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Mathematics teacher

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Alsu Gibadullina, math teacher, math teacher

Secondary and high school # 57, Kazan, Russia


In recent years Russia actively promoted and implemented the so-called inclusive education (IE). According to the materials of Alliance of human rights organizations “Save the children”: "Inclusive or included education is the term used to describe the process of teaching children with special needs in General (mass) schools. In the base of inclusive education is the ideology that ensures equal treatment for everybody, but creates special conditions for children with special educational needs. Experience shows that any of the rigid educational system some part of the children is eliminated because the system is not ready to meet the individual needs of these children in education. This ratio is 15 % of the total number of children in schools and so retired children become separated and excluded from the overall system. You need to understand that children do not fail but the system excludes children. Inclusive approaches can support such children in learning and achieving success. Inclusive education seeks to develop a methodology that recognizes that all children have different learning needs tries to develop a more flexible approach to teaching. If teaching and learning will become more effective as a result of the changes that introduces Inclusive Education, all children will win (not only children with special needs)."

There are many examples of schools that have developed their strategy implementation of IE, published many theoretical and practical benefits of inclusion today. All of them have common, immaterial character. There is no description of specific techniques implementing the principles of IO in the teaching of certain disciplines, particularly mathematics. In this paper we propose a methodology that can be successfully used as in “mathematical education for everyone", also for the development of scientific creativity of children at all age levels of the school in any discipline.

According to the author, one of the most effective methodological tools for education is a computer mathematics (SCM, SSM). Despite the fact that the SCM were created for solving problems of higher mathematics, their ability can successfully implement them in the school system. This opinion is confirmed by more than 10–year-old author's experience of using the package Maple in teaching mathematics. At first it was just learning the system and primitive using its. Then author’s interactive demonstrations, e-books, programs of analytical testing were created by the tools packages. The experience of using the system Maple in teaching inevitably led to the necessity to teach children to work with it. At first worked a club who has studied  the principles of the package’s work, which eventually turned into a research laboratory for the use of computer technology. Later on its basis there was created the scientific student society (SSS) “GEODROMhic" which operates to this day. The main idea and the ultimate goal of SSS – individual research activities on their interests with the creation of the author electronic scientific journals through the use of computer mathematics Maple. The field of application of the package was very diverse: from mathematics to psychology and cultural phenomena. SSS’s activity is very high: they are constantly and successfully participate in intellectual high-level activities (up to international). Obviously, not every SSS’s member reaches high end result. However, even basic experience in scientific analysis, modeling, intelligent  using of the computer teaches the critical thinking skills, evokes interest to new knowledge, allows you to experience their practical value, gives rise to the development of creative abilities. As a result, the research activity improves intellectual culture, self-esteem and confidence, resistance to external negative influence. It should be noted, however, that members of the scientific societies are not largely the so-called "gifted", than ordinary teenagers with different level of intellectual development and mathematical training. With all this especially valuable is that the student is dealing with mathematical signs and mathematical models, which contributes to the development of mathematical thinking.

From 2007 to 2012. our school (№. 57 of Kazan) was the experimental platform of the Republican study SKM (Maple) and other application software in the system of school mathematical education under the scientific management of Professor Yu. G. Ignatyev of Kazan state University (KF(P)U).

Practical adaptation of computer mathematics and other useful information technologies to the educational process of secondary schools passed and continues to work in the following areas:

  1. The creation of a demonstration support of different types of the lessons;
  2. The embedding of computing to the structure of practical trainings;
  3. In the form of additional courses - studying of computer applications through which you can conduct a research of the mathematical model and create animated presentation videos, web-pages, auto-run menu;
  4. Students’ working on individual creative projects:
    • construction of computer mathematical models;
    • creating author's programs with elements of scientific researches;
    • students create interactive computer-based tutorials;
    • creation of an electronic library of creative projects;
  5. The participation of students  in the annual competitions and scientific conferences for students;
  6. The accumulation and dissemination of new methodological experience.

