In recent decades, several Russian schools have been implementing a world-unique 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 comprehensible than traditional arithmetic. The range of activities also expands due to, for example, the introduction of strategies for enumeration, game winning, and algorithms (also operating in a visual environment). At the same time, the student’s position changes: they independently discover and build mathematics and constantly solve personally new, but feasible tasks that are “not-known-how-to-solve.” The student’s resources are saved by using a computer to perform routine arithmetic operations that have already been discovered and understood. The implementation of this approach is discussed in detail and is illustrated by examples of actual tasks that are representative of the program under consideration and the whole approach.

The paper describes in detail a class of educational problems from an elementary school course of mathematics and computer science. This course has been implemented over the past decades by a team led by Academician of the RAS A.L. Semenov. In the problems, it is necessary to find, build, or list all objects that satisfy a certain system of conditions. The student conducts these activities in a visual world of 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 in computational combinatorics (counting the number of options), with search problems in the theory of computational complexity, with “big ideas,” and general cognitive strategies for their formation in education are considered. The content of education in our approach is more adequate to the context of the modern world.

This 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, by changes in its role in the modern digital world, and by changes in the world. The basic system of concepts is described, starting from which the whole modern mathematics of the finite, as well as primary mathematical education, is built. For all levels of mathematical education and for all learners, the ability to solve problems whose solutions are not known (to the student) and to do it independently with the help and support of a motivating teacher is a fundamental skill. A significant role is played by a computer mathematical experiment and the transfer of tasks with solutions having been found by the student to the computer. This article also describes the actual state of affairs in the Russian mass education system and proposes the necessary actions to implement the formulated principles of modern mathematical education in Russia. Most of the articles presented in this issue demonstrate and detail the implementation of the formulated principles.

We consider the content and methods of teaching mathematics in primary school in the context of digitalization. The necessity of a significant updating of the curriculum is shown. In particular, this implies the use of digital tools as an object of study and a way of significantly improving the content and effectiveness of learning mathematics to prepare children for the future life and work.

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.

A mathematical experiment has always been a key source for a 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 mathematical education closer to mathematical research for the majority of students. This tendency is especially desirable in the modern world, where it becomes possible thanks to digital technologies. The article discusses the results of the authors’ work 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 discussed. A project concerning schoolchildren’s work on generalizations of Napoleon’s theorem is considered in detail.

The words “Programming is the second literacy” were coined more than 40 years ago [13], but never came to life. The paper develops and details that old slogan by proposing that the mainstream mathematics education in schools should merge with education in computer science/programming. Of course, this means a deep structural reform of school mathematics education. We are not talking about adapting the 20th century mathematics to the 21st century—as it outlined in [10, 19], we mean the 21st century mathematics education for the 21st century mathematics. To the best of our knowledge, this paper is perhaps the first known attempt to start a proper feasibility study for this reform. The scope of the paper does not allow us to touch the delicate socio-political (and financial) sides of the reform, we are looking only at general curricular and didactic aspects and possible directions of the reform. In particular, we indicate approaches to development of a Domain Specifiic Language (DSL) as a basis for all programming aspects of a new course.

This paper considers an approach to mathematical education adequate to the task of developing mathematics and its applications in the 21th century. This approach is based on improving the efficiency of the educational process by maintaining the motivation of students of various categories. The basis for the formation of motivation is, on the one hand, independent design; invention of mathematical objects, methods of action, and models of the world around us; and the discovery of facts of mathematical reality and, on the other hand, solving of new, unexpected, and feasible tasks for the student. The student’s work is similar to the work of a mathematician–researcher and programmer. The possibilities of research work in educational mathematics are significantly expanded due to computer-based intramathematic experiments. Debugging a computer program is a special kind of mathematical experiment.

The paper discusses an example of an educational project in modern mathematics in which school students create mathematics that is new to them. The mathematical results produced by the students in the theory of definability also have an “absolute” novelty, i.e., are the basis for professional publications. The described course was based on recent results of this article’s authors in the theory of definability. The new results were obtained by a group of 10 schoolchildren from different regions of Russia.