**SCIENCE AND EDUCATION A NEW DIMENSION**

**PEDAGOGY and PSYCHOLOGY**

**Vol. 5. 2013**

**Tarasenkova N. Peculiarities of encoding geometrical concepts**

Abstract: The article focuses on the problem of securing understanding and proper assimilation of geometrical concepts by the students. With an example of the concept of an isosceles triangle the complexity of students’ mental activity in the process of assimilation of the concept is revealed. The features of verbal, verbal and graphic, and content and graphic codes of geometrical concepts are characterized. The conclusion is made that it’s of importance to use the content- graphic interpretation of geometrical concepts in the process of teaching geometry at all stages, and that only in this case one of the ultimate goals of the formation of each concept, namely – creating the folded structure of this knowledge, can be achieved. The contents of the folded structure of the concept, take an isosceles triangle as an example, include the following components: a) shell structures – the term "an isosceles triangle" together with the code content- graphic interpretation of the concept (it is an image of an isosceles triangle with a horizontal base and a vertex located in the upper half-plane relative to the line containing the base of the triangle, equal sides are symbolically marked by an equal number of lines), b) the core of the structure – the meaning of the figure in which, firstly, the reflection of the essential properties of the concept of an isosceles triangle is concentrated and this is enclosed in its definition, and secondly, the very image of one representative of the multitude of isosceles triangles in one of the possible locations in space, and c) the structure operator – a one-to-one correspondence between the image and its meaning. A comparison of cognitive acts in the process of applying geometrical concepts with folded structures (codes) which are characteristic for the teacher, and extended structures which are typical of the students at the initial stage of learning geometrical concepts, results in the necessity of developing a special method of concept formation in the students. This method comprehensively considers all semiotic peculiarities of any learning objects, including geometrical concepts. The teacher should master encoding and decoding as the type of signs and symbolic activity, correctly implement it, and be able to generate the appropriate knowledge and skills in students. Teaching how to encode the concept on the content and graphic level (and not just on the terminological and verbal-graphic level, as is common) should be considered as a separate methodological problem.

*Keywords: geometry, concept acquisition, signs and symbolic means, encoding, understanding, Geometry skills*

tarasenkova_n._peculiarities_of_encoding_geometrical_concepts.pdf | |

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**Garov Kosta Andreev, Todorova Elena Hristova A System of Classes on the Topic of Spreadsheets Meant to Cultivate Key Competences**

Abstract: This work is aimed at sharing a set of classes dedicated to the topic of Spreadsheets taught within the Information Technologies subject to students from the 5th-7th grades in the Bulgarian schools. This system will contribute to the formation of some key competences in students as stipulated in the European Framework of Reference. These competences have been adopted by the European Parliament and the Council of the European Union. We demonstrate herein an approach to cultivating digital competences and their relationships to other competences. Interdisciplinary relationships are identified between Information Technologies and other subjects on the curriculum related to the teaching of the topic of Spreadsheets.

*Keywords: knowledge, skills, key competences, Information Technology, spreadsheets.*

garov_kosta_andreev_todorova_elena_hristova_a_system_of_classes_on_the_topic_of_spreadsheets_meant_to_cultivate_key_competences.pdf | |

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**Malova I.E., Gureeva I.L. The basics of creating workbooks for situations for use of computer presentations in the educational process**

Abstract:The article considers the advantages of using information technologies in the educational process from the standpoint of students, and from positions of training: save time; help improve the quality of education; enable the efficient organization of the future work. There are two groups of situations using a computer: the computer is used at the lesson (lesson in high school); the computer is used outside of the lesson (lesson). The first group includes the situation: you use one computer for demonstrations (for example, used a computer presentation, demonstrated by a teacher; demonstrates the effects of and/or the process of work with the computer program or studying educational one); students work for multiple computers, doing individual tasks requiring special computer programs; students work together with the network program (such as NetMeeting). The second group includes the situation: the pupils (students) working with electronic resources provided by the training (for example, with computer presentation on a particular subject); the pupils (students) working with electronic resources of the Internet (for example, carry out tests in a mode on-line, search for information); the pupils (students) themselves create electronic resource (for example, in preparation of homework, reports, projects). Based on the requirements of the subject completeness, self-pupil attainment of reflection reveals the target, substantive and organizational basis for creation of pages of the workbook in a situation of using computer presentations in the educational process on the lesson in a school, or a lesson in school. Examples of school and higher courses of mathematics (computer science), methods of teaching mathematics, methods of teaching Informatics providing three objectives for the use of work exercise: the inclusion of students in the process of goal setting and planning to study a topic; to explore new theoretical material; identify ways to work with the tasks of a certain type.

*Keywords: workbook, information technology, computer presentation, methods of teaching mathematics, methodology of teaching Informatics.*

malova_i.e._gureeva_i.l.__the_basics_of_creating_workbooks_for_situations_for_use_of_computer_presentations_in_the_ educational_process.pdf | |

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**Millousheva-Boykina Dobrinka Vassileva, PhD & Milloushev Vassil Borisov, DSc.**

The Activity of “creating” mathematical problems.

The Activity of “creating” mathematical problems.

Abstract: In the paper is motivated the necessity to study the activity of creating mathematical problems in theoretical plan (with the analogy to the main activity in the education in Mathematics – solving problems). For this purpose there are presented the views of a number of scientists specified in the field of methodics of education in Mathematics in Bulgaria and abroad. An analysis of a lot of cited publications is made, as a result of which there are made appropriated conclusions. In these conclusions is presented the actually of the topic – the construction of mathematical problems is a suitable activity in education in solving problems, moreover a lack of full and targeted investigations on the following two aspects of the problem is found in these publications: first, essential and structural characteristics of the activity of creating of educational math problems; second, its functional characteristics and in particular those of them which refer directly to the mastering the methodics of education in solving problems. All this requires regular study (in terms of current theoretical aspects) of the activity of creating mathematical problems (ACMP). The main purpose of this paper is the study of this activity in order to make a detection, complementarity and systematization of its basic and functional characteristics. As a result of this study is constructed a model of the content of the activity of ACMP, which is presented schematically. The scheme reviews also the relations between its stages and sub-stages, so that the model represents the structure of the ACMP.

*Keywords: problem, model, the activity of creating mathematical problems.*

millousheva-boykina_dobrinka_vassileva_phd__milloushev_vassil_borisov_dsc.__the_activity_of_creating_mathematical_problems..pdf | |

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**Turgunbaev R.M., Allambergenov I.H. Some approaches implement the elements of continuity in the teaching of**

mathematical analysis in the system of academic high schools, university.

mathematical analysis in the system of academic high schools, university.

Abstract. Effective implementation of the continuity of educational programs is a necessary condition for the formation of intellectual, personal and behavioral qualities, knowledge and skills of students, creating the potential for further personal development. In the articles, which deals with the problem of continuity in the implementation of the teaching of mathematics in the college system (schools, lyceums) school educational process of high school and college (schools, lyceums) are treated separately. These works do not represent the methodologies of rational logic of construction and content of the material accounting hierarchy process of assimilation. The article deals with the technique training elements drafting table using the approach described V.P.Bespalko, which shows the level of knowledge and level of abstraction learning educational information on the elements of mathematical analysis in academic lyceums. The example of table training elements on "Sequences and their limits." Specifying the required mastery of educational information helps to clarify the standards, training programs can be used to write textbooks, manuals, books of problems for schools, colleges and universities, to create a single educational space in mathematics.

*Ключевые слова: учебные программы, преемственность в обучений, качество усвоения, система задач, математический анализ, Узбекистан*

turgunbaev_r.m._allambergenov_i.h._some_approaches_implement_the_elements_of_continuity_in_the_teaching_of_ mathematical_analysis_in_the_system_of_academic_high_schools_university.pdf | |

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**Ivanova V., Boykova K.**

Intensification of the learning process through using information and communication technologies

in the English courses

Intensification of the learning process through using information and communication technologies

in the English courses

Abstract: in the article there is described the use of information and communication technologies (ICT) in the process of teaching English at the Faculty of Mathematics and Informatics (FMI) at Plovdiv University “Paisii Hilendarski”. ICT is used for grading, assigning project tasks and communicating with students. However, by using ICT, undergraduates learn not only a foreign language but also competencies useful for their future careers such as accessing information efficiently and effectively, evaluating information critically and competently and using information accurately and creatively. English language learning at FMI is assisted by a dedicated website, which contains information about the language classroom policy, English classes schedule, tasks and deadlines for project assignments and others. Communication between teachers and students is achieved through announcements, email and discussion forums as well as online debates and real-time chats. The need for intensification of the learning process is explained in the article and the meaning of the term is specified. There is an outline of the blended learning approach applied in the education of English at FMI in which the basic structure of the traditional course is retained and technology resources are used to supplement traditional teaching in the classroom. This approach makes the process of learning interactive and caters for students’ different needs and learning styles so students develop a greater interest in the subject matter of the course. The learning environment is more flexible and the teaching methods are responsive to the diverse needs of students so they become more motivated and active in class. Also, there is defined the use of tests in English for self-study, which are developed on the basis of every textbook unit studied in class, with the purpose of obtaining long-term knowledge and skills in the foreign language. The results of a survey conducted with students from FMI are analyzed regarding the use of ICT in the learning process.

