Том 511, № 1 (2023)

The article formulates the foundations of modern mathematical education that meet the new goals and objectives of mathematical education dictated by: changes in mathematics itself over the last century, changes in its role in the modern digital world, changes in the world itself. The basic system of concepts is described, starting from which all modern mathematics of finite as well as primary mathematical education is built. For all levels of mathematical education and for all learners, independent, with the help and support of a motivating teacher, solving problems that are not known (to the student) how to solve, are fundamental. A significant role is played by a computer mathematical experiment and the transfer of tasks to the computer, the solution of which is found by the student. The article also describes the real state of affairs in the Russian mass school and proposes the necessary actions to implement the formulated principles of modern mathematical education in Russia. The results achieved in this direction are described. Most of the articles in this issue demonstrate and detail the implementation of formulated principles.

In recent decades, several Russian schools have been implementing a unique for the world education program of mathematics for elementary schools. In it, the landscape of school arithmetic is radically expanded due to the basic objects of modern mathematics and computer science. These objects and their operations are visual, making them much more accessible than traditional arithmetic. The range of activities is also expanding due to, for example, the introduction of strategies for enumeration, winning the game, algorithms (also operating in a visual environment). At the same time, the student’s position is changing – they independently discover and builds mathematics, constantly solves personally new, but feasible tasks, which are “not-known-how-to-solve”. The student’s resources are saved by using a computer to perform routine arithmetic operations that they have already discovered and understood. The paper presents in detail the implementation of this approach, illustrated by real examples of tasks representative of the program under consideration and for our vision.

The paper is describing the class of educational problems from the course of mathematics and computer science for elementary school. This course has been implemented over the past decades by a team led by A.L. Semenov. In the problems it is required to find, build, list all objects that satisfy a certain system of conditions. The student conducts these activities in the visual world of the basic objects of discrete mathematics and computer science: strings (finite sequences of symbols), bags (multisets), tables, trees, and statements containing quantifiers. The connections of these problems with problems of computational combinatorics (“counting the number of options”), search problems of the theory of computational complexity, with “big ideas”, general cognitive strategies in their formation are considered. The content of education in our approach is more adequate to the context of the modern world.

Mathematical experiment has always been a key source for mathematical discovery. Over the past 50 years, thanks to digital technologies, its role in mathematical research has grown significantly. Digital technologies have opened up fundamentally new opportunities for experimentation in mathematics education, bringing the majority of students of mathematical education closer to mathematical research. Such an approximation is especially desirable in the modern world, where it becomes possible thanks to digital technologies. The article discusses the results of the work of the authors over the past decades on the application of a computer mathematical experiment at different levels of school and university education. Particular attention is paid to dynamic geometry environments. The possibilities of using computer algebra systems are also considered. The project of schoolchildren’s work on generalizations of Napoleon’s theorem is considered in detail.

Mathematical education, both mass education, and university education of non-mathematicians, are in an abominable state, and rapidly degrading. We argue that the instruction of non-mathematicians should be dramatically reformed both as substance and style. With traditional approach, such a transformation would take decades, with unclear results. But we do not have this time. The advent of Computer Algebra Systems gives the mathematics community a chance to reverse the trend. We should make a serious attempt to seize this opportunity. In the present paper we describe one such project of reform implemented at the St Petersburg State University.