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PRIORITIES IN THE TEACHING OF MATHEMATICS
FOR THE FUTURES
Antoni Pardala Rzeszow University of Technology pardala@prz.edu.pl
Roza I. Kadirbayeva, Madina A. Jamankarayeva South Kazakhstan State Pedagogical University roza-1961@mail.ru, d_madina08@mail.ru
Проблематика статьи касается анализа избранной современной литературы, который направлен чтобы найти приоритеты обучения математики для будущее. Эта работа имеет тоже характер case study примеров из учительской практики Польши и других стран. И ее авторы делают попытку определить своего рода диагнос и прогнос для улучшения качества результатов процесса матема-тического образования, в частности с математически талантливыми детми и учащимися.
Ключевые слова: приоритеты в преподавании математики; диагнос и прогнос улучшенмя резултатов математического обучения.
The topic of the article concerns the analysis of selected modern literature, which is aimed at finding priorities for learning mathematics for the future. This work also has the character of a case study of examples from the teaching practices of Poland and other countries. And its authors are trying to define a kind of diagnosis and prediction for improving the quality of the results of the process of mathematical education, in particular with mathematically talented children and students.
Keywords: priorities in teaching mathematics; diagnosis and prognosis of improved results of mathematical teaching.
1. Introduction
Favorable opinions about a school or university are not relevant solely in their vicinity, as such information continues to pervade and fascinate particular national and international environments. Among other things, it concerns the efficiency and effectiveness of their students' mathematical education. The contemporary dilemmas of the mathematical education at schools and universities regard as follows:
1) development of memory, the ability to think in children, pupils and students,
2) mastering the material, competences and habits in solving the mathematical exercises, tasks and problems,
3) effectiveness and quality of their mathematical education,
4) developing their mathematical culture,
5) searching for mathematically gifted children and students, programs and methods for developing their mathematical talent, see: Г. В. Томский (2017).
All-encompassing aspects of aforementioned problems in the interdisciplinary terms are presented in the lectures and papers of the participants of the scientific conferences, in the articles and monographs, see: A. Pardala(2011), Г. Г. Малинецкий(2013), А. М. Вони (2018) and others. For instance, N. Woolcock (2008) refers to the reports from the GCSE (General Certificate of Secondary Education) exams from the subject of mathematics, that is the British state exams for the students between the ages of 14 and 16, and puts forward critical and firm opinions of the report's authors: lowering of the teaching standards threatens the future of the economics, people should be ashamed of their mathematical illiteracy, instead of being proud of it, what is taking place these days. Significant diversification of the level and quality of didactics of mathematics in teaching the pupils all around the world confirms the results of the international examination PISA (Programme for International Student Assessment), organized every three years by OECD (Organisation for Economic Cooperation and Development. Л. Ясюкова (2013) also states that the results of the intellectual monitoring of pupils and students show a deepening gap between apt and the inept ones, and between the currently functioning systems of education in the examined countries. The quality and effectiveness of teaching pupils and students to a large extent depends on:
1) the quality of education and preparation to work of the mathematics teachers, their knowledge in the field of mathematics and methodology, qualifications, competences and necessary habits;
2) the exchange of experiences between the teachers within the framework of self-improvement or collaboration with universities.
These aspects mainly concerns acquaintance with the experience of using the methods in working with children, examples and the results of the practice of forming their logical, computational thinking. G. V. Tomski's original concept of working with mathematically gifted pupils, students and methods for increasing the efficiency of
their mathematical abilities were analyzed in detail by А. М. Вони (2018). In particular, he stresses, that an increase in the efficiency of the development of analytical thinking and creative abilities is provided by the intellectual, creative and mathematical game of JIPTO developed by G. V. Tomski. But the JIPTO game allows even schoolchildren to practice creating mathematical models and theories, starting with the simplest ones with a further increase in complexity.
GRIGORI TOMSKI GRIGORI TOMSKI
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ELEMENTARY GEOMETRY OF PURSUIT
MATHEMATICS OF JIPTO and research topics
2. Methodology of work on teaching mathematics for the future
The topic and problems of work concern one of the key research directions of mathematics didactics defined as: Priorities and problems of informatization of modern mathematics education in different countries - experience and acquaintance with best practices for the future. Following this, both in the world, in Poland and in the Russian Federation, international scientific conferences and studies are organized concerning didactic concepts and the benefits of integrating mathematics with computer science.