Traditionally, the training system has the structure: explanation of a new →  the solution of tasks→ check, self-test and control → planning of the new unit  with using analysis. However, the main task types: 1) elementary, 2) basic, 3) combined, 4) integrated, 5) custom. With the increasing the level of training a number of basic tasks are growing and some integrated tasks become a class of basic. Thus, the library for basic operations is generated. The decision of the educational task occurs on the way of mastering the theoretical knowledge of mathematical modeling: 1) analysis of conditions (and construction drawing), 2) the search for methods of solution, 3) computation, 4) the researching.

To introduce computer mathematics in this training system, you can:

  • At demonstrations. For example, with Maplе facilities you can create a step-by-step interactive and animated images, which are essentially the exact graphic interpretation of mathematical models.
  • If we have centralized collective control.
  • If students have individual self-control.
  • In the analysis of the conditions of the problem, for the construction and visualization of its model, the study of this model.
  • In the computations.
  • In practical training of different forms.
  • In individual projects with elements of research.

In the learning process with the use of computer mathematics in the school a library of themed demonstrations, tasks of different levels and purpose, programs, analytical testing, research projects is generated. With all this especially valuable is that the student is dealing with mathematical signs and mathematical models. Addiction to them processed in the course of working with them it’s unobtrusive, naturally, organically.

Mathematical modelling (MM) is increasingly becoming an important component of scientific research. Today's powerful engineering tools allow to carry out numerous computer experiments, deep and full enough of exploring the object, without significant cost painless. Thus provided the advantages of theoretical approach, and experiment. The integration of information technology and MM method is effective, safe and economical. This explains its wide distribution and makes unavoidable component of scientific and technical progress.

Modeling is a natural process for people, it is present in any activity. The introduction with nature by man occurs through constant  modeling of situation, comparing with the basic models and past experience by them. Method for modeling, abstraction as a method of understanding the world is therefore  an effective method of learning. Training activities associated with the creative transformation of the subject. The main feature of educational training activities is the systematic solution of the educational problems. The connection of the principles of developmental education, mathematical modeling, neurophysiological mechanisms of the brain and experience with Maple leads to the following conclusions: the method of mathematical modeling is not only scientific research but also the way of development of thinking in general; computer and mathematical environment (Maple), which is a powerful tool for scientific simulation can be considered as the elementary analogue of the brain. These qualities of computer mathematics led to the idea of using it not only as an effective methodological tool but as a means of nurturing the thinking and development of mental functions of the brain. To study this effect the school psychologist conducted a test, which confirm the observations: the dynamics of intellectual options students  who working with Maple compares favorably with peers. In the process of doing computer math, in particular Maple, are involved in complex different mental functions. It is in the inclusion of all mental functions is the essence of integration of learning, its educational character. And this, in turn, contributes to the solution of moral problems.

Long-term work with computer mathematics led to the idea to use it as a tool for psychological testing. One of the projects focuses on the psychology and contains authors Maple–tests to identify the degree of development of different mental functions. Interactive mathematical environment  gives a wide variability and creative testing capabilities. Moreover, Maple–test can be used not only as diagnostic but also as educational, and corrective. This technology was tested in one of psycho neurological dispensaries a few years ago.

Currently, one of the author's students, the so-called "homeworkers", the second year is a young man with a diagnosis, categories F20, who does not speak and does not write independently. It was impossible to get feedback from him and have basic training until then author have begun to apply computer-based tools, including system Maple. Working with the computer tests and mathematical objects helps to see not only the mental and even the simplest thinking movement, but also emotional movement!

In general, the effectiveness of the implementation in the structure of educational process of secondary school of new organizational forms of the use of computers, based on the application of the symbolic mathematics package Maple, computer modeling and information technology, has many aspects, here are some of them:

  • goals of education and math in particular;
  • additional education;
  • methodical and professional opportunities;
  • theoretical education;
  • modeling;
  • scientific creativity;
  • logical language;
  • spatial thinking, the development of the imagination;
  • programming skills;
  • the specificity of technical translation;
  • differentiation and individualization of educational process;
  • prospective teaching, the continuity of higher and secondary mathematics education;
  • development of creative abilities, research skills;
  • analytical thinking;
  • mathematical thinking;
  • mental diagnosis;
  • mental correction.

