INTERACTION OF KINDERGARTEN AND FAMILY IN THE MATHEMATICAL DEVELOPMENT OF PRESCHOOLERS

The relevance of the study of the interaction of kindergarten and family in the mathematical development of preschoolers is due to the fact that the unification of the family and kindergarten in one educational space implies the cooperation of teachers and parents. The mathematical development of preschool children will be successful if the teacher interacts with parents. Only cooperation between the kindergarten and the family can ensure the effectiveness of educational activities in the mathematical development of a preschool child. It is necessary to select the forms and methods of organizing this interaction, taking into account the age characteristics of children and the settings of the Federal State Educational Standard (complex tasks, integration of different types of activities, individualization of the child's mathematical development, etc.) in the mathematical development of preschool children. The work of a preschool teacher on the mathematical development of children is particularly time-consuming and requires a lot of pedagogical attention. Mathematical knowledge of preschoolers, such as counting down to ten in ascending and descending order; the ability to recognize numbers in a row and randomly, quantitative (one, two, three...) and ordinal (first, second, third...); to study and present the basic geometric shapes (triangle, square, circle, and others); the basic forms of measurement: the ability to measure length, volume, width, height using a conditional measurement measure; comparison of objects: heavier-lighter, larger-smaller, wider-narrower, higher-lower; the ability to navigate in space and time, etc., will later be the basis for school education.

Measuring dimensionality and purity of high-dimensional entangled states

High-dimensional entangled states are promising candidates for increasing the security and encoding capacity of quantum systems. While it is possible to witness and set bounds for the entanglement, precisely quantifying the dimensionality and purity in a fast and accurate manner remains an open challenge. Here, we report an approach that simultaneously returns the dimensionality and purity of high-dimensional entangled states by simple projective measurements. We show that the outcome of a conditional measurement returns a visibility that scales monotonically with state dimensionality and purity, allowing for quantitative measurements for general photonic quantum systems. We illustrate our method using two separate bases, the orbital angular momentum and pixels bases, and quantify the state dimensionality by a variety of definitions over a wide range of noise levels, highlighting its usefulness in practical situations. Importantly, the number of measurements needed in our approach scale linearly with dimensions, reducing data acquisition time significantly. Our technique provides a simple, fast and direct measurement approach.