Sad, Serbia; 2Elementary school of Nis; 3Faculty of Mechanical Engineering Nis
The idea for this work is based on the visualization of the limit process in elementary school. The circumference of a circle is obtained visually starting from the perimeter of regular polygon. The volume of cylinder is obtained visually starting from the volume of prism.
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| 4:45pm - 5:45pm | AKMUI IV: Arbeitsgruppen | |||
| 4:45pm - 5:45pm | Collaborative: Collaborative use of DGS and KETpic (working group) Session Chair: Setsuo Takato | |||
| VSP 1.04 | ||||
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KETCindy - Fine Combination of KETpic and Cinderella 1Toho University, Japan; 2Kogakuin University, Japan; 3Nagano National College of Technology, Japan; 4Kisarazu National College of Technology, Japan; 5Graduate School of Information Science, Nagoya University, Japan
According to the result of our questionnaire survey, one major opinion of collegiate mathematics teachers in Japan is that there is no necessity to use high-quality graphics in education. However, from our experience, graphics use seems to play a crucial role in some classroom situations. Though we have been attempting some statistical efficiency assessments for teaching materials containing graphics, it is not so easy to separate the effect of using graphics from other factors like context of classroom or communications between students. The aim of this research is to verify the effect of using high-quality graphics in collegiate mathematics education through some cognitive scientific experiments. The experiments were designed in the following two ways:
(1) Detecting the change of students’ brain activity after their seeing effective figures through EEG (ElectroEncephaloGram) measurement (2) Observing the difference in time needed for students to answer a question between before and after their seeing effective figures In fact, we picked up the case of the comparison of growth degree between exponential function y=2^x and power function y=x^4. We prepared some graphs of these functions by gradually changing the scale in y direction, so that students can recognize that the growth of y=2^x is greater than that of y=x^4 when x becomes sufficiently large. We showed these graphics step by step to three students and detected their brain activities through EEG (ElectroEncepharoGram) measurement. As a result, the judgment of these students changed when they saw a triggering figure, and some change in the trend of EEG signal and solution time was observed at that time. These results indicate that using effective figures should have great influence on students’ reasoning processes.
A Collaborative Laboratory for Geometry: A Case Study at Portugal and Serbia 1CISUC/University of Coimbra, Portugal; 2Faculty of Mathematics/University of Belgrade, Belgrade, Serbia; 3CISUC/Department of Mathematics, University of Coimbra, Coimbra, Portugal; 4School of Science and Tecnology/University of Trás-os-Montes e Alto Douro, Vila Real, Portugal
The Web Geometry Laboratory} (WGL) platform is a collaborative blended-learning Web-environment for geometry, it integrates a dynamic geometry systems (DGS) and it provides a collaborative environment for students and teachers. Its use is possible in the context of a classroom or remotely. Apart from its development its evaluation is being done through a series of case studies, sustained through a qualitative approach (interpretive research), being conducted in Portugal and Serbia (Prototype in hilbert.mat.uc.pt/WebGeometryLab).
An initial case study in Portugal, with groups of secondary students (17 years old) was done, using various gathering information techniques: quizzes; tests; direct observation; record interactions on the platform; challenges. We analysed the use of the WGL collaborative environment by the students. Another case study, in Serbia, was conducted in the context of remote access to the platform (homework). The study included 50 secondary students (15 years old). All students attended the traditional classes in school. Half of the students used WGL platform for homework and the other half did their homework the traditional way. We investigated the impact of collaborative work to the motivation level and level of achievement. Using an action research approach, the platform is being developed. These studies revealed some aspects that could be enhanced, e.g. a chat feature. More and wider case studies are being prepared allowing the validation and further development of the WGL platform. These studies also indicates that there is a significant improvement in the motivation of students and a slight improvement in their achievement when using the WGL platform. The WGL platform will include in future stages of development the implementation of an adaptive environment allowing the construction of students' profiles and learning paths. A final stage will be the integration of a geometric automated theorem prover and its use in the learning process.