*Keywords: information and communication technologies (ICT), intensification, blended learning, self-study tests, interactivity*

ivanova_v.__boykova_k.__intensification_of_the_learning_process_through_using_information_and_communication_technologies_in_ the_english_courses.pdf | |

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**Chashechnikova O. Teaching orientation mathematics on development of creatiev personality**

Abstract: Development of the system of forming and development of creative personality, creative thought in the conditions of the differentiated teaching of mathematics foresees the analysis of basic concepts in the context of research, researches of pedagogical pre-conditions of construction, organization and realization of introduction of this system. Theoretical analysis of the researches of problem testifies to the ambiguousness of interpretations of key concepts, about the necessity of clarification of the proper concept-category vehicle. The specific of display of creative thought in educational-cognitive activity during the study of mathematics determines a necessity to examine creative thought as higher form of independent thought of man in the process of implementation of untraditional tasks, untraditional decision of traditional tasks, as creation in the process of thought of new facilities, methods and receptions the system of which can be applied in subsequent for implementation of more wide circle of tasks use, realization of intellectual, creative capabilities during intellection. The modern state of problem of development of creative personality is analysed in the article, creative thought of student in the process of teaching mathematics. The creative thinking characteristics, which can be diagnosed in teaching of mathematics, have been distinguished. The ways of decision of problem are offered in the conditions of the differentiated teaching. The differentiated going is examined near the study of mathematic, during realization of which quality of knowledges and abilities of schoolсhildren; efficiency of forming of creative thought. Possibilities to forming creative thought are considered in the process of learning mathematics. Differentiating is necessary strategies and teaching tacticians, that foresees differentiation of rate and teaching style. Experience of creative activity, purchased a schoolboy in the process of teaching mathematics, does more effective a study by him other educational objects.

*Keywords: creative thought, teaching mathematics, educational-cognitive activity*

chashechnikova_o._teaching_orientation_mathematics_on_development_of_creatiev_personality.pdf | |

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**Nestoruk N. Experimental research in the preparation system of students majored in electromechanics**

Abstract: The paper highlights qualitative preparation of students majored in electromechanics to conduct experimental research in studying engineering sciences, which is determined by presence of the basic element – availability of knowledge in electromechanics, gained in studying general technical subjects, that can be shifted onto technical subjects by students majored in electromechanics and use it while studying common electro-mechanical devices when conducting experimental research. The paper presents a summarized structure of knowledge in electromechannics of students, who specialize in the given field, defines its place and role while conducting experimental research in the process of studying technical subjects, specifies the meaning of some notions related to the field of polytechnic training in electromechanical context and emphasizes that one of the basic components of professional training for students majored in electromechanics must be principal polytechnic training that combines common scientific and common technical subjects while the key role is given to electromechanics. One of the main tasks of electrical engineering is to give general knowledge and notions to a future engineer electrician, without which he will not be able to use effectively electromechanical devices necessary to provide reliable and economic operation of various technical facilities in his professional activity. To fulfil professional functions that are connected with installation, repairs and maintenance of complicated equipment, specialists in this field of industry should know not only the basics of operation of electrical mechanical devices, operating rules and repairs procedure, but also know how to analyze different parameters of electromechanical devices, fulfil logical and mathematical processing of instrument readings and signals provided by controlling equipment. According to this, successful mastering of the course in electrical engineering facilitates both deeper learning of

other technical courses and qualitative arrangement and conducting experimental studies in the process of learning technical courses.

*Keywords: engineer-pedagogue, students majored in electromechanics, electro-technical knowledge and skills, experimental*

studies, technical subjects.

studies, technical subjects.

nestoruk_n._experimental_research_in_the_preparation_system_of_students_majored_in_electromechanics.pdf | |

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**Akulenko I. To the problem of investigating the profile school teacher’s methodical preparation**

Abstract. Teacher education has become an area of considerable interest among policymakers in many countries over recent years, a development that underlines the central importance of teacher knowledge to quality learning. The work associated with teacher preparation as well as experience gained in many of IEA’s studies, such as TIMSS, led to a request for an in-depth investigation of teacher preparation and training, particularly in terms of the subject area of mathematics. The first IEA study of teacher preparation and training of primary and lower secondary mathematics teachers (TEDS-M 2008), was focused on how teachers are prepared to teach mathematics in primary and lower secondary schools. Thou Ukraine weren’t involved to this investigation, the research on this point was organized. The main purpose was to study what is the level and depth of the mathematics and related teaching knowledge attained by prospective Ukrainian secondary teachers. The research targeted future teachers who are in their final year of training before they are eligible to become practicing teachers of mathematics in secondary school. The task was to measure the intended and achieved knowledge of mathematics, mathematics pedagogy and general knowledge for teaching of future teachers in their last year in the sampled teacher education programs. The questionnaire was similar to the TEDS-M research. Four subject areas assessed at the future teachers of secondary school: number, algebra, geometry, and data. Sub-domain numbers included: whole numbers, fractions and decimals, number sentence, patterns and relationships, integers, ratios, proportions, and percent, irrational numbers. Sub-domain algebra included: patterns, algebraic expressions, equations/formulas and functions, calculus and analysis, linear algebra and abstract algebra, number theory. Subdomain geometry included: geometric shapes, geometric measurement, location and movement. Sub-domain data: data organization and representation, data reading and interpretation, chance. Framework for the mathematics pedagogical content knowledge included two sub-domains “mathematical curricular knowledge” and “knowledge of planning for mathematics teaching and learning. Some analyses of the results of prospective Ukrainian secondary teachers are given in the article.

*Keywords: prospective secondary teachers, teacher preparation and training in mathematics, mathematics pedagogy*

akulenko_i._to_the_problem_of_investigating_the_profile_school_teachers_methodical_preparation.pdf | |

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**Biliunas A. Some form of training pupils in the proof of mathematical statements**

Abstract. This article is devoted to some form of training pupils in the proof of mathematical statements. We consider the notion of „a form of organization learning” as a way to organize learning activities, which is governed by a predetermined schedule; outward expression of a coherent work of the teacher and the pupils carried out in a certain order and in a certain mode. Forms of training provide the answer to the question of how to be organized academic work in view of the who, where, when, and for what purpose is learning. Generally accepted in the forms of organization of educational work is the collective – the lessons in the school, lectures and seminars in high school, field trips, extracurricular activities, etc. and the individual – individual work, degree and course projects, etc., which differ significantly in the degree of autonomy of the cognitive activity pupils and the degree of leadership training activities on the part of the teacher. In our research we develop a methodical system of mathematical statements, so in this article we focus precisely on those forms of learning, which we consider is more suitable for the implementation of our method. And senior pupils to develop skills to prove mathematical statements are effective forms of organization of training pupils to the proof of mathematical statements: front-work, group form of organization and bringing independent search for proof. Front work of the organization should be the proof of statements by the heuristic conversation and use it in the event that the new algorithm or a new method (the method of reception) proof. To understand the use and retention of skills composed algorithm, a rule-reference, a new method or way to proof it is advisable to use a form of group Independent search of proof or conducting independent the proof of should be made by analogy to the plan or the known proof. The content of education and age characteristics of pupils require adequate, appropriate forms of education, determine its character. The main forms of organization and appropriate forms of education pupils for proofs of mathematical statements is the front-work, group form of organization of the evidence and an independent search for evidence or conducting independent proof. But the combination of various components, such as in the learning process, duration, dynamic structure, methods of organization, methodological support, allow the creation of the most diverse forms of learning.

