Application of new strategies of positive pedagogy in modern mathematical education

Raikhelgauz Leonid ORCID: 0000-0001-9797-794X PhD in Physics and Mathematics Associate Professor, Department of Partial Differential Equations and Probability Theory, Voronezh State University 394030, Russia, Voronezhskaya oblast', g. Voronezh, ul. Universitetskaya Ploshchad', 1 jikol_85@mail.ru

The challenges facing the education system today are due to the peculiarities of the organization of the learning process in a post-industrial society. In the 19th and 20th centuries, the criterion of maturity was considered to be the solid assimilation of knowledge, skills, and responsible fulfillment of a clearly defined range of tasks, today the effectiveness of social processes increasingly depends on creativity, initiative, and the search for new solutions to old tasks and to those that arise for the first time. These changes actualize the departure from the knowledge-oriented paradigm, didactic approaches aimed at enriching the student with information. Now the emphasis is changing from the assimilation of information to the development of a strategy for working with it, the ability to reproduce these strategies regardless of the learning conditions, control situations in spite of complicated situations becomes significant. We agree with E.I. Kazakova that the integration of the education system and the digital world should go not only (and not so much) along the path of introducing digital solutions into the learning process, but rather along the path of modernization of offline educational practices. The main question is not what is in the computer, but how the teacher and student interact about this content ^{[3, p. 50]}.Another significant result of education is the socialization of personality – educational results are considered not as a final value, but as an opportunity for a person to achieve social success.

An important emphasis in this direction is the emphasis on transferring the knowledge, skills, and competencies acquired in training to solving life situations. This trend is manifested in the increase in the importance of functional literacy in school education, which is defined by PISA as the ability of a person to understand and use written texts, reflect on them and engage in reading in order to achieve their goals, expand their knowledge and capabilities, participate in social life ^{[10, p. 21]}. The main expectations from the school in terms of the formation of functional literacy are: the development of the ability to transfer knowledge to extracurricular situations; the formation of the skill of effective relations with the outside world; the ability to adapt quickly in changing conditions. All this gives rise to a lot of discussions about the relationship and hierarchy of subject, meta-subject and personal results. Thus, there is a need for a new humanistic paradigm in which knowledge is considered as a means to achieve individual goals, and the end result is aimed at the formation and development of fundamentally new personality traits of the student, so that the process of his self-determined development is as effective as possible. Authoritarian teaching strategies show their inefficiency in solving this problem, which actualizes the search for new didactic solutions due to the humanization of pedagogical practice. The main research question, the resolution of which our research is aimed at, is formulated as follows: how can the ideas of positive pedagogy be applied in mathematical education to form a stable, personally significant educational result for a student?The ideas of positive pedagogy have analogues in many pedagogical concepts: humanistic pedagogy, pedagogy of success, pedagogy of nonviolence, personality-oriented pedagogy.

The closest approach to modern views on positive pedagogy is the concept of felixological education by E.P. Pavlova and N.E. Shchurkova, the essence of which is to educate a person's ability to be happy in this life on this earth ^{[7, p. 3]}.

In the paradigm of continuing education, a personality-oriented approach is a key mechanism for the formation of lifelong learning attitudes, and neglect of its postulates leads to the appearance of academic barriers and their preservation throughout life ^{[5, p. 132]}. In the context of a personality-oriented approach, the purpose of education is to create conditions for the development of the individuality of a student of any age. Relations between all participants of the educational process are characterized by manifestations of subjectivity: so the teacher does not influence the student authoritatively, but creates conditions for students to realize the goals and objectives of mastering a particular material, accepting these tasks as personally significant and making efforts to achieve them. The student is also perceived as a subject of his own educational activity: he can set academic tasks himself that differ from those formulated by the teacher, develops his own learning strategy based on understanding the optimal ways for himself to learn new things and overcome difficulties that have arisen in the course of achieving the result. Accordingly, there should not be a fixed content of education with a personality-oriented approach – it is formed during the interaction of participants in the educational process. As priority teaching methods, the dialogue between the teacher and the student, the student's independent work with information, the performance of creative works, the need for which is determined by the student himself, are put forward. Forms of learning also acquire the characteristics of interactivity, practice-oriented and self-organizing formats are most often used, self-education plays an important role. I.M. Osmolovskaya believes that "a purely personality-oriented approach currently does not exist in the practice of teaching, some of its elements are used" ^{[6, p. 36]}.