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| 4:45pm - 5:45pm | Creativity: Creative Mathematical Thinking and Digital Tools (working group) Session Chair: Péter Körtesi | |||
| VSP 1.03 | ||||
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Convex hull of the maximum volume of a space curve in the special case Deoma, Russian Federation
Let СN be the closed three-dimensional polygon with 2N edges (N > 3), the perimeter L(СN) and the convex hull of СN volume V(CN). We want to find maximum V(CN) for given L(СN) in the special case when the convex hull may be divided into tetrahedra having one common edge. Let C be the rectifiable closed three-dimensional curve with the length L(C) and the volume of the convex hull V(C). C may be obtained using the СN limit at infinity. We want to find maximum V(C) for given L(C).
We assume that the convex hull maximum volume is achieved if two conditions are satisfied: at first, the slope angle θ between the curve and the Z-axis is constant, the segment which intersects the Z-axis is perpendicular to it. The second condition is: the projection of the convex hull to the XY plane has the maximum area. For V(L) there is an exact evaluation. The sign of the equality holds if and only if the curve is congruent to the curve obtained in the paper. There are two solids of equal volume. One solid is axisymmetric, the second solid is centrally symmetric. The area of the convex hull projection on the XY plane has an exact evaluation. The sign of the equality holds if and only if the curve is congruent to the curve obtained in the paper. Finding the maximum area of the convex hull of CN projection in case N = 4 + 2n is reduced to finding the function extremum under the obtained conditions. These equations may be solved analytically for n < 12 and numerically for an arbitrary n. The solution has been found and checked using DGS GInMA. There are some examples of the maximum volume convex hulls V(CN) and maximum V(C) for given L(C). | |||
| 6:00pm | Dinner: Conference dinner | |||
| Bergschenke (conference dinner location) | ||||
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| 9:30am - 10:30am | Keynote IV: Keynote (Marcelo de Carvalho Borba) | |||||||
| Jacob-Volhard-Hörsaal | ||||||||
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Math Problem, Facebook and Emergent Classrooms Universidade Estadual Paulista, Brazil
“If production of knowledge is understood in this way, what constitutes a ‘‘problem’’ will depend on the nature of the humans-with-media collective. A problem that needs to be solved, or that puzzles someone, may not be a problem when a search software tool like Google is available. Similarly, a real problem for collectives of humans-with-orality may not constitute a problem for a collective of humans-with-paper-and-pencil.” (p. 804, Borba (2012)
In this talk I will unpack the above quote from a recently published paper on ZDM. I will discuss first the way Internet and mobile telephones in particular, and digital technology in general, are changing the nature of what it means to be a human being (Castells, 2009; Borba, 2012). I will present to the reader my view regarding four phases of the use of digital technology in mathematics education (Borba, 2012) in order to discuss how interaction occurs in presence of such technology. I will then discuss what can be labeled “emergent classrooms” within the fourth phase. I will focus on how social networks such as Facebook and other features of this phase are transforming interaction in the classroom, and perhaps even creating new images of what a classroom may be. Examples from pre-calculus/early calculus will be provided. References Borba, M.C. Humans-with-media and continuing education for mathematics teachers in online environments. ZDM Mathematics Education 6(44) p. 801-814. (2012). Borba, M. C. Potential scenarios for Internet use in the mathematics classroom. ZDM, 41(4), 453–465, 2009. Borba, M. C., Villarreal, M. E. (2005) Humans-with-media and the reorganization of mathematical thinking: information and communication technologies, modeling, visualization, and experimentation. New York, Springer. Castells, M. (2009) Communicating Power. London: Oxford University Press Levy, P. (1993) As Tecnologias da Inteligência: o futuro do pensamento na era da informática. Rio de Janeiro: Editora 34. | |||||||
| 10:30am - 11:00am | Coffee Su AM: Coffee break and Poster presentation | |||||||
| Georg-Cantor-Haus | ||||||||
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Mathematical Modeling of Material Experiments in High School 1University of Novi Sad, Serbia; 2Eklementary and hogh School Ruski Krstur
Mathematical modeling can be applied in high school trough different real world situations. In this paper we present how can material science experiments be used for introducing students with interdisciplinary scientific approach. Using high school mathematics knowledge and Geogebra students can easily discover contents of contemporary Science.
