Natalya V. Yezhova

In the context of the digital transformation of education and information overload, traditional teaching methods no longer meet the needs of society and the individual. There is a contradiction between the need for students to develop a deep critical understanding of the world and the need to develop skills in navigating dynamic, distributed information networks. In our opinion, the resolution of this contradiction is possible through the synthesis of two main modern pedagogical theories of connectivism and constructivism. In this study, we made their comparative analysis. The paper reveals their philosophical and theoretical foundations, contrasting the inner meaning generation of an active student (constructivism) and navigation through distributed external networks of knowledge (connectivism). The authors compare the basic principles of the theories, the role of the teacher and the student, as well as practical methods for implementing each theory in the educational process. Special attention is paid to their responses to the challenges of the digital age. The paper systematizes criticism of both concepts, identifies their weaknesses and strengths. In conclusion, the position on the complementary potential of theories is substantiated. The conclusion is made about the need for synthesis: constructivism provides depth of understanding and critical thinking, while connectivism offers tools for working with dynamic information flows. The article emphasizes that an effective educational model of the future should integrate the ability for deep reflection (constructivism) and the competence of communication management in the digital environment (connectivism). The research uses methods of theoretical analysis of the principles of connectivism and constructivism, modeling and network analysis. The conducted complete comparative analysis of the two most important pedagogical theories makes it possible to streamline and clarify their conceptual fields in relation to digital didactics. It is shown that they are not mutually exclusive alternatives, but represent complementary levels of the educational process organization. The work expands the theoretical basis for designing adaptive educational trajectories, showing that adaptivity should not be limited only to the selection of tasks according to complexity (as in classical adaptive learning), it should include the possibility of choosing different ways of meaning formation (constructivism) and networking (connectivism). The results of the study are of direct practical importance for teachers, methodologists and developers of digital educational products.

A pressing problem in modern education, confirmed by both official data and local studies, is schoolchildren's inability to explore various problem-solving approaches, evaluate intermediate results, and select optimal solutions. The aim of this study is to demonstrate that engaging high school students in well-organized mathematical dialogue based on solving a series of modified tasks helps solve this problem. By task modification, we mean a partial change in the task data or the addition of new data, leading to qualitative changes in the solution process, increasing the number of solution options, or leading to the possibility of no solution. By solving and analyzing such problem series, students learn to identify patterns and build mathematical models. The empirical study, conducted during the 2024-2025 academic year at the Natural Sciences Lyceum of Peter the Great St. Petersburg Polytechnic University, involved the systematic inclusion of modified tasks series in the educational process. Sixty-four 10th-grade students participated in the experiment. Classes were conducted using both traditional methods and modified and dialogue-based learning. Testing, workbook analysis, and observation of student activity in class were used to evaluate the effectiveness of this approach. Logarithmic equations and inequalities formed the basis for the developed learning materials. The article provides an example of a series of tasks resulting from the modification of a simple example. Methods for gradually increasing the complexity of modified tasks, allowing for differentiated instruction, are discussed. The study demonstrates that dialogue-based learning, incorporating elements of task modification, leads to a deeper understanding of the material and promotes the development of students' research skills. The theoretical significance of the study is determined by the development of methods in which work with modified tasks is transformed from a discrete element of in-depth training into an integrated educational model implemented throughout the entire mathematics learning process. The study clarifies the role and methods of organizing educational dialogue in solving non-standard problems, demonstrating its importance for finding and reasoning about solutions. Since the implementation of the proposed methodology improves the quality of schoolchildren's mathematical training by purposefully developing their ability to solve not only standard problems but also creative, research-based ones, this study has practical significance.