*Keywords: forms of learning, proof mathematical statements, front work, group form, an independent search.*

biliunas_a._some_form_of_training_pupils_in_the_proof_of_mathematical_statements.pdf | |

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**Vasilenko I. The influence of the historical and cultural mathematical quest for the formation of cognitive students' interest in mathematics**

Abstract. The article highlights the necessity to conduct extracurricular work for рupils who have chosen humanitarian profile of training. They should solve the tasks related to literary, historical, social problems connected with mathematics. The article analyzes and proves the need of such a type of extracurricular work in a modern society as a quest, i.e. the amateur intellectual competition, basedon a subsequent fulfillment and execution of tasks prepared for teams or individual players. The publication defines the goal of the quest. The preparatory stages and organizational recommendations for the historical and cultural mathematical quest «The Golden Horseshoe of Cherkasy region» are analyzed. In the course of the quest the research experiment of investigating the dynamics of students' cognitive interest in mathematics was implemented. The poll (in the form of diagrams, tables) and the analysis of the quest at various stages are displayed, and the results of the survey of experimental research are analysed. The results of students' cognitive interest in mathematics are implemented in such a form of extracurricular work as a historical and cultural mathematical quest. The subject of further experimental research is formulated. An example of one of the fragments containing historical in formation and tasks relating to the town of Cherkassy is given the article. To formulate the problem, we have taken historical information relating to the local history and underlined certain facts (about the life and personality of prominent figures, dates of important events, etc.) and created a problem, the solution of which enables to perform the tasks.To solve the problems the students previously get acquainted with local historical material concerning a specific point of interest, then work on their own tasks, isolating in it the known and unknow nhistorical and mathematical facts, then start solving a number of mathematical problems. Their successful solution enables to perform the tasks as well. Extracurricular work in mathematics for students in school plays an important role in the educational process that is why the necessity of completing additional tasks (problems) related to the themes of the native land, provides emotional, scientific and comprehensive knowledge of the native land, improves cognitive interest and level of educational achievements in mathematics. The experiment showed a good impact on a students' cognitive interest in mathematics of such a form of extracurricular work as a historical and cultural mathematical quest on the process of forming a students' cognitive interest in mathematics.

*Keywords: cognitive interest, extracurricular work, historical and cultural mathematical quest.*

vasilenko_i._the_influence_of_the_historical_and_cultural_mathematical_quest_for_the_formation__of_cognitive_stu-_dents_interest_in_mathematics.pdf | |

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**Voloshena V. Mathematical modeling in the process of formation of practical competences of students**

Abstract. One of the main tasks of modern natural-mathematical education is forming pupils’ scientific world view. An important role in this process is given to teaching students mathematical modelling. Having universal method of scientific cognition features, mathematical modelling in school studying plays a role of integrating component of a subject content of nature-studying disciplines. The analysis of scientific and methodical literature sources and school studying process shows that, in spite of wide usage of mathematical modelling method in studying program subjects, forming proper students’ abilities takes place mainly at mathematics lessons. An our opinion this fact has a big negative influence on didactic effectiveness of usage of this method during studying process. We are sure that overcoming such narrowness is possible under the condition that forming mathematical modelling abilities takes place not only during mathematic lessons, but also during studying all nature-studying subjects. If during studying natural events and laws and during studying problems solving focus students’ attention on the hierarchal process of mathematic model building, the process of mathematic modeling studying becomes more effective, and studying of mathematic modeling as building and researching of models’ succession helps forming students’ understanding the modeling character of cognition and knowledge scantiness, helps developing students’ cognitive and creative abilities, on this basis, and because of the need to fully explore the most important elements of the method of mathematical modeling in the senior school, prompted a separate elective course on study of the method of mathematical modeling.

*Keywords: mathematical design, mathematical model, applied tasks, competence.*

voloshena_v._mathematical_modeling_in_the_process_of_formation_of_practical_competences_of_students.pdf | |

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**Godovaniuk T.L. First classes of methods of teaching mathematics**

Abstract. The article highlights the features of methodical preparation of future teachers of mathematics during first classes of methods of teaching mathematics in Pedagogical University. It is determined the place of the course «Methods of teaching Mathematics» in the system of methodical training of future Math teachers in pedagogical university and argues that qualitative methodology of preparing students for teaching mathematics in secondary school is an important aspect of professional training of future teachers. Reveals the structure of the course and submits its brief description. The first part - the general methods of teaching mathematics (108 hours studied for one semester at the third year), special methods of teaching mathematics in elementary school (135 studied during two semesters at the fourth year), special methods of teaching mathematics in high school (135 hours, studied for two semesters at the fifth year). Overall, 432 hours are given for the course, 1/3 to 2/3 of total hours is given to independent studying. The purpose of the article is to reveal features on the first classes on methods of teaching mathematics in pedagogical university. On first classes students become familiar with the methods of teaching mathematics as a teaching science and an educational discipline. It is proposed to include historical overview on the development of methods of teaching mathematics in Ukraine and the countries to which it belonged before (considered 4 periods). We describe how students highlight the structure, content, purpose and objectives of the course. The necessity to become familiar future teachers with the documents is shown in regulating the learning process on Mathematics in secondary school. Revealed the ways students become conscious in studying regulations and school textbooks. Marked expediency of disclosuring the main methodological approaches to teaching mathematics in modern school (active, self-oriented, developing, axiological and competence) and ways to implement them are illustrated on specific examples. The essential characteristic of the first classes on methods of teaching mathematics is using innovative forms, methods and means of education.

*Keywords: methodological training, methods of teaching mathematics, first classes, students, future teachers of mathematics, methodological approaches to studying.*

godovaniuk_t.l._first_classes_of_methods_of_teaching_mathematics.pdf | |

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**Homenyuk A.V., Zabransky V.Ya. Competence-based approach in the process of teaching algebra for secondary school students and means of its implementation**

Abstract. The article realizes the content and nature of the concept of "competence approach" and the means of its implementation in the process of teaching algebra. Competence-based approach in the process of teaching algebra for secondary school students is treated as the focus of the learning process to the formation and development of general and specific subject competences in accordance with the content of education and the requirements to the results of students' learning activity as defined by the State Educational Standards and Programs in Mathematics, while ensuring the development of key personality competencies. The basic purpose for the practical implementation of competencebased approach in teaching school algebra is to define different levels of the competencies hierarchy: trans subject competencies, general subject competencies and special subject competencies. Trans-subject (cross-curricular) competencies can be represented in the form of "umbrella" over all learning process, they are often called the "key", or "base" competencies. In teaching algebra in secondary school, for example, they might be as follows: to think laterally (outside the box), to understand belonging to different sets, to acquire and, respectively, use modern technology. General subject competencies are acquired by the students during the study of a subject in all forms of secondary school. They are as follows: relation to Algebra as an integral part of the general culture of mankind, a necessary condition for a full life in a modern society based on acquaintance with the ideas and methods of mathematics as a universal language of science and technology, an effective means of modeling and investigation of processes and phenomena of the world, mastering the mathematical language by the students, their understanding of mathematical symbols, mathematical formulas and models that allow to describe the general properties of objects, processes and phenomena. Special subject competencies are those which are acquired by the students in the process of algebra studying. They are: to have knowledge about the number (from natural numbers to real), to perform oral, written, and instrumental calculation; to learn the system of functional concepts; to know how to use the functions and their graphs to describe dependencies between variables, descriptions of events and processes. Among the means of the competence-based approach, the following methods are proposed: the formation and development of reflection at the lessons of algebra, the use of project techniques in the process of learning algebra, the implementation of an applied pattern in teaching algebra.

*Keywords: competence, competency, competence-based approach to teaching algebra, reflection, project technology, applied pattern in teaching algebra.*

homenyuk_a.v._zabransky_v.ya._competence-based_approach_in_the_process_of__teaching_algebra_for_second-_ary_school_students_and_means_of_its_implementation.pdf | |

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**Gorbachev V.I. Methodolody of the number world-image**

Abstract. The article deals with the categories of the number world-image. It studies the principles and methods of teaching number systems.