Undoubtedly, positive psychology has also contributed to the emerging positive pedagogy ^{[11, p. 280]}, which touches on issues related to positive emotions, personal characteristics and attitudes, and is aimed at understanding and "training" the state of happiness and hope. A distinction can be made between positive psychology as a perspective where research and practice are consistent with the values and intentions of positive psychology, and as a discipline whose research and practice are intentionally located within certain boundaries of the scientific field. As a perspective, positive psychology emphasizes striving for what is good, virtuous, and perhaps with less emphasis on avoiding what is bad, immoral, and problematic. The focus is on strengths, not weaknesses, and thriving over (or in spite of) suffering.

The first wave of positive pedagogy ideas was represented by the application of positive psychology to teaching in schools. Various frameworks, approaches and curricula have been developed aimed at creating happy, engaged students ^{[14, p. 9]}. Over time, it became clear that while focusing on the well-being and happiness of students could be beneficial, it was clearly not enough. The understanding has come that the environment matters. Thus, the second wave of positive education emphasizes going beyond specific programs that develop specific skills to create learning conditions that support the well-being of everyone in the school community, both within the framework of the discipline taught and within the curriculum as a whole ^{[13, p. 15]}.

Today, positive psychology, which, according to T. Lomas, is at the stage of the "third wave" of its development, is characterized by a general movement of expansion "beyond the individual", a movement towards greater complexity in terms of: research orientation (increasing interest in superindividual processes and phenomena); objectivity (becoming increasingly interdisciplinary); culture (becoming increasingly multicultural and global); and methodologies (covering other ways of cognition) ^{[12, p. 7]}.

The "didactics of optimism" justified by Yu.V. Andreeva corresponds to our methodological positions and ideas of positive pedagogy ^{[1, p. 48]}. This direction of didactics solves an important research problem: how to develop optimism within the framework of training and how to effectively use pedagogical resources and means of success situations? The author's concept is based on a number of provisions consonant with our ideas: the understanding of pedagogical optimism as an acquired personal and professional quality of a teacher, based on the innate desire of the individual for success and the basis of the student's academic optimism on the situations of success lived by him in which such significant factors of academic success are formed as: motivation to achieve success, overcoming educational difficulties on the way to it, positive thinking and the ability to constructively analyze failure.

At the same time, the "didactics of optimism" is presented by the author as a theory of teacher training, it is about educating the future teacher of an optimistic view of the student and the learning process. In the concept of positive pedagogy, the learning space is considered more broadly "with an understanding of the two-sidedness of the pedagogical process:  designing and providing special conditions for positive changes both in the pedagogical activity of the teacher (setting the goal of developing a positive personality, searching for appropriate principles, content, technologies, criteria and indicators of pedagogical activity) and in the personality of the student: his style of thinking, attitude and activity" ^{[4, p. 235]}.

For us, the essence of positive pedagogy is revealed in the etymology of its name - "positum" (from Latin - given, actual, existing). In the context of mathematical education, this is taking into account the actual academic experience of the student (I know how to solve). And here we integrate the provisions of positive pedagogy and metacognitivistics: the student is a cognizing subject, has individual prerequisites for academic success and an arsenal of means for solving mathematical problems. Awareness, understanding and application of mathematical skills allows the student to develop not only their own cognition, but also metacognitive strategies (I think about how I solve the problem). In teaching mathematics, it is necessary to take into account not only the abilities and interests of the student, but also to diagnose difficulties arising in time, considering any problems in mastering the metematic content as individual ways of responding. In the process of studying complex mathematical knowledge, learning errors are marked with the meaning "here you can improve your abilities", and successes and achievements show the student the resources developed by him, on the basis of which he will be able to build his educational strategies in the future.