       According to the author, the unique experience of the Kazan 57–th school suggests that computer mathematics (Maple) is the most effective also universal tool of new methods of inclusion. In recent decades, there are more children with a specific behavior, with a specific perception, not able to focus, with a poor memory, poor thinking processes. There are children, emotionally and intellectually healthy, or even ahead of their peers in one team together with them. High school should provide all the common core learning standards. It needed a variety of programs and techniques, as well as specialists who use them. Due to its remarkable features, computer mathematics, in particular Maple, can be used or be the basis of the variation of methods of physico-mathematical disciplines of inclusive education.


Research work

The fractal structure’s researching.

Modeling of the fractal sets in the Maple program.

Municipal Budget Educational Establishment “School # 57” of Kirov district of Kazan

    Author:  Ibragimova Evelina

    Scientific advisor:  Alsu Gibadullina - mathematics teacher

    Translator:  Aigul Gibadullina

In Russian


In English



     ( Images - in attached files )

Table of contents:


I. Studying of principles of fractals construction

II. Applied meaning of fractals

III. Researching of computer programs of fractals construction



We don’t usually think about main point of things, which we have to do with every day. Environmental systems are many-sided, ever–changing and compli­cated, but they are formed by a little number of rules. Fractals are apt example of this – they are complicated, but based on simple regulations. Self – similarity is the main attribute of them.  Just one fractal element contains genetically information about all system.  This information have a forming role for all system. But some­times self – similarity is partial.

Hypothesis of the research. Fractals and various elements of the Universe have general principles of structural organization. It is a reason why the theory of frac­tals is instrument for cognition of the world.

Purpose of the research. Studying  of genetic analogy  between fractals and alive and non-living Universe systems with computer-based mathematical mod­eling in the Mapel’s computer space.

Problems of the research. 

  1. studying of principles of fractal’s construction;
  2. Detection of  general fractal content of physical, biological and artificial sys­tems;
  3. Researching of applied meaning of fractals;
  4. Searching of computer programs which can generate all of known fractals;
  5. Researching of fractals witch was assigned by complex variables;
  6. Formation of innovative ideas of using of fractals in different spaces;

The object of research.  Fractal structures, nature and society objects.

The subject of research. Manifestation of fractality in different objects of the Universe.

Methods of the researching.

  1. Studying and analysis of literature of research’s problem;
  2. Searching of computer programs which can generate fractals and experimenta­tion with them;
  3. Comparative analysis of principles of generating of fractals and structural or­ganizations of physical, biological and artificial systems;
  4. Generation and formulation of innovation ways to applied significance of fractals.

Applied significance.

Researching of universality of fractals gives general academic way of cognition of nature and society.


I. Studying of principles of fractal construction

We can see fractal constructions everywhere – from crystals and different accu­mulations (clouds, rivers, mountains, stars etc.) to complex ecosystems and bio­logical objects like fern leafs or human brain. Actually, the idea that frac­tal principles are genetic code of our Universe has been discussed for about fifteen years. The first attempt of modeling of the process of the Universe construction was done by A.D. Linde. We also know that young Andrey Saharov had solved “fractal” calculation problem – it was already half a century ago.

Now therefore, fractal picture of the world was intuitively anticipated by human genius and it inevitably manifested in its activity.

Fractals are divided into four groups in the traditional way: geometric (constructive), algebraical (dynamical), stochastical and natural.

The first group of fractals is geometric. It is the most demonstrative type of fractals, because we can instantly observe the self-similarity in it. This type of fractals is constructed in the basis of original figure by her fragmentation and real­izing of different transformations. Geometrical fractals ensue on repeating of this procedure. They are using in computer-generated graphics for generating the pic­tures of leafs, bush, dimensional structures, etc.

The second large group of fractals -  algebraical. This fractals are constructed by iteration of nonlinear displays, which set by simple formulas. There are two types of algebraical fractals – linear and nonlinear. The first of them are determined by first order equates (linear equates), and the second by nonlinear equates, their na­ture significantly brighter, richer and more diverse than first order equates.