*Keywords. Methods of teaching mathematics, number systems,world-image.*

gorbachev_v.i._methodolody_of_the_number_world-image.pdf | |

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**Gorda I.M. Computers technologies as an integral component in monitoring of higher agrarian educational establishments**

Abstract. Execution of processing, storing and presentation of results of monitoring is time consuming for a teacher. As a result there is a need to use computers technologies in monitoring in higher agrarian educational establishments. Many scientists study the issue of usage of computer technologies in education but there are still not many researches using computers technologies for processing of results of monitoring in teaching Math in higher agrarian educational establishments. The article describes using of computers technologies in two types of monitoring in higher agrarian educational establishments: monitoring of students’ academic level in Math and monitoring of sub-faculty. In particular during monitoring of students’ academic level in Math we used electronic journal developed in chart MS Excel processor. The structure of electronic journal and ways of using it are described in the article. The journal allows a teacher of Math to do survey of students' grades during academic year; to watch grades of each student at the regular bases in order to

find out what affects efficiency of students’; in provides automatic calculations of students’ points in Math for every module in particular and Math as a subject. The chart MS Excel processor is useful in usage in sub-faculty monitoring. The experience proves that it gives opportunity to store and systematize the results of activity each sub-faculty member during academic year. It also helps to determine automatically number of the points of every teacher for determination of quality level of his work and to print out the results. Besides, Microsoft Access was used by us during administrative sub-faculty monitoring, in which "Teachers" database was created. In allowed to store electronically general information about sub-faculty staff which is useful for useful for higher school administration. The experience of using computers technologies in monitoring of higher agrarian education establishments proves that they enhance monitoring.

*Keywords: quality of mathematical training of students, the quality of activity of Mathematics teachers, monitoring, collecting, processing, storage, computer technology.*

gorda_i.m._computers_technologies_as_an_integral_component_in_monitoring_of_higher_agrarian_educational_es-_tablishments.pdf | |

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**Yevtushenko N.V. History of Mathematics as part of the general cultural content competence of teachers of Mathematics**

Abstract: The article is devoted to the theoretical aspect of the history of Mathematics study as part of the general cultural competence of the Mathematics teachers. The notion of general cultural competence of the Mathematics teachers, its content and structure are considered. In particular, the overall cultural competence of the Mathematics teacher is defined as a professionally significant integrative quality of personality that combines motivationally significant, cognitive, and emotional components, which ensures the unity of common pedagogical, mathematical, information culture, determines the ability of a subject to participate in educational activities and focus in contemporary social and cultural space. The state of common cultural competence by means of the history of Mathematics was determined:

– ability to improve and develop one’s intellectual and cultural level through the history of Mathematics;

– knowledge of the basic points of the history of Mathematics;

– ability to reproduce the wealth of the actual content of the historical development of Mathematics, skills and experience

of purposeful and logical interpretation of the history of Mathematics. The general cultural potential of the history of Mathematics, which includes the basic functions of Mathematics in culture: the worldview (formation of the view of the world), cognitive (satisfaction of human needs for knowledge), methodological (providing methods, techniques, which made the study of processes and phenomena in the various sciences), social and practical (the ability to change and transform the world) was described. The role of the history of Mathematics as an effective mechanism for the development of common cultural competence of Mathematics teachers, due to its characteristics: interdisciplinarity, complexity, integration and dynamism was discovered. The urgency of historical and mathematical material use for the development of the main features of key competencies - multifunctionality, overobjectification, interdisciplinarity is mentioned. The peculiarities elements of history of mathematics being used as the component of common cultural competence with are mentioned:

1) motivationally significant – revealing interest in the history of mathematics as a part of human culture;

2) cognitive – knowledge of the basic provisions of the history of mathematics;

3) acting – the ability to reproduce the richness of the actual content of the historical development of mathematics;

4) emotional – the ability to perceive and express emotion details the history of mathematics.

Special attention is paid to the need of a teacher to have very high level of general cultural competence and a clear understanding of the cultural role of mathematics.

*Keywords: history of Mathematics, general cultural competence, Mathematics teachers*

yevtushenko_n.v.__history_of_mathematics_as_part_of_the_general_cultural_content_competence_of_teachers_of_mathematics.pdf | |

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**Karupu O.W., Oleshko T.A., Pakhnenko V.V. Analizing practice of teaching higher mathematics to Ukrainian and foreign students in National Aviation University**

Abstract: Foreign students in National Aviation University may choose to study in English, Russian or Ukrainian. We present the results of our 12-year experience of teaching Higher Mathematics in English to foreign and Ukrainian students in National Aviation University. As English is one of the official languages of ICAO (International Civil Aviation Organization), this trend of education is urgent and favors the further improvement of professional skills of the university graduates. Beside this, foreign applicants may study without a preparatory course of Ukrainian or Russian. The modular technology of studying which is one of the main components of the European Credit Transfer System is applied. This discipline is cоmрlеtеly covered with textbooks and manuals published in our university. Syllabuses of these courses in English which are compliant with ECTS are fully developed. Therefore we study the specific problems of methodical, didactic and organizational nature which arise while teaching higher mathematics in English-speaking groups. Some of these problems arise as a result of different approach to teaching mathematics in secondary schools in Ukraine and countries native for our students. These include, first of all, a very low level of knowledge on trigonometry and stereometry, which is detected for many students. Another essential problems are insufficient skills of foreign students in the techniques of differentiation and integration, their inadequate skills in solving practical problems of a technical applications. We address specific issues that arise while teaching the discipline “Higher Mathematics“ to studentswho are not native speakers and therefore studied mathematics in their native languages in secondary school. We will discuss general issues of planning and giving lectures and practical trainings to such students, namely, the need for stating out and clarification of mathematical terminology, teaching them to read formulas, appealing their attention to semantic inconsistencies of mathematical terms in different languages. In addition, we analyze the specific features of linguistic and organizational grounding of students from different countries, who study at NAU, and give our recommendations for dealing with them.

*Keywords: mathematics, higher mathematics, teaching mathematics.*

karupu_o.w._oleshko_t.a._pakhnenko_v.v._analizing_practice_of_teaching_higher_mathematics_to_ukrainian_and_foreign_students_ in_national_aviation_university.pdf | |

File Size: | 749 kb |

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**Kolchuk T.V.**

Formation of mathematical competence of secondary school students in the process of distance learning

Formation of mathematical competence of secondary school students in the process of distance learning

Abstract: This article deals with the problem of formation of mathematical competence of secondary school students in distance learning of geometry. Distance learning is becoming increasingly more and more popular as it allows to study throughout the whole life without giving up work. Therefore increasing of the number of people who need high-quality and available education is observed. In Ukraine the problem of wide use of distance learning arises during the quarantine periods, for children and adults with physical disabilities, in work with talented children, particularly in preparation for the writing of research papers and olympiads in different subjects. Distance learning has gained particular importance in connection with the adoption of a new state standard of basic and upper secondary education in Ukraine on the principles of competency and enhancement of applied nature of mathematics teaching. The author proposes to use distance technology as one of the ways of mathematical competence forming. In order to validate the optimal pedagogical conditions for increasing the effectiveness of mathematical competence formation in geometry teaching in secondary school distance learning courses "Geometry. Form 7-9" have been developed. The main ways of the components of mathematical competence formation have been considered. Software products which are used at schools during studying geometry and the ability to use them for distance learning have been analyzed. A set of pedagogical conditions providing the effectiveness of the mathematical competence formation in the process of distance learning geometry in secondary school have been revealed and grounded. The author refers to them such conditions as the development of cognitive interest, creating achieving success situations, taking into consideration age peculiarities of students, providing the differentiation of levels, the use of modern information and communication teaching technology which couldpromote the development of students personality and others.

*Keywords: distance learning, distance learning course, mathematical competence, information and communication technology, geometry, pedagogical conditions.*

kolchuk_t.v.__formation_of_mathematical_competence_of_secondary_school_students_in_the_process_of_distance_learning.pdf | |

File Size: | 580 kb |

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**Krylov V. Young professionals are able to compete in the labor market information technology**

Abstract: The market of information technology (IT) is rapidly growing. To compete for young professionals in the IT job market requires the ability to acquire additional knowledge and learn as much sought-after technologies. This ability can be developed, developing professional readiness of the young, the taste for continuing self-education. To this should be included in the educational process results of the most modern achievements in applied research, especially interdisciplinary research results is important. Where there is not yet formed a unified system of data collection and analysis patterns, unified interpretation of the results. It is on such material on the basis of acquaintance with different points of view, even contradictory views on the issue, the intern most effectively acquire skills formation proper representation of the problem. Thus obtains the necessary set of skills to compete successfully in the IT job market.