The positive results are analyzed in detail, as well as the discovered shortcomings of the recent changes and innovations in the educational system of this country or the reform of mathematical education introduced in recent years according to the requirements of the Bologna system, see: Pardala A., Uteeva R. A., Ashirbayev N.K. (2015); Pardala A., Ashirbayev N.K., Rakhymbek D. (2015), napgana A., nrax P.(2018); napgana A.(2018.
The methodology for the development of this topic is based on the indication of practical experience and an attempt to analyze examples from specially selected publications on an effective method of informatization of education. The development of the theme of the work has the character of a case study of the experience of the author and his staff from the practice of informatization of mathematical training for students of mathematics, computer science, as well as similar experience of teachers working with children and students. The research objective of this work is to synthesize, approximate and describe the results of selected literature, experience and examples from the teaching practice of informatization of general education, as well as from the integration of teaching mathematics with other school subjects. The author of the work tries to answer the following research question: What didactic and qualitative benefits does the application of experience and examples, familiarization with the best practices of informatization in the process of mathematical education of children, pupils and students? Trying to answer this question here, I use the statement attributed to I. M. Gelfand: theories come and go, but examples remain that concern the didactic work of teachers not only in elementary education. For children, pupils and students, examples of exercises and tasks from the practice of subject teaching are essential. Then they can provoke the right questions for them:
1) How to find the right answer for the question or the right solution for this problem?
2) What method or methods can be used to find the right solution? In particular and using ICT will it be possible to find this solution for this problem here.
On the other hand, always proven examples from the practice of teaching mathematics at this stage of education will be useful in favor of: a) arousing curiosity and cognitive interests from them, intrinsic motivation for informatization of subject-specific knowledge and learning, b) the need to erect "cognitive bridges" mathematics with other subjects of study.
3. The examples of the research and practice of the mathematical education
In Polish schools, for a long time now, winning streak of work of many mathematics teachers continues. Undoubtedly, the reason for pride is provided by the results of the PISA examinations in the past few years, or by the 6 medals won at the International Mathematical Olympiad in Thailand in 2015: gold, silver and four brown ones! Currently, there is an increasing worry in many countries - including Poland -
about the effectiveness of general education, improvement in the concept of teaching mathematics of children, pupils and students, their results and achievements on a certain level of this education, or the implementation of the accepted standards, see: www.ifispan.waw.pl . It is impossible not to notice the importance of mathematics and the role that it plays, the application, mathematical modelling, the quality of education in this field, the mathematical culture, the preparation of the staff for the needs of certain countries, their proper functioning, for their further economic and intellectual development, as well as their societies, in the modern world. In this matter, mathematics is an unquestionable phenomenon of the world-class culture, in which, among others, the history of its development and the achievements of some of its distinguished representatives are reflected, as well as from other fields of knowledge or creativity. Some of them are implemented also into the process of teaching mathematics or general education of children, pupils and students.
Computer programs and the Internet, innovative and computer technologies, Information and Communication Technology (ICT) and other technological resources assist in the traditional teaching methodology and learning methods, the perception of the material to learn and also methods of mathematical modelling of phenomena and problems. The contemporary results of research confirm that the reasonable use of ICT, Edumatrix as a new didactic aid and a new method of teaching the beginnings of mathematics and programming elements, modern modes of presentation and visualization of the material to learn, learning platforms and methodology of e-learning should not present any danger for the traditional methods of science education, see: G. Siemens (2010), www.edumatrix.pl . On the other hand, it can present challenges in the development of their memory and internal motivation to learning, not only mathematics. It can also be a reason for discarding their need to learn the ability of logical thinking or their ability to explain the correct solution of a mathematical exercise or statement, formulation of a hypothesis, when a computer, the Internet or ICT will be their only 'authority' when it comes to education. The concept of G. V. Tomsky is also correlated with the theme of this work: JIPTO - from kindergarten to university, see: [19]. At present, the most common dynamic game exercises and tasks are used in the practice of teaching mathematics, as well as computer programs for example: GeoGebra, CAS -computer algebra systems, methodological and monographic developments, results of international scientific conferences and others.