The third known group of fractals – stochastical. It is generated by method of random modification of options in iterative process. Therefore, we get an objects which is similar to nature fractals – asymmetrical trees, rugged coasts, mountain scenery etc. Such fractals are useful in modeling of land topography, sea–surface and electrolysis process etc.

The fourth group of fractals is nature, they are dominate in our life. The main difference of such fractals is that they can’t demonstrate infinite self-similarity. There is “physical fractals” term in the classification concept for nature fractals, this term notes their naturalness. These fractals are created with two simple opera­tions: copy and scaling. We can indefinitely list examples of nature fractals: hu­man’s circulatory system, crowns and leafs of trees, lungs, etc.  It is impossible to show all diversity of nature fractals.


II. Applied meaning of fractals

Fractals are having incredibly widespread application nowadays.

In the medicine. Human’s organism is consists from fractal structures: circulatory system, bronchus, muscle, neuron system, etc. So it’s naturally that fractal algorithms are useful in the medicine. For example, assessment of rhythm of fractal dimension while electric diagrams analyzing allows to make more infor­mative and accurate view on the beginning of specific illnesses. Also fractals are using for high–quality processing of  X–ray images (in the experimental way). There are designing of new methods in the gastroenterology which allows to ex­plore gastrointestinal tract organs qualitative and painlessly. Actually, there are discoveries of application of fractal methods for diagnosis and treatment of cancer.

In the science. There are no scientific and technical areas without fractal calcu­lations nowadays. It happens due to the fact that majority of nature objects have fractal structures and dimension: coasts of the continents; natural resources alloca­tion; magnetic field anomaly; dissemination of surges and vibrations in an elastic environments; porous, solid and fungal bodies; crystals; turbulence; dynamic of complicated systems in general, etc. Fractals are useful in geology, geophysics, in the oil sciences… It’s impossible to list all the spaces of adaptability.

Modeling of chaotic processes, particularly, in description of population models.

In telecommunications. It’s naturally that fractals are popular in this area too. Natan Coen is person, who had started to use fractal antennas. Fractal antenna has very compact form which provides high productivity. Due to this, such antennas are used in marine and air transport, in personal devises. The theory of fractal an­tennas has become an independent, well-developed apparatus of synthesis and analysis of electric small antenna (ESA) nowadays. There are developments of possibility of fractal compression of the traffic which is transmitted through the web. The goal of this is more effective transfer of information.

In the visual effects. The theory of fractals has penetrated area of formation of different kinds of visualizations and creation of special effects in the computer graphics soon. This theory are very useful in modeling of nature landscapes in computer games. The film industry also has not been without fractal geometry. All the special effects are based in fractal structure: mountain landscape, lava, flame, fog, large flows and the same. In the modern level of the cinema creation of the special effects is impossible without modeling of fractals.

In the economics. The Veirshtrass’s function is famous example of stochastic fractals. Analysis of graph of the function in interactive mathematic environment Maple allows to make sure in fractal structure of function by way of entry of dif­ferent ranges of graphic visualization. In any indefinitely small area of the part graph of the function absolutely looks like area of this part in the all . The property of function is used in analysis of stock markets.

In the architect. Notably, fractal structures have become useful in the architect more earlier than B. Mandelbrode had discovered them. S.B. Pomorov, Doctor of Architecture, Professor, member of Russian Architect Union, talks about applica­tion of fractal theory in the architect in his article. Let’s see on the part of this arti­cle:

“Fractal structures were found in configuration of African tribal villages, in an­cient Vavilon’s ziggurats, in iconic buildings of ancient India and China, in gothic temples of ancient Russia .

We can see the high fractal level in Malevich’s Architectons. But they were cre­ated long before emergence of the notion of fractals in the architect. People started to use fractal algorithms on the architect morphogenesis consciously after Mandel­brot’s publications. It was made possible to use fractal geometry for analyzing of architectural forms.

Fractals had become available to the majority of specialists due to the comput­erization.  They had been incredibly attractive for architectors, designers and town planners in aesthetic, philosophical and psychological way. Fractal theory was per­ceived on emotional, sensual level in the first phase. The constant repression lead­ing to loss of sensuality.

