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| 1 | The teaching of mathematics should reflect the global trends in the development of mathematical and pedagogical scientific thoughts. The authors develop guidelines of studying mathematics in the 1st grade on the basis of the Federal State Educational Standard, taking into account the partitions into knowledge elements. Thus, the structure and the formation of the being developed textbook content by chapters are described. The theoretical part should allow to students to get answers to control questions, univariate and multivariate tests, problems and exercises. Each chapter of the being developed textbook is divided into paragraphs, each paragraph is divided into items. One item of theoretical material contains either one new idea to study, or a set of interrelated concepts that define one new idea to study. According to this it is possible to form the mathematical knowledge trajectories. So, there are broad opportunities to transform knowledge elements into multimedia forms, including presentations, videos, test modules and other newly emerging means of representation and visualization, using the achievements of artificial intelligence. There can be defined such main directions as natural numbers counting, acquaintance with geometric shapes, comparison of objects in the value and quantity, acquaintance with a squared paper and measurement standards, the study of objects position, the sets consideration, introduction to the algorithm concept. The conceptual directions of teaching mathematics in the 1st grade determined by the authors provide the foundation for the vertical of mathematical education in the 1–11th grades of general education schools. Keywords: primary education, teaching, mathematics, knowledge element | 1089 | ||||