In our opinion, positive pedagogy involves the adaptation of traditional forms of education focused on the formation of academic competencies with scientific knowledge of psychology and cognitive science about the optimal development of cognitive abilities and the well-being of the student. It is aimed at developing students' abilities for well-being (well-being in the role of a student and effective educational functioning). This meets the basic needs of the modern generation of youth identified by I.Y. Tarkhanova and M.A. Zaitseva ^{[9, p. 341]}. As the priority generational characteristics, the authors designate: positive thinking (the expectation of demonstrating positivity and brightness, readiness for challenges, orientation to relationships on equal terms, freedom of self-realization);  the desire for emotional safety of interaction (the search for positive qualities of interaction partners, the request for emotional safety and comfort in communication, the demonstration of a benevolent attitude); the desire for independence (the need for adults to accept their point of view, unwillingness to follow norms and regulations), the latter the authors attribute to age, not generational characteristics.

Thus, positive pedagogy is at the stage of its active formation and is difficult to separate from other concepts and scientific trends. But it is positive pedagogy that is completely consonant with the ideas of metamodern. We agree with S.V. Kolesova that "positive pedagogy is a vivid example of an integrative direction in the development of pedagogical science" ^{[4, p. 237]}.

With regard to the problems of mathematical education, the application of the ideas of positive pedagogy also seems to be effective, since our long-term observations allow us to state students' anxiety and anxiety, if necessary, to solve mathematical problems and perform non-elementary numerical actions, this situation is fixed by us both in the school and university process of studying mathematical disciplines. But mathematics is an exact science aimed at the formation of knowledge, which is based on working with complex knowledge. Is the emotional background, success situations and personal satisfaction of the student with the educational result so important when studying it? In our opinion, the introduction and study of mathematical material creates a precedent for the actualization of universal educational actions (UDS), including in conditions of uncertainty of means and choice of content, multiplicity of setting goals and the possibility of interpreting results, non-standard meta-subject use of techniques and methods for the study of "problem areas" of the development of subject activity. Thus, consistently working with complex mathematical knowledge, giving analogies of its application in life, we create a situation of success. The student learns to work with a mathematical text, analyze the solution of a problem, formulate questions, he analyzes where it is clear, where it is not clear, and the teacher helps to build an individual educational route for his needs. 

Mathematical education today, from the point of view of educational results, is considered by us not so much as the study of subject content. Mastering sufficiently complex knowledge in mathematics to the extent necessary for passing the final state exam usually does not require that the student is absolutely free to navigate the entire school course of mathematics, but a necessary condition for the successful development of knowledge is personal interest and a desire to understand the logic of mathematical material. The study of mathematics requires constant work, constant solving of mathematical problems, the result of which every mathematics teacher at school strives for is the ability to study the subject independently, apply mathematical skills in life to solve everyday and cognitive tasks. This is the element of functional literacy formed by means of mathematical education. It is no coincidence that the concept of "mathematical literacy", which has been discussed so much recently in the scientific and pedagogical community, in addition to the subject component, contains an essential meta-subject component related to the communicative, informational, reading and social competencies of the individual ^{[8, p. 175]}. An important meta-objective result of mathematical education seems to us to be the ability of the student to consciously approach the process of achieving an educational result and demonstrate its stability regardless of the complexity of the mathematical material, the attitude and support of the teacher, the emotional state, changes in the conditions and factors of verification. The survey of school teachers conducted by us (n=78) confirms the importance of the above-subject factors of the stability of the educational result in the subject area "mathematics".

So the factors of the stability of the results of teaching mathematics teachers called: abilities 32%; diligence 38%; technology proficiency 41%; mastering knowledge, skills 69%; teacher competence 28%; parental control over homework 14%. As you can see, most school teachers believe that the sustainability of educational results depends on the development of knowledge, skills and abilities. However, often, a child who has successfully graduated from school and has shown high results on the unified state exam shows a sharp decline in academic performance at the stage of studying at the university, that is, his academic results are not stable.