Electronic Trainers for Successful Math Teaching to Pupils of Primary School G.S.Scovorodu Kharkiv national pedagogical university, Ukraine
Primary school is undergoing significant changes, which are associated with rapid updating of information technology and high level of informational activity of children. In primary school, they focus on the formation and improvement of subject knowledge and general study skills. One of the modern ways of forming general study and subject skills by younger pupils is use of electronic trainers. Electronic trainers feature an ability to provide real variability of tasks, uniqueness of exercises, operative assessment of correctness of each task, adjustment of task difficulty, ability to provide a shade of competitiveness and gaming to the exercises. To develop electronic trainers teacher can use programs that are part of the integrated Microsoft Office package and designing environments. There are some examples of electronic trainers for younger pupils.
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| 11:00am - 1:00pm | AKMUI V: Vorträge & Abschlussveranstaltung Session Chair: Ulrich Kortenkamp | |||||||
| VSP 1.03 | ||||||||
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Vergleich Dynamischer Raumgeometrie-Systeme (DRGS): Stand eines Forschungsprojektes
In Computerumgebungen ausführbare Raumgeometrieprogramme können als Werkzeuge eine aktive Auseinandersetzung raumgeometrischer Inhalte in der Schule unterstützen. Hierzu existieren zahlreiche Programme, innerhalb derer DRGS idealisiert eine eigene Klasse bilden. Obwohl neben Cabri 3D und Archimedes Geo3D weitere für die Schule konzipierten DRGS existieren, erfahren sie – zumindest in der deutschsprachigen mathematikdidaktischen Literatur - kaum Beachtung. Wie lässt sich dies rechtfertigen?
Im Rahmen eines Forschungsprojektes werden als zentrale Ziele verschiedene DRGS theoretisch und empirisch miteinander verglichen und Empfehlungen für ihren schulpraktischen Einsatz erarbeitet. Im Vortrag werden erste Ergebnisse zum Stand des Forschungsprojektes dargelegt. Podcasts und Screencasts im Lehramtsstudium
In welcher Weise können Podcasts und Screencasts für den mathematischen Lernprozess in einem Lehramtsstudium genutzt werden? Für den Grundschullehramtsstudiengang an der Goethe-Universität Frankfurt/Main werden derzeit unterschiedliche Lehr-Lern-Szenarien entwickelt und erprobt. An ausgewählten Beispielen werden im Vortrag die Potentiale dieses Werkzeugeinsatzes herausgearbeitet.
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| 11:00am - 1:00pm | Edu. Resources: Open Educational Resources in Mathematics (working group) Session Chair: Paul Libbrecht | |||||||
| VSP 1.04 | ||||||||
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Adaptations to a DGS learning resource University of Education Weingarten, Germany
Learning resources have been created to represent digital units of exchangeable materials that teachers and learners can "pull from" in order to support the learning processes. Leveraging the web, one can often find learning these resources; but what characteristics does it need in order to be easily exchangeable? Although several investigations have explored the ways to publish and, in particular, the ideal best practice to do so, few have considered realistic use cases, where the technical competency of the teacher is counterb
In this paper we consider a simple language-crossing situation which was stimulated by the i2geo.net platform. A learning resource has been re-purposed from a French context to a German context, from one school and its software reality to another (Cabri to DynaGeo, Windows French to Windows German, Google Earth to Google Maps), from one set of learners' competency to another. In neighbouring fields, practices such as that of the PhET repository of physics simulations, which collects simulations as "center-points" of the learning resources' exchange process, while "scenarios" are satelite resources, are described as candidate sharing mechansims. This need for adaptations is not uncommon. Until a large diversity in the available resources can be achieved and searched through, it will remain the mainstream practice of teachers that attempt to employ learning resources. However, quite an amount of learning resources' publishers do not make it fully possible. This research highlights a few learning resources' characteristics that support this, from the legal and technical point of views, from the ease of perception to the ease of adaptation.
The Use of Computers for Calculus Teaching Pontifícia Universidade Católica, Brazil
This paper aims at analyzing the use of computers when teaching differentiability and continuity in real functions and real variable. The relation is approached in the case of a non-differentiable continuous function in the interval of straight lines. This example is found in an article written by David Tall and is used to evidence a way in which a computer helps the learning and teaching of concepts of Differential and Integral Calculus when didactic and meaningful materials are produced. Elements of Tall’s theory on the advantages of the use of computers in Education, as well as the historical importance of the development of an example of a continuous non-differentiable function are presented in this paper. Also, a case of a function defined as limit to a series of functions is explored. In addition, commands and tools which are available in the software named GeoGegra are presented. As a result, we present tools which will hopefully contribute to the practice as well as advancements in Mathematics Education at Higher Education level.