Application of fractal structures is effective on the microenvironment designing level: interior, household items and their elements. Fractal structures introduction allows creating a new surroundings for people with fractal properties on all levels. It corresponds to nesting spaces.

Fractal formations are not a panacea or a new era in the architect history. But it’s a new way to design architect forms which enriches the architectural theory and practice language. The understanding of na­ture fractal impacts on architectural view of urban environment. An attempt to de­velop the method of architectural designing which will base in an in-depth fractal forms is especially interesting. Will this method base only on mathematics? Will it be different methods and features symbiosis? The practice experiments and re­searches will show us. It’s safe to say that modern fractal approach can be useful not only for analysis, but also for harmonic order and nature’s chaos, architect which may be semantic dominant in nature and historic context.”

Computer systems. Fractal data compression is the most useful fractal applica­tion in the computer science. This kind of compression is based on the fact that it’s easy to describe the real world by fractal geometry. Nevertheless, pictures are compressed better than by other methods (like jpeg or gif). Another one advantage is that picture isn’t pixelateing while compressing. Often picture looks better after increase in fractal compressing.

Basic concept for fractal computer graphics is “Fractal triangle”. Also there are “Fractal figure”, “Fractal object”, “Fractal line”, “Fractal composition”, “Parent object” and “Heir object”. However, it should be noted that fractal computer graphics has recently received as a kind of computer graphics of 21th century.

 The opportunities of fractal computer graphics cannot be overemphasized. It allows creating abstract compositions where we can realize a lot of moves: hori­zontal and vertical, diagonal directions, symmetry and asymmetry etc. Only a few programmers from all over the world know about fractal graphics today.  To what can we compare fractal picture? For example, with complex structure of crystal or with snowflake, the elements of which line up in the one complex composition. This property of fractal object can be useful in ornament creating or designing of decorative composition. Algorithms of synthesis of fractal rates which allows to reproduce copy of any picture too close to the original are developed today.


III. Researching of computer programs of fractal construction

Strict algorithms of fractals are really good for programming. There are a lot of computer programs which introduce fractals nowadays. Computer mathematic systems are stand out from over programs, especially, Maple. Computer mathe­matics is mathematic modeling tool. So programming represents genetic structure of fractal in these systems and we can see precise submission of fractal structure in the picture while we enter a number of iterations . This is the reason why mathematic fractals should be studied with computer mathematics.  The last dis­covery in fractal geometry has been made possible by powerful, modern com­puters. Fractal property researching is almost completely based on computer cal­culations. It allows making computer experiments which reproduce processes and phenomenon which we can’t experiment in the real world with.

Our school has been worked with computer mathematics Maple package more than 10 years. So we have unique opportunity to experiment with mathematic fractals, thanks to that we can understand how initial values impact on outcome   (it is stochastic fractal). For example, we have understood the meaning of the fact that color is the fourth dimension: color changing leads to changing of physical char­acteristics. That is what astrophysics mean talking about “multicolored” of the Universe. While fractal constructing in interactive mathematic environment we re­ceived graphic models which was like A. D. Linde’s model of the Universe. Perhaps, it demonstrates that Universe has fractal structure.



Scientists and philosophers argue, can we talk about universality of fractals in recent years. There are two groups of two opposite positions. We agree with the fact that fractals are universal. Due to the fact that movement is inherent property of material also we always have fractals wherever we have movement.  

We are convinced that fractal is genetic property of the Universe, but it is not mean that all the Universe elements to the one fractal organization. In deployment process fractal structure is undergoing a lot of fluctuations (deviations) and a lot of points of bifurcation (branching) lead grate number of fractal development varie­ties.  

Therefore, we think that fractals are general academic method of real world re­searching. Fractals give the methodology of nature and community researching.

In transitional, chaotic period of society development social life become harder. Different social systems clash. Ancient values are exchanged for new values literally in all spaces. So it’s vitally important for science to develop behavior strategies which allow to avoid tragic mistakes. We think that fractals play important role in developing of such technologies. And – synergy is theory of evolving systems self- organization. But evolution happens on fractal principles, as we know now.


P.S.  Images - in attached files


Hare in the forest

The rocket flies



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