Our analysis of academic performance in mathematical disciplines of first-year students of the humanities of Voronezh State University showed that the majority of students have difficulties in mathematical activities included in the programs of basic general education. Students of humanitarian fields of training often have a desire to avoid mathematical actions, a desire to complete the solution of the problem faster, regardless of the correctness of the result obtained. Perhaps such manifestations are associated with negative experiences received at school. The fear of being negatively evaluated, doubt in one's mathematical abilities leads to avoidance of mathematical disciplines at the university and unwillingness to choose a natural science and engineering education.

Diagnostics of first-year students (n = 128) using the questionnaire "Mathematical anxiety" (Anxiety rating Scale (SMARS) confirms the presence of the problem of fear of mathematical content. The questionnaire includes 25 questions about situations related to mathematics, in relation to each of which the subject must assess the degree of his anxiety on a five-point scale (from "not at all" to "very much"). In the course of the study, high average scores on the scale of test mathematical anxiety were found, indicating that the majority of subjects have a fear of tests, control papers and exams. There was also an increased anxiety about the need to attend mathematical courses, especially characteristic of students of humanitarian fields of training. Observation of these manifestations suggests that the stability of the results of mathematical education is influenced by such a factor as mathematical anxiety formed during school education. Thus, modern pedagogical practice faces the task of finding approaches to teaching mathematics that prevent the development of anxiety.

An important role in this is played by the formation of educational motivation, the value of studying mathematics and using it to solve life problems. The consideration of teaching mathematics as a process of constructing personal values of a learning subject is based on the definition of value-semantic units of mathematical content that have personal significance for the subject of learning. With the help of pedagogical support, the student finds such a meaning, internalizes it and fixes it in his own life orientations related to learning and achieving social success. In our opinion, the main role in overcoming mathematical anxiety belongs to the teacher. It is teachers who should take more active actions in encouraging anxious students, instilling in them confidence in their abilities. The effectiveness of any pedagogical activity, both within the framework of formal education and in private tutoring practices, largely depends on the teacher's ability to create an atmosphere of trust, cooperation and mutual understanding, take care of the child's personality, provide him with conditions for the disclosure of abilities and the development of opportunities for self-realization in educational activities, including when solving mathematical problems.Within the framework of the concepts of positive pedagogy, a number of approaches have been developed that can be used to preserve the stability of educational results and relieve mathematical anxiety: a facilitative approach aimed at building relationships between the teacher and the student, based on a helping teaching style, on the empathy of the teacher; a collaborative approach based on the joint activity of students (individual growth with group interaction); development of functional literacy (in the light of our topic – mathematical literacy), based on the development of the ability to transfer mathematical knowledge into an interdisciplinary and life context.

Contextual learning (application of mathematical statistics to psychology); a reflexive approach to learning that contributes to the formation of motivation, management of activity as a process and the ability of the individual to assess the impact of the efforts made on the educational result.

The igrodidactic approach deserves special attention. Its use in mathematical education has not yet gained popularity, since for a long time it was believed that game moments only distract from studying. And today, criticism of the gamification of education continues to prevail in pedagogical discourse. It is discussed that gamification unnecessarily simplifies the motivational task of learning, reducing it to primitive reinforcement with entertainment stimuli; game culture in the civilizational aspect can displace educational culture. However, a number of studies in the field of positive pedagogy consider the results of learning using gamification elements as the most positive, for example, in terms of increasing motivation and involvement in learning tasks, as well as pleasure from them ^{[2, p. 36]}. In our opinion, it is necessary to explain to teachers-mathematicians the didactic meaning of the introduction of gamification elements into the education system in general and the management of the learning process of mathematics in particular; to familiarize them with new forms and ways of organizing a motivationally secured educational space through immersion in the game space, including virtual. It is also an urgent task to identify and evaluate the didactic effects of this learning technology.Thus, the application of humanistic constructs of positive pedagogy in mathematical education manifests itself in the development of metacognitive and personal-motivational characteristics of cognitive activity of students on the development of standards and samples of mathematical thinking and the choice of ways of applying mathematical knowledge to solve life problems.

The search and analysis of ways to correlate the skills of solving mathematical problems and the mechanisms for implementing interdisciplinary and extra-academic relationships based on a personality-oriented approach allows you to form a student's attitude to self-development and self-organization, which, in turn, contributes to the growth of the stability of the educational result.