Using dynamic geometry in problem-based engineering courses University of West Bohemia, Czech Republic
It is a modern trend today when formulating the curriculum of a geometric course at the technical universities to start from a real problem originated in technical praxis and subsequently to define which geometric theories and which skills are necessary for its solving. Nowadays, interactive and dynamic geometry software plays a more and more important role in this scenario as it helps to think algorithmically, enables to discuss the solvability of the whole class of geometric problems from different point of views and mainly serves as a first step to variational geometry needed later in geometric modelling. This makes the teaching and learning process more efficient and also more interesting for students. In our contribution we will present this approach on several particular examples taken mainly from the courses for mechanical engineering, of course with emphasis on the application of dynamic geometry. All examples are characterized by a unique structure. First, we present a real engineering problem, then we subdivide it to a number of elementary sub-problems and finally we show how the curriculum and the corresponding teaching procedures are chosen to satisfy the goal to teach the students to solve the real-life problems. We will present our experience with the problem-based teaching of geometry using dynamic geometry software and discuss some aspects of this approach.
Isogonal and isotomial transformations of a triangle University of Miskolc, Hungary
The median and symmedian lines, the centroid and the symmedian point of a given triangle present interesting properties. Part of these properties can be formulated in a more general context for isogonals and isotomials, based on the trigonometric and algebraic forms of the Ceva’s theorem.
In a triangle the isogonal of a line passing through one of the vertices of the triangle is a line symmetric to the bisector of the given angle. Similarly a line passing through one of the vertices of a triangle and its isotomial intersect the opposite side of the triangle in two points which are symmetric to the midpoint of the given side. It can be proven that the three isogonals, respectively isotomials of three concurrent lines which pass through the three vertices of the triangle, are concurrent as well. This property serves as definition for the isogonal respectively isotomial transformation, the image of a given point in this transformation will be the intersection point of the three isogonals, respectively isotomials of the three lines which pass through the given point and the vertices of the triangle. The lecture is aimed to present some of the properties of the isogonal and isotomial transformations, and to visualize them using GeoGebra.
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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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| 1:00pm - 2:15pm | Lunch III: Lunch (incl. Coffee break VIII) | |||||||
| Georg-Cantor-Haus | ||||||||
| 2:15pm - 3:15pm | Workshop 6 | |||||||
| VSP 1.03 | ||||||||
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Interactive GInMA textbooks in creative geometry teaching Deoma, Russian Federation
This workshop is focused on the aspect of visualization in geometry teaching. We use visualization as a basic tool in the study of all major geometric topics. On the workshop, we consider the samples of visualization with GInMA software for teachers and students.
We introduce participants with free DGS GInMA http://deoma-cmd.ru/en/Products/Geometry/ There are many GInMA electronic textbooks in Russian and some of them are translated into English. All the pictures in these books are interactive. We will show how to get the interactive solutions of problems by clicking on the textbook Figures after the GInMA software have been installed from the website. At first, we show the basics of working, simplest instruments, tooltips using and moving inside GInMA textbook. As an example we take GInMA textbook "Transformations". In this 120 minutes workshop we show - different types of the symmetry: with respect to the point, to the line, to the plane, to the circle, - the concepts of polar correspondence and inverse correspondence, - the homography (projective transformation), - we consider triangles centers properties using different transformations. We use properties of the triangle centers to construct the mapping in a convenient way. - we show Steiner's mapping of the straight line in common case and in some special cases, - projective transformations of the planes which transform a group of triangles to the group of regular triangles. Then we consider the samples of visualization with GInMA in solids geometry, show some flexible polyhedrons and possibilities of there moving. Visualization in geometry with the use of GInMA allows to make transformations in a comfortable pace, to perform the necessary intermediate transformations, to provide repetitions. Students make the entire logical chain of transformations and change the parameters of these transformations. Such regime helps students to understand the topic, not mechanically memorize. If you want to become acquainted with GInMA or if you plan to attend the workshop with your laptop, please install free GInMA software from the website http://deoma–cmd.ru/en/Products/Geometry/GInMA.aspx Prior experience with GInMA is not necessary. The knowledge on how to use GInMA and its tools will be introduced. At the end of the workshop you can create yourself the interactive geometric draft. | |||||||
| 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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| 2:15pm - 3:15pm | Workshop 8 | |||||||
| VSP 1.04 | ||||||||
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Dynamic Visual Resources for 16-19 Mathematics Institute of Education, University of London, UK
A resource using dynamic geometry software is being co-developed by teachers in Ontario and England for the new International Baccalaureate Mathematics at standard and higher level, but also useful for other mathematics courses at this level. The aim is to promote student exploration of mathematics in dynamic and visual ways. We will present some of the resources for Geometer's Sketchpad and Cabri - come and look at new ways to explore sequences and series, functions, vectors, calculus...
