SPECIAL FOCUS PAPER INTERACTIVE MATHS WITH GEOGEBRA Interactive Maths with GeoGebra doi:10.3991/ijet.v6iS1.1620 Daniela Velichová Slovak University of Technology, Bratislava, Slovakia Abstract—This paper brings information on dynamic multi- Cognitive tools allow users to explore mathematical con- platform mathematical software GeoGebra produced by cepts dynamically and have been increasingly discussed Markus Hohenwarter in 2002 for learning and teaching in the recent years, as can be seen in [1] – [5]. Authors mathematics, available free on web in 45 languages. GeoGe- Luis Moreno-Armella, Stephen J. Hegedus, and James J. bra enables production of self-standing dynamic worksheets Kaput provide in [4], p. 103 an historical overview of as interactive java applets embedded in html pages present- symbolization, which can be considered as a kind of the ing dynamic constructions and interactive calculations on evolution of cognitive tools. They identify five stages of Internet. Thus it is enhancing available powerful electronic the evolution of symbolization “from static, inert inscrip- tools suitable for production of instructional materials in tions to dynamic objects or diagrams that are construct- on-line maths education. Several possibilities are presented ible, able to be manipulated and interactive”. on how this useful utility might be used in e-learning solu- The “static inert stage” of symbolization in mathemat- tions as a dynamic interactive platform for calculations and ics consists of a wide spectrum of textbooks and handouts drawings. prepared by teachers for ages and still widely used. The Index Terms—dynamic mathematical software, cognitive second stage denoted as “static kinaesthetic/aesthetic tools, cognitive connections, symbolization in mathematics. stage”, dwells in the use of chalk and marker pens on black-green-white boards and slides with overhead pro- I. INTRODUCTION jectors and allows users to erase and change the inscrip- tion. Besides the flexibility to make changes, these tools Research in cognitive psychology indicates that our also provide opportunities to use several colours to em- brains store knowledge using both words and images. phasize particular notations. In the third “static computa- Instruction that targets and engages both of these systems tional stage” users get a static presentation of the input of representation has been shown to significantly increase they provide to some certain devices such as calculators students' comprehension and retention. Explicitly engag- and graphing tools. Gabriel Salomon [6] favours this type ing students in the creation and usage of non-linguistic of interaction with cognitive tools because he believes representations has even been shown to stimulate and that users can have more opportunities to deal with high increase activity in the brain. Manipulatives are concrete order thinking tasks if they leave low level cognitive ac- or symbolic artefacts that students interact with while tivities to the tools. The last two stages of this evolution learning new topics. They are powerful instructional aids framework focus on the dynamic perspectives of maths because they enable active, hands-on exploration of ab- presentation. In the fourth, “discrete dynamic stage” a stract concepts. Research has shown that computer-based discrete co-action between user and environment is de- manipulatives are even more effective than ones involv- scribed. The users have more control over the process and ing physical objects, in part because they can dynamically the output obtained from the media, because they have link multiple representations together. the ability to change and to manipulate the input dynami- At the beginning of the 3rd millennium we are facing a cally through the first interactive tools that have recently dramatic change in the basic nature of teaching and learn- appear in some mathematics software solutions. In the ing strategies caused by the massive use of new technol- fifth stage named “continuous dynamic stage” users can ogy. We can benefit from this development in general, be actively interacting with the real objects through dy- and in mathematics especially, as currently available dy- namic interactive devices and therefore they can get in- namic and visual learning environments could affect our stantaneous and continuous feedback. perspective in terms of the content and comprehension of The dynamic worksheets prepared in GeoGebra enable mathematics education as well. There is an increasing users to create dynamic mathematical objects such as importance of dynamic linking of multiple representa- graphs and functions and to interact with these objects. tions in facilitating students’ visualization because stu- Assuming that these mathematical objects are the real dents can explore, solve, and communicate mathematical objects of this platform, although the platform itself is concepts in various ways, such as using dynamic multiple virtual in nature, then the continuous interaction between representations and mathematical modelling. Simply these objects and users could be considered as a continu- showing pictures or figures is not sufficient to encourage ous dynamic interaction. Similarly, GeoGebra can be students to visualize or use various representations. Ex- considered as a haptic device detecting the movements of tensive usage of new communication technologies has the slider (that is controlled by the tactile connection of enlarged the pool of cognitive tools and possibilities of the user and the computer mouse) and adjustable objects their application in teaching and learning processes with described in this platform, as mentioned by Karadag and visible consequences and feasible improvements. McDougall in [7]. Cognitive tools have their irreplaceable position in the didactics and educational theory and there exist a lot of ideas on how we can understand their role in education. iJET – Volume 6, Special Issue 1: "VU2009", May 2011 31 SPECIAL FOCUS PAPER INTERACTIVE MATHS WITH GEOGEBRA II. GEOGEBRA FEATURES AND USAGE Teachers can guide students to understand how these GeoGebra is an intuitively controllable application two representations of mathematical objects are con- suitable for all users without any extra mastership in nected to each other. Changes in the superposition of the computer literacy. It can serve for development of in- conic section determining points are reflected in the form structional materials in mathematics in many different of its equation. Following these changes in the algebraic forms, types and styles, and for all levels of mathematical window students can realise the role of the coefficients of education. It is free to be downloaded form a website the bi-quadratic form. Existence of the tangent line to the [10], while nothing but a Java 6 platform is necessary for conic section passing through the chosen point can be its full operation. GeoGebra seems to be particularly easy investigated by changing its position, while equations of and intuitive to learn. Files can be saved in “.ggb” format, the tangent lines appear immediately in the updated form. or as dynamic web pages. GeoGebra can output files as In this way, students are engaged to study relations of pictures (.png) or as encapsulated postscript for publica- mathematical objects in more complex form, in both tion quality illustrations. GeoGebra user interface offers a ways, geometric - synthetic and algebraic - analytic rep- rich graphics menu for drawing various objects, while the resentations receiving thus comprehensive knowledge. complete construction protocol is saved and it appears in After having developed an understanding of the rela- any chosen language from the available 45 versions, tion between the particular type of determined conic sec- therefore no translation is necessary and free sharing of tion and its equation, and tangent line to a conic section developed instructional materials is genuinely supported and its equation, students have the opportunities to create all over the world. Users are encouraged to visit GeoGe- cognitive connections between their previous knowledge bra webpage and GeoGebra user’s forum GeoGebraWiki, and the outcomes of the current exploration. They can a free pool of teaching materials for this dynamic mathe- create various cognitive connections, and by further ex- matics software where everyone can contribute and up- plorations they can be engaged in asking what-if ques- load materials. tions, as: What if would two of the conic section deter- GeoGebra has a built-in Cartesian coordinate system, mining points coincide? What do I have to change in the and accepts both geometric commands (drawing a line 5 determining point’s superposition in order to receive through two given points, a conic section determined in ellipse, parabola, or hyperbola? What if I move point R to different ways, in Fig 1. by 5 points) and algebraic ones be the point on the conic section or to be its interior (drawing a curve with a given equation). Among its more point? interesting features is the ability to draw tangent lines to Many various questions that can be created by teachers algebraic and even transcendental curves at given points, and students are leading to heuristic learning approaches, while equation of this tangent line is available immedi- hands-on exploration of abstract concepts and cognitive ately too. This double representation: the geometric – connections. GeoGebra represents in this sense the exam- synthetic one and the algebraic-analytic