*Keywords: information technology, labor market, young professional competition*

krylov_v._young_professionals_are_able_to_compete_in_the_labor_market_information_technology.pdf | |

File Size: | 547 kb |

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**Kulishov V.S. Theoretical substantiation of methods of teaching pupils basis of the economy in the senior profile school**

Abstract. The article is devoted to the theoretical foundations of methods of teaching pupils basis of the economy in the senior profile school. We found out the discordance of the development of national methodical science to the modern market realities of society and psychological and pedagogical features of the young generation. The definition of concept “methods” is analyzed and disclosed the essence of category “methods of teaching basis of the economy”. We exploredthe relationship of methods of teaching basis of the economy with some of related courses, such as economy, pedagogy, psychology, pedagogical art and aging psychology. The main criteria of structuring the content of economic training of pupils are defined based on the person-centered approach that dominates in modern school teaching. These criteria are characterized by the assimilation of the experience of previous generations, upbringing of typological qualities of the personality and intellectual development of the person. We proved the contest of profile economic training of senior pupils is system which consists of system of economic knowledge; set of intellectual, social and practical abilities and skills; forming of socially and psychological qualities, some individual competencies of person; detection of creative features and experience of creative activity of person. The requirements that relate to the construction of the content of profile economic training of pupils are clarified. Among them: compliance of the content goals and objectives of modern economic education and its scientific foundations; compliance of the content pedagogically justified ratio between theoretical and practical basis of economic training; ensuring systematic and the logical sequence of presenting of educational material; practical orientation and professional direction of the contest of economic education; attentionto the psychological characteristics of pupils age; connection of content of educational material with real life; compliance of the content on its designated study time. The content of the experimental profile course “Basis of the economy” is chosen by systematic component of the methods of teaching basis of the economy. The structure of the course consists of five parts: “Introduction to the economy”, “Functioning of the market mechanism”, “Applied economy”, “National economy” and “International Economy”. According to the principle of implementation this program within the educational sphere “Technologies”, its contest introduces program minimum of the course “Basis of the economy” and also chapters that are connected with the applied character of economic knowledge and their professional implementation. Accordingly to all of the above, we defined the goal of introducing the course “Basis of the economy” and its main objectives.

*Keywords: economics education, methods of teaching, contest of the education, profile economic training, educational program*

kulishov_v.s._theoretical_substantiation_of_methods_of_teaching_pupils_basis_of_the_economy_in_the_senior_profile_school.pdf | |

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**Lenchuk I.G. Structurally-genetic method in tasks «on crossing»**

Abstract. Today’s teachers don't reveal to the students the importance of innovative pedagogical techniques for mastering the discipline of constructive science-based «Geometry» approach. This is not conducive to the professional and personal formation of future teachers. An article particularly shows graphic and graphic-analytical capabilities of constructivegenetic method based on the examples of positional and metric problems «on the intersection» and their solutions (conjunctive with a computing method). For the purposes of realization of general skills through the course of the problem solving, core issues of successful definition and redefinition cutting plane in the figure are disclosed in details. There is no need for digging into the nature of structural-genetic method, one would ponder. However, two types of structures are differentiated as operating strictly according to the rules of stereometric figures of the parallel projection: positional and metric. In the higher education institutions they were not determined with proper attention. At the same time, the spatial problemof the calculation is effectively solved only on an image with a certain position. Tasks that require the visual setting of the relative position and fixation of geometric shapes incidence are, called positional (affine). The theorem on the line, which is owned by the plane has substantial component of the third paragraph of § 1 of the textbook O.V. Pogorelov for High School «Geometry: Stereometry» with a meaningful title «Crossing the line with the plane», only includes a theorem of straight plane accessories. The evidence is concluded: «From Theorem 1.2 it follows that the plane and the line, which does not include it, either do not overlap, or intersect at one point». The logic of a creative person would follow: «Does the name of the item correspond to its contents?», «Why was not the essence of the question portrayed in image?». A pedagogical professional would be able to provide answers to the incompleteness in the presentation of important facts, as «read between the lines». We suggest to start learning stereometry from the 1-st and 2-nd major positional problems, and their visual presentation and with further application of the common problems in the calculation. Extra attention needs to played to the essentials of stereometry while training the future teachers. Studying ways to use graphic and graphic-analytical methods regarding the nature of the tasks of the metric is one possible way to continue the study of the problem.

*Keywords: internal projection, model, cutting plane, a figure crossing, redefinition, graphic (graphic-analytical) method*

lenchuk_i.g._structurally-genetic_method_in_tasks_on_crossing.pdf | |

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**Lovianova I.V. Retrospective analysis of the problem of differentiation of educating to mathematics at general school**

Annotation: A concept "Differentiation of educating" is analysed in the article. One of types of differentiation is distinguished, namely profile differentiation. Under the differentiation they understand the multiplicity and variety of individual measures to achieve socially agreed goals of education (A. Furman). The differentiation of training is allowed to organize educational process on the basis of the account of individual characteristics of the personality. It is provided learning the content of education every student when the invariant part is allocated. The profile education is the democratic and wide form of the furcation of school on the senior stage. The profile educating is a type of the differentiated educating that envisages the account of educational necessities, inclinations and capabilities of students and conditioning, for educating of senior pupils according to their professional self-determination, which is provided due to changes in aims, maintenance and structure of organization of educating. It is defined the objectives of the teaching of mathematics in terms of the profile of differentiation. It is analyzed the historical aspect of the problem in a comprehensive school. Namely, there are presented the following periods: pre-revolutionary Russian school the end of IX – the beginning of XX century, the achievements of the Soviet school in the 50-60s and the end of 80s-beginning of 90s of the twentieth century; the current experience of the Russian and Ukrainian schools in solving the problems of differentiation of educating. The problem of differentiation of training was the focus of the pedagogical community and found a solution in Russian pre-revolutionary school through furcation at senior levels of education. In the early twentieth century, the idea of differentiation of teaching mathematics has been strengthened. In the works of the progressive teachers and in the «minds» of officials of education has become increasingly clear need for differentiated education and primarily the profile education. A new impetus to the idea of profile education has received in the process of preparation education reform in 1915-1916. Not less important for the development of the idea of a specialisation are the achievements of the Soviet school in 1917-1990. The end of 1980 – beginning of 1990 years in Ukraine there are new types of educational establishments (gymnasiums, lyceums, colleges), in this period the idea of differentiated education mathematics has been acquired focused and associated with the creation of the so-called planned mandatory learning results. The idea of profile education were relevant both in the domestic and foreign pedagogy during all periods of its development.

*Keywords: differentiation of education, profile differentiation, education reform, the concept of profiles education.*

lovianova_i.v.__retrospective_analysis_of_the_problem_of_differentiation_of_educating_to_mathematics_at_general_school.pdf | |

File Size: | 580 kb |

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**Osipа L.V. The role of computational problems in the process of high school students’ algorithmic culture formation**

Abstract: The article defines the role of computational problems in the process of high school students’ algorithmic culture formation; describes the technology of solving computational problems using software tools; the main types of simple computational problems are identified; the stages of solving computational problems with the use of software tools are disclosed. The computational problems solution with the use of software tools is one of the adequate ways and means of effective intellectual development and the formation of algorithmic culture of high school students. Such training is primarily aimed at the development of intellectual abilities, logical and algorithmic thinking, the acquisition of skills and experience in algorithmic activity. The ability to comprise the algorithms is an important element in the process of solving computational problems. This ability is of key importance in the process of solving computational problems with the use of software tools. The process of high school students’ algorithmic culture formation involves teaching activity aimed at disclosing the algorithmic content of the material, computational problems selecting, familiarizingstudents with common algorithmic techniques, basic algorithmic structures, algorithm types, properties, and methods of their description. It also provides the formation of students’ ability to comprise, use, analyze and evaluate the effectiveness of the algorithm for solving computational problems, and the analysis of capabilities and effectiveness of using software tools for solving computational problems. Using software tools in the process of solving computational problems requires the development of the relevant methodology which implements in the construction of the complete sequence of calculations for a certain algorithm and the creation of calculators in the software tools environment. The last but one are considered to be the implementers of the algorithms for solving problems of a certain type. Further research on this issue will be associated with the development of the curriculum of the elective course "The computational problems solution with the use of software tools," and the writing of a textbook.

*Keywords: algorithmic Culture, calculator, computing tasks, software tools.*

osip_l.v._the_role_of_computational_problems_in_the_process_of_high_school_students_algorithmic_culture_formation.pdf | |

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**Pavlyuchenko L.S. Test and Evaluation activities – an important component of the training**

of future teachers of informatics

of future teachers of informatics

Abstract: The article deals with the topic assistance of computer technology in the implementation of approaches to monitoring and evaluate of participant’s knowledge, the transition from traditional evaluation methods to test control methods to use. Test control makes estimates meaningful, objective and differentiated.