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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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| 3:15pm - 3:45pm | Tea. & Lea. III/III: Teaching and Learning (mixed general topics) Session Chair: Roman Hašek | |||||||
| VSP 1.03 | ||||||||
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Using GeoGebra software to typeset mathematical text Faculty of Education, University of South Bohemia, Czech Republic
We often have to typeset mathematical text accompanied by visual images when producing materials to support distance learning. Choosing how the necessary images are to be created is an important decision to be made. One possible method is the use of dynamic geometry software. This article will deal with the possibilities of exporting images from GeoGebra software and their subsequent import to the LaTeX typesetting environment and WYSIWYG text editors. Although GeoGebra enables convenient creation of dynamic geometry figures and offers a range of tools for viewing images on the computer monitor, there are limited possibilities for exporting completed figures in the form of images. It is particularly problematic to achieve the uniform appearance of images created with various graphics zoom options. However, one of the basic requirements is that line thickness and font size of text labels among images should be uniform in a particular publication. After examining images created by exporting figures with various settings, we have proposed and tested a procedure that secures a uniform format for images exported from GeoGebra. We have gone on to propose and program an improvement to the module securing the export of figures from GeoGebra software. This modification enables convenient creation of images with the required uniform appearance.
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| 4:00pm - 8:00pm | Excursion | |||||||
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| 9:30am - 10:30am | Keynote V: Keynote (Predrag Janičić) | |||||||
| Jacob-Volhard-Hörsaal | ||||||||
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Challenges for the Next Generation Mathematics Education Software University of Belgrade, Serbia
The next generation mathematics education software should take advantages of the state-of-the-art research in the fields of automated reasoning. The new tools should be able to automatically solve different sorts of mathematical problems, provide understandable solutions, guide the users through the solving process, check if their solutions are correct, provide an appropriate support for interactive theorem proving, etc. In this talk, we will discuss these and other challenges for the next generation mathematics education software, primarily for geometry. For geometry education software, some of the specific challenges are defining appropriate foundations for high-school geometry, automated proving of theorems with human-readable proofs, automated solving of construction problems, linking theorem proving with dynamic geometry tools, automated discovery of theorems, automated discovery of loci, etc.
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| 10:30am - 11:00am | Coffee Mo AM: Coffee break and Poster presentation | |||||||
| Georg-Cantor-Haus | ||||||||
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Computer Modeling in Mathematics Training of Future Civil Specialists 1G.S.Scovorodu Kharkiv national pedagogical university, Ukraine; 2Department of Physics and Mathematics Sciences National University of Civil Protection of Ukraine
Computer modeling is a significant indicator of professional competence for future civil safety specialists, whose professional activities include an ability to apply computer modeling to solve various professional problems by choosing appropriate computer systems.
The above said determines the content of educational process of future civil protection specialists in computer modeling teaching, which goes beyond the applying of ready-to-use computer models and involves thorough improvement of computer modeling skills required for the implementation of all phases of construction of the model and its research. The system of computer modeling skills formation, which covers all periods of educational process (basic mathematics level, interdisciplinary level, professional level). Development and use of computer simulation has a positive impact on the training of future professionals in general, bringing the motivational component in the learning process, facilitating the mapping of interdisciplinary connections, the integrated use of knowledge of the various sciences, enhancing the importance of self-learning and cognitive and research students. A dynamic introduction to fractional calculus 1University of Szeged, Hungary; 2University of Novi Sad, Serbia
Fractional calculus, i.e., calculus of derivatives and integrals of fractional order are getting more and more important in applications, in particular in oscillation theory, biology, etc… However these notions are not part of any standard university curricula, mainly due to the deep mathematical theories needed.