Today's competencies of children and students in the field of reading, writing and calculation already require the expansion of their competence in computational thinking. And also expected to strengthen the programming skills of students in European schools. It is assumed that the students who started today in elementary schools will carry out a profession in the future that does not yet exist. It is assumed that the students who started today in elementary schools will carry out a profession in the future that does not yet exist. Most of them will be related to the ICT industry. For example, in the United States, it is estimated that by 2020 there will be more than one million free jobs for programmers! Similar trends are observed in Europe. In this
regard, in Poland, in early September 2017, classes on the basics of computer programming were introduced in the classes of primary school students. The skill of programming becomes an element of their general education; this is in favor of their future. To achieve this goal in educational practice, there is a useful Polish didactic, educational aid from Edumatrix and the teaching method is also integrally connected with it, also called method teaching Edumatrix, which can be addressed to children from kindergarten, see: www.edumatrix.pl .
Example 1: Some benefits and aspects of Edumatrix
What is Edumatrix ? This is a set of corresponding cubes prepared and developed by teachers who are passionate about ICTs so that it can support the natural needs of children and students in their intellectual development. You can work with him at different stages of training. Edumatrix as a learning method allows you to combine mathematical, logical and ICT problems. In the beginning, this education of children, pupils is conducted in a picturesque, easy and even fun way, using their full spontaneous activity and creativity. With the help of the Edumatrix method, you can soon open the path for them and continue the expected path to mathematical training and algorithmic thinking, which is facilitated by the visualization of their activities. Using Edumatrix, you can teach children and students not only the elements of programming itself, but also to form the skills and abilities they need to solve problems and problems of varying complexity in different ways and methods that help and develop their analytical and logical thinking. Of course, Edumatrix helps the student to get acquainted with the basic concepts of a programming language: entering and reading data, processing data, making the final decision, repeating the necessary actions. In addition, it also has other advantages and useful functions, see: napgana A., nrax P.(2018).
Example 2: European project - Junior Code Academy Project
An innovative extension of the Edumatrix method for students is the Junior Code Academy educational project, see: www.juniorcodeacademy.edu . Its main goal is to develop and test a new curriculum program, develop skills and abilities, expand knowledge concerning programming in young students (aged 10-15 years) in European schools. So that they can meet the needs of tomorrow, implementing a learning strategy as part of the requirements established by the core competencies of the 21st century, in particular, such as knowledge, skills and abilities, methods and ICT. To be able to conduct logical reasoning or solve problems correctly. This international initiative is funded jointly by the Erasmus + Commission and is implemented by seven partners from four different countries, including Greece, Italy, Portugal and Poland. Current results from the implementation of this project see: www.juniorcodeacademy.eu/wp- content / uploads / 2018/03 / JCA_Final_Publication_PL.pdf .
4. Research questions and priorities in mathematical education for the future
The topic of this work as well as the questions presented below touch upon important aspects of the theory and modern practice of mathematical education:
1) How to "open" and motivate those pupils and students who are having difficulties with the subject, such as they "cannot think" or "cannot see" in mathematics; "cannot absorb" the knowledge and presented material in mathematics?
2) How to "open" and motivate those, who are learning it unwillingly or those who are reluctant to the subject and learning mathematics altogether?
The race for the accurate determining and establishing the priorities regarding the concept of mathematical education for children, pupils and students and for their future, is still ongoing. What is more, this current "game" for the concepts and educational standards, methodology and practice of teaching mathematics targeted for the needs of future is one of the interdisciplinary key issues of research of mathematical education under the conditions of the innovative development of the world's countries.
