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

“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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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 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.

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...

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.

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.