 S_SESSIONS_BROWSE_HIDE_PRESENTATIONS |  S_SESSIONS_BROWSE_TABLE |
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S_SESSIONS_BROWSE_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) S_SESSIONS_BROWSE_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) S_SESSIONS_BROWSE_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) S_SESSIONS_BROWSE_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) S_SESSIONS_BROWSE_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 | |||||||
