Fifth Central- and Eastern European Conference
on Computer Algebra- and Dynamic Geometry Systems
in Mathematics Education

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.

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.

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.

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.

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 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.

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.

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.

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.

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 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.

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).

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.

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.

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.

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.

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.

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.

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.