Mathematics, Computer Mathematical Systems, Creativity, Art

Mathematics, Computer Mathematical Systems, Creativity, Art

Sergiy Rakov (G.S. Skovoroda National Pedagogical University, Ukraine), Viktor Gorokh (G.S. Skovoroda National Pedagogical University, Ukraine) and Kirill Osenkov (G.S. Skovoroda National Pedagogical University, Ukraine)
DOI: 10.4018/978-1-60566-352-4.ch015
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The chapter discusses the possibilities modern IT opens for Mathematics and its applications to real life, in particular to Art – by an example of automated construction of caricatures. The research approach in math education which is based on computer modeling as a methodology of a competency approach is discussed as well as the future educational environment with free access to Internet resources and possibilities to operate software in distributive mode. Many examples are given – from simple illustrations to new math facts invented with original dynamic geometry package DG and proved with the help of computer algebra system Derive.
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Creativity as a common root of producing (discovering) new knowledge in Math and Science and producing new artifacts in Art (Philosophical aspects)

The tradition to oppose Math & Sciences to Art based on dichotomy “non-creative (algorithmic, reproductive) – creative (non–algorithmic, productive)” is wrong and may be concerned only in the framework of usage of the Math & Science and Art artifacts but not in the process of their creation; the processes of constructing new artifacts (both new Math & Science knowledge and new Art products) have the same base – CREATIVITY, and both these processes are similar in both cases (concentration, meditation, modeling, associative and critical thinking etc.). In wider context CREATIVITY is the main goal of a human being, CREATIVITY is the only thing that God can justify him/her (e.g. it is his/her anthropodecium (Berdyaev N., 1931)) – each personally or a mankind in a whole; only CREATIVITY transforms a human life into human being and approach human to God (Berdyaev N., 1931). It does not matter in which area you are working (are you a scientist, an artist, an engineer, a worker, a teacher etc.) – the only thing that matters is: are you CREATIVE or not. CREATIVITY is a God for mankind, the only God and the entire God, the Trinity of God: the origin, the existence and the soul of being. For mankind the human activities, which have sense only if they have the CREATIVE matter, are conditionally divided onto some distinct branches: Technologies, Science, Philosophy, Art, Ethic (in particular religion), each of which plays more or less definite role in definite historical periods of any individuum or community (tribe, nation, country etc.) but during the historical periods one of these branches in their leading role replace one other. Nowadays it seems that technologies, in particular IT, play the main role in civilization progress, stimulating other spheres: Science, Art, Ethic and that is why modern society is called an Information Society (I-Society). But technology itself can’t be the goal of civilization progress and it is obvious that this type of society will be changed by the knowledge society (K-Society), in which the main system leading role would be played by the Science, inflating and stimulating other branches of human activities. Poverty of mankind, growing the number and the scale of global ecological problems should demonstrate bounding of such models of society as I-Society and K–society and maybe Art and Ethic will be the base of a new type of society – Soul society (S–society) and the concept of Sustainable society (economic development, social development and ecological stability) is a step in this direction. IT helps to create crucially new possibilities in all other spheres of human being: human activities, human work, human leisure etc. In particular IT creates a base for new possibilities in Art, Science, Math. etc. (for example computer graphics is beneficial in a lot of IT applications). In the picture below the “magic curves” are shown and all of them are the graph of the one (!) parametric equation , depending from the only parameter – real number. What are these graphical compositions? Mathematics? IT? Art? The reader is proposed to answer these questions for himself/herself and try to estimate the approximate values of p corresponding to each of the Pictures 1– 61. Maybe these pictures are the synthesis of all these – somewhat entity of Math, IT, Art etc. and Math, IT, Art etc. are all only the sides of this entity which may exist separately only in our heads – if our heads are not too creative, but in creative heads they all compose to this entity which is the entity itself.

Figure 1.

Figure 2.

Dynamic drawing “Magic Curves” in the DG environment

Key Terms in this Chapter

Research Approach in Education: Research approach in education is a methodology of competence paradigm of education based at the principles of discovering new knowledge through learning explorations which reflect the ways of research work. Research approach in education is the reflection of educational systems on the challenges of new type of societies – knowledge society (K-society) and sustainable society (S-society) and widely use the possibilities of information and communication technologies (ICT) in computer modeling and computer based laboratories (CBL) with appropriate software for automatic and automate experiments.

Computer Algebra System: A computer algebra system (CAS) is a software program that facilitates symbolic mathematics. The core functionality of a CAS is manipulation of mathematical expressions in symbolic form. The symbolic manipulations supported typically include: simplification to the smallest possible expression, substitution of symbolic, partial and total differentiation, symbolic constrained and unconstrained global optimization,solution of linear and some non-linear equations over various domains, solution of some differential and difference equations, taking some limits, some indefinite and definite integration, including multidimensional integrals, arbitrary-precision numeric operations etc.

Social Constructivism: Constructivism is an epistemology, a learning or meaning-making theory, which offers an explanation of the nature of knowledge and how human beings learn. It maintains that individuals create or construct their own new understandings or knowledge through the interaction of what they already know and believe and the ideas, events, and activities with which they come in contact (Cannella & Reiff, 1994; Richardson, 1997). Knowledge is acquired through involvement with content instead of imitation or repetition (Kroll & LaBoskey, 1996). Learning activities in constructivist settings are characterized by active engagement, inquiry, problem solving, and collaboration with others. Rather than a dispenser of knowledge, the teacher is a guide, facilitator, and co-explorer who encourages learners to question, challenge, and formulate their own ideas, opinions, and conclusions. “Correct” answers and single interpretations are de-emphasized.

Dynamic Geometry Systems: Dynamic Geometry Systems (DGS) or Interactive geometry software (IGS, also called “dynamic geometry environments”, DGEs) are computer programs which allow one to create and then manipulate geometric constructions, primarily in plane geometry. One starts construction by putting a few points and using them to define new objects such as lines, circles or other points. After some construction is done, one can move the points one started with and see how the construction changes.

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