However, the race for priorities and successes in teaching mathematics to children and students for the sake of future continues uninterrupted. What is more, this so-called "competition" concerning the effectiveness of teaching mathematics for the purposes of future needs is a key interdisciplinary research problem in the entire world. Its purpose is to determine the conditions and strategies for the preparation and improvement of e.g. teaching staff as well as others. Its research task is to update or change the current educational system and the existing conception of preparing human capital for the needs of the future in the conditions of innovative development. It requires the modern generation of children, pupils, and youth to learn and be able to think, not just repeat texts learned during their educational period. Following this, the priorities of teaching mathematics at each level of education must be perceived, analyzed, and harmonized from the perspective of the expectations both of the individual as well as of society and the state, and then updated, taking into account the results of research projects and specific future standards. It cannot and should not be treated solely as a service provided by the state. Mathematics education should always be an investment in the development of generations of young people and adults. It should be considered necessary for their self-development and professional life. And the results of mathematical education of pupils/students should be and must be analyzed from a certain perspective:
1) What the they have learned ?;
2) What they know and what they are able to do?;
3) What they understand and what they can apply?
This is about mathematics or subject of mathematics after realization of a certain part or whole didactic module, over a period of time.
Contemporary reality is defined either as VUCA, short for the terms: volatility, uncertainty, complexity, ambiguity; or as the reality of the amount of big data received every second. It is related to the fourth scientific and industrial revolution in the world characterized by nanotechnology, nanoscience, nanospeed, nanosize, ICT. And in the nearest future, according to Bill Gates, the American computer scientist, it will also be characterized by three technological innovations in the field of energy. Its manifestations and effects concern not only the economy and national interests, but above all, society and people and their education. According to American physicist Michio Kaku, learning will change radically and will be no longer based on memorization. There is a huge role to play in the process of teaching students for the scientific and academic environment who are responsible for preparing the staff, as well as for the educational systems, schools, and teachers. Currently, the scientific community is spending an enormous amount of funds on interdisciplinary research programs such as the American research project BRAIN concerning the functioning of the human brain. The best universities of the world as well as the best scientists of various fields of science are involved.
This approach allowed me to discover, notice and analyse those new tendencies in the education of mathematics. Yet, mathematics is still one of those subjects which awakens strong feelings of anxiety, aversion and incompetence; and for most students it is perceived as complicated and senseless. This is where their belief that they are "mentally challenged" and doomed to fail in this subject comes from. On the other hand, mathematics is still perceived by pupils and students, by their parents and politicians with special aptitude and they find it to be a discipline for specially gifted ones. This is why the notion of "mathematical talent", "being naturally apt" in the mathematical thinking, or just "being a natural" in mathematics is perceived as something rare among the pupils, students and in the society, see : Г. В. Томский (2017). In this way mathematics is transformed into a sort of natural means of social selection, and what undoubtedly leads to the reluctance and anxiety towards it.
It is understood that sensory systems, i.e. the ears, eyes, nose and skin, can be likened to being the windows and doors of the human brain, because:
1) we may "hear" with our ears, but we "listen" with our brain;
2) we "look" with our eyes, but we "see" with our brain.
While researchers would be able to solve this problem, the results of such research would change what we know of the current educational process and its practical applications in a school or academic setting. Especially considering the fact that almost all of us, every parent or teacher, know and are aware of the need for proper habits, skills, and amount of time to master e.g. riding a bike. In a child's process of learning how to ride a bike, there is always the moment, sooner or later, when the learning ends as the desired ability is acquired. The child's success in learning how to ride a bike, and even its practical implications, can then be considered. The analysis of this example confirms our conviction that this step-by-step way to the success of the child was achieved in stages. The joy of the child caused by the success
has become the consequence of mastering and exercising certain necessary habits as well as the child's conscious competence (universal competence). Analogous methodology is used by students at school during the process of developing, shaping and assimilating mathematical concepts. The contemporary mission of universities and high schools of the third generation is to combine didactic and scientific activities while simultaneously transforming significant research results into the development of science and education of youth as well as economic and social development, entrepreneurship and innovation.
Two research projects undertaken in the Russian Federation also highlighted these needs. Their results can be found in the scientific monographs published in 2016, see their bibliographic description [16], [20]. The first monograph, entitled Conception of Functional Skills of Students: Mathematics and Computer Science, edited by academic RAO M.V. Ryzhakova discusses the motivation and methodology of studying the phenomenon of the Functional Skills of Pupils (FSP) in teaching mathematics and computer science at school and presents the results obtained, materials, and publications. In this monograph, the authors present: 1) the ontological basis and approach to define the FSP phenomenon and its practical aspects, 2) the concept and ways of shaping it in the process of teaching mathematics and computer science at school. In addition, the authors make a thorough analysis of the FSP phenomenon as well as the international standards of mathematics education at school in the 21st century and the pedagogical experiences related to their implementation in Switzerland, Kazakhstan, China, South Korea and Singapore. The summary of this monograph articulates the theoretical aspects of the research results obtained via the completed research project. The work also contains a presentation of their potential usage for solving the problem of individualizing mathematics and computer science at school or for developing sets of exercises or materials and methodologies for shaping the FSP phenomenon.