In our talk, we will present a series of dynamic demonstrations developed in Mathematica and Geogebra. We give an interactive introduction to different definitions, properties of “diffintegrals” by simple examples to both math and applied students. The interactive demonstrations will be available on our website www.model.u-szeged.hu.
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| 11:00am - 1:00pm | Assessment: Assessment (general topic) Session Chair: Alla Stolyarevska | |||||||
| VSP 1.04 | ||||||||
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Problem-solving according to Archimedes University of South Bohemia, Czech Republic
The article presents authors' teaching experience at a lower secondary school where teaching materials based on Archimedes' Book of Lemmas were presented to the students.
Book of Lemmas consists of 15 propositions concerning a circle/semicircle some of which are possible to use in geometry teaching at lower secondary schools. The rest of the Lemmas can be presented in higher secondary school classes. Dynamic geometry software plays an important part in these lessons. The article gives an idea of the importance of DGS in geometry teaching and also links Archimedes' propositions to geometry topics at lower secondary school, describes actual course of the classes and other findings from the lessons.
Designing human-like automated assessment to replace proportional penalties for error types University of Tartu, Estonia
We consider two kinds of algebraic exercises in Basic course of Mathematical Logic:
1) Truth-table exercises (filling the truth-table, checking of tautologicity, satisfiability, equivalence and inference, building a formula with given truth-column), 2) Formula manipulation exercises (expression using given connectives, normal forms). Starting from early nineties, our students have solved these exercises in computerized environments that check each step in the solution, give error messages and require correction before the next step. The programs diagnose and count separately errors in order of operations, truth-value/equivalence, syntax, and answer dialog. The truth-table environment also enables to establish the penalty for each type of error and counts the points automatically. The final grading, however, is done by our instructors who are able to take into account additional aspects: 1) What part of the task is solved (if the solution is incomplete), 2) Errors, 3) Solution economy/conformity with the algorithm. For this task the instructors use two additional programs to a) Find the shortest formula for a given truth-column, b) Identify and count inexpedient steps in formula manipulation tasks (24 types of inexpediency). Note also that the formula manipulation environment contains an automated Solver that provides step hints and can be used for finding the ‘ideal’ number of steps. In the paper we identify initial variables for human-like determination of grade for both kinds of exercises and show that they can be obtained by adding only fairly simple components to our existing programs. Further we describe how the teacher can specify the assessment algorithm by entering weights for parts of the task, basic penalties for error types, and spreadsheet-like formulas for possibly nonlinear calculation of penalties from the numbers of errors. Alternatively, the teacher could use a selection of pre-specified grading principles.
Student-Documentations in Mathematics Classroom Using CAS: Between Technical, Subject-Based and Everyday Language TU Dortmund, Germany
Students face many linguistic challenges in mathematics classrooms that use CAS: Not only do they need to use the mathematical language adequately, in addition to their everyday language, but they also need to master the technical language of their digital tool. These challenges become especially material when students have to document their processes and their results. There have already been important results (e.g. Ball 2003) that emphasize the extent to which CAS changes written records, and the need to learn to use the CAS syntax adequately for those written records (Ball & Stacey 2005). In this context, there has been a focus on normative questions on students’ documentation – e.g. emphasis was put on normative questions regarding what might be an adequate documentation for tests (Weigand 2013) or which means may help to structure students’ documentation (Ball 2003).
Since the distinction between CAS syntax and non-CAS syntax seems to be empirically necessary but not sufficient when looking at students documentation, there is a need for a qualitative analysis of different forms of language used in a mathematics classroom that uses digital tools. This contribution will present results of an empirical study that works out different categories that students use in order to document their work. Therefore, different forms of documentation using technical, school (subject-based) and everyday language will be descriptively analyzed. The qualitative study was conducted with 60 students in the 10th grade attending an upper secondary highschool in Germany. In different phases within a school year, after recieving a new CAS, the students worked on paper pencil tests which served as a foundation of the empirical material. Also, clinical interviews were conducted in order to find out more about the different uses of certain registers within a problem solving process. All exercises were within the context of functional reasoning.