*Keywords: control and evaluation activities, monitoring, evaluation, test control, computer technology participfnts*

pavlyuchenko_l.s.1_test_and_evaluation_activities__an_important_component_of_the_training_of_future_teachers_of_informatics.pdf | |

File Size: | 389 kb |

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**Rotaneva N. Classification is as the heuristic method in teaching on Mathematics in the 5-th forms**

Abstract: Integration of Ukraine in the global educational environment requires continuous improvement of the national education system and searching for effective ways to improve the quality of educational services. Considering the mathematical education of 5 forms students, it can be stated that it is possible to implement in case of appropriate forms, methods and tools combination in the mathematics learning process, aimed at the development of students heuristic activity, as well as the introduction of various forms of extracurricular activities in mathematics, which stirs up interest in the study of mathematics. To solve this problem, we propose to organize math study group "First acquaintance with heuristics" in extracurricular activities for students of 5 forms. Students have the opportunity to get acquainted with some special methods that are called heuristics (or heuristics) there. It is heuristics that is formed in the process of solving certain mathematical sums, and then it is applied to the search for various non-standard problems occuring not only in mathematics. As an example, the technique of organizing local study groups with students of 5-forms on “Heuristic device "classification"” is provided in this paper. The structure of group’s classes is as follows: students' mental activity enhancing and motivation (creating the atmosphere when students are interested in learning heuristics), examination of heuristics (first pupils are supposed to refer the "heuristic certificate", which contains the explanation of the heuristic device "classification" based on the "rule-goal"), consideration of sample solutions of mathematical sums on the heuristic device "classification" (a teacher gives detailed step by step heuristic sums solutions with the" rule-goal" receive and organizes collective work on doing sums) performing tasks independently on the use of the device in mathematical tasks in case of getting heuristic information or guidance opportunities (the system of tasks is provided as it plays a key role in solution process where each student chooses a heuristic information or a hint, and the teacher controls

this); recognition of heuristic techniques underlying as the basis for solution and doing the sums ( Pupils are proposed to select those tasks that can be solved by a method that is studied in this lesson.) Thus constructed mathematic heuristic study group allows a student to get acquited with a heuristic device "classification", and thus laid the foundations of heuristic techniques performance.

*Keywords: the heuristic method, the classification, the Math circle*

rotaneva_n._classification_is_as_the_heuristic_method_in_teaching_on_mathematics_in_the_5-th_forms.pdf | |

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**Skafa O. The management by the heuristic activities of the students in the extracurricular work on Mathematics**

Abstract: Heuristic teaching of mathematics system at school is investigated. Specially organized extra-curricular activities belong to organizational forms of this training. The article presents the author's technology of students heuristic performance management in extracurricular activities of 5-11 year students of mathematics schools of Ukraine, as well as a teaching manuals specially developed for courses. The structure of this paper is as follows. Heuristic study group in mathematics is offered as a heuristic activity propaedeutics for 5-6 year students. It is different from traditional classes in the way that students have the opportunity to get acquainted with some heuristics, build a rule-goal of its application, consider samples of solving mathematical tasks on the application of this technique, independently perform tasks on the use of admission to mathematical problems (in this case have the opportunity to obtain heuristic information or hint), as well as recognize a heuristic device, which lies on the basis of tasks solution. The next step - the introduction to math study of heuristic electives for students of 7-9 forms. A distinctive feature is the use of heuristic-oriented task systems. These tasks allow students to express and found hypotheses, build models, find and interpret the results. In the course of heuristic electives, students have a clear understanding of the heuristics application, heuristic skills are formed. In high school heuristic elective course in mathematics "Journey into the world of heuristics" in offered. It is a generalization and systematization of all previous extra-curricular class work in mathematics. Forms of study: lectures using heuristic tasks, including practical, heuristic conversation and dialogues, independent and group work, workshops about heuristic tasks, including application. Tasks are describeв by the situations from literature, art, jurisprudence, history, mathematics, etc., research, students’ self-esteem of their creative products. This work contributes to the development of heuristic techniques, imparts skills of independent work, stirs up interest in the study of mathematics and research.

*Keywords: heuristic elective, heuristic teaching mathematics, the management of the heuristic school activities.*

skafa_o._the_management_by_the_heuristic_activities_of_the_students_in_the_extracurricular_work_on_mathematics.pdf | |

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**Trebyk E. Innovative forms of ICT while teaching mathematics in colleges**

Abstract. The article discusses the use of innovative forms of information and communication technologies (ICT) in teaching mathematics in colleges – universities and II levels accreditation. It is noted that innovation in education is one of the important areas of creating the best educational areas. They do not occur by themselves, but are the result of scientific research, studying the best educational experience of individual teachers and even groups. Here you can find content of "College", served characteristics of the modern college student, and analyzed how to obtain them with different types of information. Describe the types of information and communication technologies that college coaches use in their teaching activities, including teaching mathematics. Such as: Internet technology (website, blog, forum, email, instant messaging, search engines, educational portals, etc.) There are multimedia software (computer simulators, multimedia presentations, training films, and educational software), electronic books and textbooks. Because of the uniqueness and diversity of ICT we can have new insights into the learning process, to create innovative options for their use in education. The content of each of the species examined information and communication technologies and the benefits of their use in teaching mathematics. In the specific examples there shown using Internet technology, multimedia software еtс. while teaching mathematics in colleges. E-register, for example, creates the conditions for parents about attendance and assessment, so that they can easily monitor their children's progress at any time and in any place where you have Internet. Form-master register make it easier the monitoring of the education. Teacher of mathematics can quickly calculate final marks of all students create comments in a register (comment on assessment, job kind, etc.), log in data at any convenient time. Children and their parents won’t be "surprise" with final marks.

*Keywords: college, information and communication technologies, innovation, math, students*

trebyk_e._innovative_forms_of_ict_while_teaching_mathematics_in_colleges.pdf | |

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**Tretyak M. Mathematical and Cultural potential of the course “Theory of measure and integral”**

Abstract: The author's model of the structure of mathematical culture of the Mathematical students has been presented. This model consists of 20 components, namely: 1) Mathematical knowledge; 2) Mathematical skills and abilities; 3) Abstract thinking; 4) Formal and logical thinking; 5) Functional thinking; 6) Probabilistic thinking; 7) Algorithmic thinking; 8) Mathematical imagination, space imagination; 9) Skill in handling of mathematical language and mathematical terms; 10) Knowledge and understanding of Mathematical methodology; 11) Knowledge and understanding of main components of Mathematics and their relationships; 12) Knowledge and understanding of place and role of Mathematics in the system of sciences and its using; 13) Knowledge and understanding of the role of science in the human and mankind life; 14) Knowledge and ability to mark out the mathematical part of the phenomena or process, to build mathematical models, to investigate such models and to interpret the results; 15) Knowledge of the history of origin and formation of Mathematics and its main chapters; 16) Knowledge and understanding of the connections between Mathematics and Art (music, painting, sculpture, architecture etc.); 17) Knowledge of the mathematical folklore, different didactic stories, jokes, funny stories; 18) Presence of the highly developed mathematical ideal; 19) Presence of the highly developed moral and ethical ideal; 20) Ability to perception and creative learning of new ideas, creativity. On the base of proposed structure of mathematical culture, the possibility of the formation and development of the elements of students' mathematical culture for the case of course "Theory of measure and integral" have been analyzed.

*Keywords: mathematical culture, elements of mathematical culture, Mathematical students, formation of the elements*

of students' mathematical culture, Theory of measure and integral.

of students' mathematical culture, Theory of measure and integral.

tretyak_m._mathematical_and_cultural_potential_of_the_course_theory_of_measure_and_integral.pdf | |

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**Filippova O. A. The development of functional representations of pupilsin the category of mathematical picture of the world**

Abstract: the concept and structure of a mathematical picture of the world of a class of functions is investigated. Basic functional models for the description of general idea of the theory of functions are investigated.

*Keywords: function, a mathematical picture of the world, a model of the theory of functions.*

filippova_o._a._the_development_of_functional_representations_of_pupilsin_the_category_of_mathematical_picture_of_the_world.pdf | |

File Size: | 619 kb |

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**Filimonova M.O., Shvets V.O. The elements of mathematical modelling in the study of geometric material in 5th – 6th forms.**

Abstract: Systematic course of geometry provides propaedeutics in learning mathematics students 5th – 6th forms. This issue is evoted to a number of dissertations and research papers, but the unknown point is the possibility of using the method of mathematical modeling to study the geometry. This article outlines the main ways of creating mathematical modeling skills in students 5th – 6th forms and suggests appropriate guidelines based on psychologo-pedagogical features young teenages. Specifically, the teaching material should be geometric: maximize visibility and draw on the experience of students, new concepts and properties of geometric shapes to explore the practical work with the elements of design and test work on the ground, the system of exercises to guide the development of spatial imagination, abstract thinking and execution skills simple logic operations, using many applied problems, challenges for the development of the ability to recognize the reality of geometric objects, perform measurements "by eye". These recommendations are elaborated on the example of the following topics: "Direct, ray, segment, broken. Corners", "Polygon. The area of are ctangle" and "Cuboid, cube. Volume."