The second monograph contains the results of a research project entitled Mathematics. Psychology. Intellect, published with the support of the Institute of Psychology of the Russian Academy of Sciences, see: [20]. The authors of the monograph describe in detail the psychological basis of the intellectual development of pupils as a result of applying a psychopedagogic approach. The proposed teaching methodology is based on specifically-prepared teaching texts that develop and enrich the cognitive, conceptual, metacognitive, and emotional experiences of students. The typology and characteristics of the so-called teaching texts for development and the related examples from school textbooks in mathematics as well as the methodological examples of the usage of such texts are described in detail.
The results of research on the mathematical education of pupils and students confirm that its implementation into teaching practice in general or academic education requires:
1) the articulation of the importance of mathematics in the development of memory and in the ability of logical thinking, in the intellectual development of a person,
2) teaching the key mathematical skills, such as: "to think logically"; "to know how to read and process"; "to know how to see"; "to know how to solve" and many others,
3) presenting examples showing that without learning mathematics and knowledge in this subject it is hard to function in the adult life, and it is even harder to achieve professional success,
4) gathering collective experience in the search of techniques, methods of solving the exercises, tasks and problems in mathematics, or in the verification.
5. Summary
In summary, the analysis of contemporary literature and the results of the research indicate that prioritization in school mathematics is unquestionably related to its quality, to the quality of school graduates, the preparation of the staff, and a society for the future. This will make it possible to refer and use in teaching practice to the work Г. В. Томский (G. V. Tomski) on the system for the early and unmistakable definition of mathematical talent, see: [18]. We believe that the improvement of the practice and results of informatization of general and subject education should be started earlier than before, which will be shown by the results and experience of the Edumatrix method. This hypothesis is also confirmed by the accumulated experience and current results from the implementation of the European project - the Junior Code Academy Project. However, there is a need to extend its verification in other countries and regions of the world. You also need to investigate the open problem: the requirements and forecasts of the informatization system for general and subject education, in particular mathematics education, children, pupils and students for the needs of the future, see : www.linkedin.com/in/alexkrol . In this regard, of course, we need an actual diagnosis and prognosis, a solid look at: 1) conditions of modernization and information technology of QS for the future development of our countries, 2) the current importance and role of the subject "Informatics" and computer programs in general and subject education. There is no doubt that the Internet, ICT and other technical tools support the traditional methods of teaching mathematics and the perception of the content of general education, as well as methods of mathematical modeling of phenomena and problems. Modern research results confirm that the correct use of modern visual means, ICT, distance learning platforms increases the efficiency of cognitive processes. This, however, does not preclude the use of traditional methods of the natural-mathematical learning cycle, see: A. Pardala (2010, 2018).
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Information about the authors of the article
Антони Пардала, д.п.н., к.м.н., вузовский профессор DSc, PhD Antoni Pardala, Emeritus Associate Professor Rzeszow University of Technology
DSc (Doctor of Pedagogical Sciences in the field of Mathematics Education), PhD (Candidate in Mathematics Sciences)
Rzeszow University of Technology
35-959 Rzeszow, street Al. Powstancow Warszawy 12, Poland
Роза Изтлеуовна Кадирбаева, д.п.н. DSc Roza Iztleuovna Kadirbayeva
DSc (Doctor of Pedagogical Sciences in the field of Mathematics Education)
South Kazakhstan State Pedagogical University
160000 Shymkent, street Baitursynov 13, Republic of Kazakhstan
Мадина Алдановна Джаманкараева, к.ф-м.н.
PhD Madina Aldanovna Jamankarayeva
PhD (Candidate of Physical and Mathematical Sciences)
South Kazakhstan State Pedagogical University
160000 Shymkent, street Baitursynov 13, Republic of Kazakhstan
