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Computer algebra systems together with dynamic geometry systems provide a user with a means of investigation of the mathematical background of various real world phenomena. The combination of symbolic capabilities with simply-made dynamic models gives us a means of introducing students to, what are in fact, complex problems. We will present selected examples of the modelling of such problems based mainly on the joint use of the computer algebra system wxMaxima and tools of the geometry and algebraic package GeoGebra.
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1 INTRODUCTION
This paper deals with the use of computers as tools of mathematical investigation in mathematics teacher training courses at the Faculty of Education where the author works. We will present selected examples that we use with students of pre-service and in-service mathematics teacher training courses in order to give them insights into some mathematical facts and phenomena as well as to promote their understanding of computers as tools of investigation. First we will provide the reader with a brief overview of various examples of the utilisation of computer algebra systems (CAS). Then we will focus particularly on the joint use of the computer algebra system wxMaxima (wxmaxima.sourceforge.net) together with the tools of GeoGebra (www.geogebra.org) as a geometry and algebraic software. In particular we will take advantage of GeoGebra's graphic functions and its built-in spreadsheet, which has proved an effective means for iterative numerical computation.
2 MATHEMATICAL INVESTIGATION BY MEANS OF COMPUTER
The concept of "mathematical investigation" is well established as a method of teaching mathematics, based on the exploration by students of a mathematical situation. Various publications, articles and lectures (for example, Burghes, 1984; Wolfram, 2010; illuminations.nctm.org; math4teaching.com) are devoted to this method as well as various projects which are aimed at the improvement of mathematics education, and give "mathematical investigation" considerable attention (for example, Baptist, Miller and Raab (2010) www.fibonacci-project.eu).
This paper aims particularly at the "use of computers as tools of mathematical investigation". This means, within the scope of the paper, the use of computer programs in such a way that reveal the mathematical essence of a problem or phenomena compared to its mere computation which reveals only a result. Computer use enables a mathematics teacher to carry out the process of the exploration...