*Keywords: geometry, mathematical modelling, model, practical work, applications problems.*

filimonova_m.o._shvets_v.o.__the_elements_of_mathematical_modelling_in_the_study_of_geometric_material_in_5th__6th_forms..pdf | |

File Size: | 550 kb |

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**Shvets L.V. Graphical representation of spatial figures in school stereometry course**

Abstract. During the studying of stereometry course a teacher has to tach students to represent stereometrical figures and their combinations, in the other words, to make students’ abilities to do these representations. Teaching students to stereometry teachers make full use of visual figure representations, which play an important role in pedagogical process. These graphical representations make easier students’ understanding of teacher’s thoughts and resume. It’s very important that they also develop students’ spatial conception of observing ratios and give them specifik geometrical forms. Thanks to it students can remember material rapidly and with more profit. On the other hand a correct sketch give possibilities to solve a sum. But an incorrect sketch can suggest an incorrect idea about students’ consideration. Of course, in pedagogical conditions graphical representations should satisfy some dictates, which are demanded by teaching specificity. Certainly, for graphical representation you can use methods of descriptive geometry, which are developed in details. But these methods fall within problems of technical kind of work, which are unexpected by teacher’s activity and unadapted to pedagogical process. These mentioned circumstances need revising of graphikal representation principles sn details in stereometry course taking into consideration sketch dictates and conditions when students san make skills to do these representations by forming off partial skills. In this article there are principles of graphical representation of spatial figures in school stereometry course. There is a comparative characteristic of principles of graphical representation of descriptive geometry and representations, which are used in school stereometry course. Also there are characteristics of parallel projecting, conception of completeness and matric precision of spatial representations to improve a level of graphic culture as to graphical representation of stereometrical figures and their combinations. There is a structure of forming and developing of senior students’ skills to represent spatial figyres and their combinations

by dividing into parts in this article, too.

*Keywords: stereometry, parallel projecting, completeness, matric precision, partial abilities, graphical representation of stereometrical figures.*

shvets_l.v._graphical_representation_of_spatial_figures_in_school_stereometry_course.pdf | |

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**Shulga N. Synergetic model of learning stochastics future economists**

Abstract: This article substantiates the synergetic approach to the construction of models of learning as a complex, open, non-equilibrium, non-linear system. This system is based on the principles of synergy: homeostasis, hierarchical, non-linearity nonclosure, instability, a dynamic hierarchy, observability. Structural diagram the learning process is the interaction of three hierarchical levels of the educational process: the mega-level, which reflects the interaction of information, material and socio-economic parameters of management; macro-level, the interaction of the order parameterStudent ↔ Educational Environment ↔ Teacher of which can be described by a system of nonlinear differential equations, the micro-level, containing in itself all the short variables describing the learning process. The system of nonlinear differential equations shows the dependence of the order parameters of their time derivatives. Each equation of the system also includes control parameters held constant. Each variable that is included in the differential equation depends

also on the short-lived micro-level variables. The dynamic model shows the emergence of a system education of the chaos, the structuring of its levels, orderly transition to a phase of development and a return to chaos as a result of internal and external random factors. The author notes that the social and economic relations in society are influenced by a large number of random factors. Therefore the preparation of future economists must be accompanied by the study subjects that will enable to assess the possibility of rescheduling. One of these disciplines is the theory of probability and mathematical statistics. The learning process stochastics may also be presented in terms of synergy. The control parameters should guide the process for the formation of the skills required for their activities in the face of uncertainty. The order parameters must meet the requirements that push them to the control parameters. Micro-level contains necessary for the implementation of the learning process of students of economic universities stochastics short field variables.

*Keywords: the process of learning, synergy, stochastics, model, modeling the learning process, training of economists*

shulga_n._synergetic_model_of_learning_stochastics_future_economists.pdf | |

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**Antonenko T.L. Modern approaches in psychology to studying of values and senses**

Abstract. The author of the article examines the features of modern civilisation that are characterised by certain crises and new information technologies. The characteristic is given to the epoch of postmodernism, postnonclassical psychological science, interval epistemology and constructivism elements. Using synergetic approach to the investigated problem, the author of the article defines the concept of “ideal” as attractor that is the vector of the life course. On the basis of scientific literature analysis the essence of the concepts “sense”, “value”, “dialogue” is revealed. The phenomenon “multidimensional person” is analized from positions of interval epistemology which has allowed to examine the features of “internal” and “external” mode of being of the person tending to self-affirmation. The features of designing of the language personality private world, dialogicality as the leading methodological idea of values and senses formation, psychotherapeutic practice of dialogical method usage are disclosed. In the article the attempt has been made to characterize human nature, its integrity and antagonism. On the basis of the used technique (A. Etkind’s colour test of relations) the emotional relation of students to the spiritual phenomena – values such as good, evil, conscience, belief, truth has been defined. In the technique content certain dichotomy based on mutually exclusive concepts of good and evil has been set. After analyzing the received results, our assumption that such phenomena as good and evil appear to be the centre of value-sense sphere of the personality (kernel) and form its pre-reflexive, unconscious basis has been proved. The found out dichotomy good-evil not simply sets structure of value-sense sphere, but gives value-sense meanings to the phenomena and objects of life. In the article the influence of culture on person’s choice of values and senses, on formation of the image of man of culture, his cultural identification which reflects connection of the personality with universal cultural wealth is shown.

*Keywords: values, sense, value-sense sphere, personal value, personal sense, dialogue, postmodernism, interval epistemology, multidimensional person, culture, synergetics.*

antonenko_t.l._modern_approaches_in_psychology_to_studying_of_values_and_senses.pdf | |

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**Bulgakovа O. Personality-typical and action-situational As opposed to as a part of social interaction**

Abstract. We consider the strategic aspects of interaction, based on personal experiences and personal meanings of the subject. Determine the psychological structure in the social interactions that influence the course of interactive relationships. Psychological structure must be such that are relevant to determining the strategic and tactical aspects of interaction. If the first back to personal experience and the related system of personal meanings of the subject, the latter involving the use of its intellectual potential. In this case we are talking about the emergence of a functional system that forms a typical student-and action-situational as opposites. Consider situational experience as conscious creation of models in the sense of tactical basis, which indicates the significance of the principle of additivity of the system (system enclosure) to describe the mechanism of change through appropriate awareness of the relevant causal relationships. To create a conscious strategic models and hence personal-typical settings, using data on the ratio of actual and ideal "I" obtained by T.Liri test as an indicator of trends. Understanding trends by streamlining their interactive circumstances in relation representations of themselves in the actual and ideal aspects are, in our opinion, the main mechanism that determines the change in policy (student-model) constructs. Strategic and tactical installations that under certain conditions require correction can be converted into those that meet the requirements of constructive social interaction, based on the principles of rationalization, transfer, additivity by using data on trends in personal development in the parameter space of actual and ideal "I". Tactical aspects of interaction are determined intellectual potential of its participants. Formulated opinion on strategic and tactical participants as an interactive installation based psychological interaction.

*Keywords: social interaction, personality, activity settings.*

bulgakov_o._personality-typical_and_action-situational_as_opposed_to_as_a_part_of_social_interaction.pdf | |

File Size: | 552 kb |

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**Kuznetsova O.V. Individual and typological distinctions in the adaptivity of personality**

Abstract. In the article some results of theoretical and empirical research of individual and typological differences on adaptivity are presented. Adaptivity is examined as a property of personality, characterizing a capacity for internal and external (behavioral) transformations, aimed at preserving or restoring the equilibrium relationship of the individual with the micro-and macro-environment for changes in its characteristics. This is a complex integral property of the individual, holistic formation, including formally-dynamic, meaningfully-personal and socially-imperative levels. A formally-dynamic level is presented by those descriptions that contain information about psychological essence and form of display of the studied phenomenon. Here structure of adaptivity is a three-component formation, including cognitive, emotionally-motivational and behavioral components. Each of them performs a specific function, totality of that provides an adaptive effect. The components of adaptivity are presented by the row of specific parameters among that as basic can be distinguished: breadth – narrowness of scope of signals of society, lightness – hardness of their understanding and hierarchy constructing, exactness – inaccuracy of orientation in social expectations (cognitive component); stability – changeability of the emotional experiencing, readiness – absence of willingness to change (emotionally – motivational component); readiness – absence of readiness to realization of the structural actions, readiness for overcoming failures, readiness or absence of readiness to realization of the actions directed to gaining the purpose (behavioral component). Prevailing of one or a few components (disposition of indexes) is examined as a "type" of adaptivity. Psychological characteristics of representatives of types of adaptivity on the factors of personality of R.Cattell are investigated. Every "type" of adaptivity is characterized by original combination of factors of personality, presenting the specificity of its individually-psychological properties. The type of adaptivity can be examined as founding for research of individual distinctions in this property of personality that considerably allows extending the range of problems of differential - psychological study of adaptive possibilities of personality.

