Viktoria I. Snegurova

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.

Educational chat-bots are becoming important tools for supporting students’ independent learning. In case of teaching higher mathematics, the architecture of chat-bots becomes particularly significant: moving from rigid linear scenarios that ensure a strict sequence of topic study to more flexible navigational models. It allows students to independently build their own learning flows. However, fully free navigation is associated with the risk of fragmented and superficial assimilation of the material, disruption of the course logic, and loss of methodological integrity. This highlights the necessity of transitioning to a graph-based model that combines the structure of the course with the possibility of personalized choice and adaptation depending on the level of knowing the material. The aim of the study is to substantiate the methodological and technological feasibility of implementing a graph-based chat-bot architecture capable of taking into account semantic links between concepts, typical student's difficulties, and natural language queries. The theoretical part builds on contemporary research in the field of personalized higher mathematics education, the use of graph structures in pedagogy, and natural language processing technologies. The empirical study, conducted in 2024–2025 at the Yerevan branch of Plekhanov Russian University of Economics, included a comparison of the effectiveness of two types of educational chat-bots – those with sequential (linear) and free navigational organization. The collected data showed that chat-bots with free navigation flow were perceived as more convenient and structurally flexible, but they also came with the risk of insufficient learning of the material due to the lack of reliance on internal course logic. Based on the analysis of student feedback and theoretical premises, the necessity of introducing a flexible graph-based model has been substantiated. This model ensures not only variability in navigating the learning material but also contextual returns to key topics, adaptation of the pace and content of learning, and support for self-assessment processes. Such a model allows achieving a balance between freedom and logical coherence of the learning material, which is particularly important for disciplines with a hierarchical structure of concepts, such as mathematical analysis. The results of the proceedings can be used in designing digital solutions for teaching higher mathematics and in creating chat-bots aimed at personalizing the educational flow.