Gains and Pitfalls of Quantifier Elimination as a teaching tool Goethe Uni Frankfurt, Germany
Tarski has shown that formulas of first order predicate logic over certain fields can be decided algorithmically and algorithmic progress, especially the method of algebraic cylindrical decomposition . Tarski himself noted that this leads to a decision procedure for elementary geometry as well. Furthermore it gives a systematic way to solve systems of polynomial inequalities over R. Many notions from calculus that are expressed in terms of quantifiers can be formalized and decided for purely algebraic functions. This shows that the method of quantifier elimination is suited for several classes of problems that are relevant in math education at various levels. Thus the question arises, whether this method can be used as a teaching tool. One may hope that having access to quantifier elimination in a computer algebra system may give students the opportunity to explore the mentioned fields of application. Especially one may hope that this may provide a playground to exercise the formalisation step in mathematics. E.g. one may have an intuitive idea of what it means for a function to be convex on an interval but it is a crucial further step to be able to formalize this in the language of predicate calculus. We give examples of all kinds of didactically relevant applications and especially example on the formalizations of notions. Based on this example set we systematize the potential and the inherent problems of quantifier elimination a s a teaching method.
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| 11:00am - 1:00pm | Curriculum: Curriculum (general topic) Session Chair: Csaba Sárvári | |||||||
| VSP 1.03 | ||||||||
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Teaching numerical methods using CAS University of Pécs, Hungary
University of Pécs launched Information Technology (IT) engineer MSc program in 2013. The curriculum involves Numerical Methods as a facultative subject. In my lecture our experience during the teaching of this subject is summarized.
The focus of the subject was solving model problems using Maple, a Computer Algebra System (CAS), sometimes substitute the exact mathematical proofing. During the solutions we have tried to take advantage of opportunities offered by the used Maple computer algebra system. While composing the topics of the course, the rapid development of computer algebra systems was taken into consideration (eg. different methods of solutions of linear equation systems). On the other hand, this way students with limited mathematical skill are also able to understand more complex tasks, such as solution of multivariate interpolations and regressions, or that of partial differential equations. In our lecture we present some real-life examples from the course material.
Extremal Polynomials with Computer Algebra: An Elementary Approach University of Szeged, Hungary
Most math students are familiar with classic Chebyshev polynomials T_n. They are usually introduced as a class of orthogonal polynomials wrt a certain weight function on a closed bounded interval [a,b]. However, it turns out that they can be also introduced as a class of extremal polynomials: namely, the nth (scaled) Chebyshev polynomial is the polynomial which deviates least from the zero constant polynomial on an interval among the monic polynomials with degree at most n. In this talk we investigate the explicit characterization of the generalized Chebyshev polynomials of low degree on some particular subsets of the complex plane: To give the coefficients, roots and norms of these polynomials can be computationally difficult. We consider some approaches to attack the problem by computer algebra and we sketch a pool of possible student projects that can be built around this topic.
The illustrative computational and graphical tools are developed in Mathematica by the author.
Software Support of Functional Line in Precalculus Studies Holon Institute of Technology, Israel
The functional line penetrates and closely interlaces with all areas of mathematics at different levels of studies, often determining their content and methods. Precalculus plays a special role in formation of the corresponding conceptual vocabulary. It is here that students become familiar with different properties, types and operations on functions, master the skills of "reading" graphs of functions, learn to recognize and take advantage of the functional dependencies "hidden" in a problem.
An adequate software, which concentrates students' attention not only and not so much on the demonstration of examples of the concepts being studied, as activates independent creative activity in detection and use of the corresponding properties of the studied material and the connections between them can significantly increase the strength and depth of understanding of the studied matter. This report presents an approach to development and use of such software and its implementation in the author’s non-profit program VisuMatica. Various examples illustrate the technique of dynamic creation and evolutionary development of generalized models as result of live collaborative analysis of the needs and characteristics of the studied material, and proper activities of the students. | |||||||
| 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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| 1:00pm - 1:30pm | Closing: Closing ceremony | |||||||
| Jacob-Volhard-Hörsaal | ||||||||
| 1:30pm - 2:45pm | Lunch IV: Lunch | |||||||
| Georg-Cantor-Haus | ||||||||