*Keywords: adaptability, structure of adaptability, type of adaptability, factors of the person*

kuznetsova_o.v._individual_and_typological_distinctions_in_the_adaptivity_of_personality.pdf | |

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**Kazanzhy M.I. The structure of personal facilitative potential: creation experience**

Abstract. The article is devoted to the problem of facilitative ability phenomenon. Such issues as succession of stages in the facilitative ability study, peculiarities of reconsideration and extension of some fundamental ideas, further understanding of the given phenomenon are examined in this paper. A certain unreasonableness of defining facilitative ability as a personal “trait” or “virtue” is shown. A person who can be conditionally considered “healthy” is not a bearer of facilitative-inhibitive ability as a personal virtue (or trait) but just able to do facilitative or inhibitive actions under the circumstances which implies the presence of potential to do these actions. The term “facilitative ability” means a psychological property displaying personal facilitative potential. This property is different from a personal trait (virtue) by less stability as it is being actualized in the period of deliberate purposeful facilitative activity. This property may “contract”, become latent and convert to potential in the periods when there is no need to help others in their development. The same thing can be said about inhibitive ability. In fact inhibitive-facilitative ability is one and the same event with a contrary sign; both these events can be in the potential state; inhibitive ability is also a temporarily actualized potency. This paper represents the results of theoretical creation of personal facilitative ability structure. Having analyzed the available literature it was possible to establish that facilitative potential can be defined as unity of two integral constituents: actualizing and resourceful. The actualizing constituent includes sense-valuable, requiremental-motivational and behavioral components. Sense-valuable and requiremental-motivational components are motivational by their nature and their division is rather conventional. The actualizing constituent is responsible for actualization, demonstration of the resourceful component in order to help other people’s development. The resourceful constituent involves a range of emotional, cognitive and strong-willed properties manifested in the ability to influence on others and can be used as “inhibitive” if the actualizing constituent of facilitative potential is not developed enough. To sum up, it should be mentioned that this study provides important information about the problem of interaction between facilitative-inhibitive ability and personal facilitative potential.

*Keywords: facilitative ability, facilitative potential, facilitative potential components, virtue (trait).*

kazanzhy_m.i._the_structure_of_personal_facilitative_potential_creation_experience.pdf | |

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**Lototska (Ilina) I. The psychological phenomenon of «The Death Vallay of Past Experience»**

Abstract. This article is a brief of the issue of success anthology, the author developed the concept of success, argued a direct link of success with the activities of intelligence structures. Briefly described the main stages of the experiment "Psychology of life success and failure." The basis of the article is a description of the psychological phenomenon of "The valley of death of past experience" in personal transformations. We considered the algorithm described by Prochazka and Norcross, E. Kübler-Ross (adapted by Lototska) as a platform for management changes in adults. In the researching process of our experiment, it was noted that the most of our participants strongly passed the first three stages, some of them went out into the 4th stage and then back to the start again. After 14 months of pilot work, we confirmed that a new psychological phenomenon preceding the onset of changes was found. We divided the preparation stages for changes into three parts: the identification, awareness, change. We have identified and described seven stages of passing "The valley of death of past experience" – from the choice of idea– reference, to the consciousness – resolution: 1. «Idea»; 2. «Quick win»; 3. «Collapse»; 4. «Fear. Back traction»; 5. «Motivational escape»; 6. «Awareness»; 7. «Permission» The statistical data of the experiment, in which the results of "The valley of death" passing in the project "The psychology of life success and failure" is described in % in a sample "normal" (live and remote format) and a sample "mental disorders". And also data on the maximum recorded "The valley of death» among our members are presented in the article. The phenomenon also explains one of the reasons for unsuccessful transformations in adults. It is important to permit yourself to see, to reveal a contradiction that will cause another round of development, or to refuse, that is not to enter into the transformation. It is interested for us to research in details the characteristics of those who refused to enter the VD to work out the means of help and self-help.

*Keywords: psychology of success, success, failure, the phenomenon. «The Death Valley of Past Experience»*

lototska_ilina_i.__the_psychological_phenomenon_of_the_death_vallay_of_past_experience.pdf | |

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**Оrishenko O.A. Cognitive-empathic person type: individual psychological characteristics**

Abstract. Empathy is studied from the positions of the continuum of care approach, which allows to consider it as a complex multi-level, but at the same time holistic education. In the context of this approach, theoretical and empirical research are indicators of empathy and quality (emotional, cognitive, predictive and effective empathy) and the substantial levels (empathic orientation). At a substantial level examines the characteristics of empathy, which, in General, reflect an empathetic orientation of the personality. Empathic orientation is understood by us as a manifestation of empathic preferences ' choice ' is the subject of those objects, those spheres of life, in relation to which, first of all, arise Оrishenko O.A. Cognitive-empathic person type: individual psychological characteristics empathic experience of the subject and where the most pronounced the individual identity of empathy. In the context of a continuum of care approach as indicators of empathic orientation (the substantive structure of empathy, we select the following options: empathy to the family, relatives, friends; empathy to colleagues, students, clients; empathy to the strange and unfamiliar people; empathy subject itself; empathy for the characters artistic works; empathy to the representatives of the animal world; empathy to vegetable world; empathy towards nature, towards the world in General; empathy to past events that occurred in the life of the subject or in the lives of those around him; empathy to the events of a possible future. In the present communication sets out the results of an empirical study of individual-typological peculiarities of one of our dedicated types of empathy (cognitive- empathic type), which amounted to face with sustainable domination (disposition) of cognitive empathy (empathy, based on the intellectual processes: a comparison, analogy, understanding and awareness of the emotional States of other person, including the experience, knowledge and skills). Found that the cognitive empathic personality type is characterized by a unique combination of personality factors and indicators, included into the structure of empathic orientation, sociability, and psychological insight. Studied and described the psychological portrait of personality typical representative of cognitive empathic type in the context of the wide range of properties of the person.

*Keywords: empathy, empathic orientation, cognitive empathy, cognitive- empathic type.*

rishenko_o.a._cognitive-empathic_person_type_individual_psychological_characteristics.pdf | |

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**Yefymchuk H. O. The Structure of Linguocultural Concept ARROGANCE in English**

Abstract: The article deals with the structure of linguocultural concept ARROGANCE. The problem is relevant as there is necessity to study the concept in English conceptual sphere, which is impossible without taking into consideration its constituents. As nowadays there are different approaches to studying the structure of concepts, the main of them are analyzed and compared in the article. In particular, it discusses the approaches to the study of the concept structure suggested by linguists I.A. Karasyk, I.A. Sternin, Yu.S. Stepanov, V.V. Kolesov, G.G. Slyshkin. Based on the position that the concept as part of a conceptual system is expressed by semantic structures of different levels, all approaches provide multiple layered concept structure. Thus, such components as etymological, associative, notional and figurative were defined within the structure of the concept ARROGANCE. Each of these is described in more details in the article, with reference to the examples of verbal representation of the concept in the English language community. The etymological origin of the key representative of the concept has been traced. An approach to the study of associative field of concept structure have been suggested, as it requires more detailed research. The notional component has been determined, with the interpretation of the basic vocabulary definition. The semantic core and peripheral areas have been defined. This helps to reveal the peculiar characteristics of the concept that are common for the members of different language communities. Particular attention is drawn to the figurative component of concept structure. In particular, the examples of conceptual metaphors that implement the phenomenon of superiority in discourse have been suggested in the article. Historical level of the structure of concept ARROGANCE has been considered using examples of phraseology, which demonstrate the diachronic development of the notion of superiority. Based on the analysis of all elements of the concept the conclusions about the prospects for further research have been drawn.

*Keywords: concept, linguocultural concept, structure of concept, conceptual metaphor, conceptual sphere*

yefymchuk_h._o._the_structure_of_linguocultural_concept_arrogance_in_english.pdf | |

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