Mathematics is a science historically based on solving problems about quantitative and spatial relations of the real world by idealizing the necessary properties of objects and formalizing these problems. The science dealing with the study of numbers, structures, spaces and transformations.

People usually think of mathematics as just arithmetic, that is, the study of numbers and actions with them, such as multiplication and division. In fact, mathematics is much more than that. It is a way of describing the world and how one part of it combines with another. The relationship of numbers is expressed in mathematical symbols that describe the universe in which we live. Any normal child can excel at math because “number sense” is an innate ability. True, it takes some effort and some time.

Being able to count is not everything. A child needs to be able to express himself well in order to understand tasks and make connections between facts that are stored in memory. It takes memory and speech to learn the multiplication table. This is why some people with brain damage have difficulty multiplying, even though other types of counting are not difficult for them.

Other kinds of thinking are also required to be good at geometry and to understand shape and space. We use math to solve problems in life, such as dividing a chocolate bar equally or finding the right shoe size. Through knowledge of math, a child knows how to save pocket money and understands what he or she can buy and how much money he or she will then have left over. Math is also the ability to count the right amount of seeds and sow them in a pot, measure the right amount of flour for a pie or fabric for a dress, understand the score of a soccer game and many other everyday activities. Everywhere: at the bank, in the store, at home, at work, we need to be able to understand and deal with numbers, shapes and measures. Numbers are only part of a special mathematical language, and the best way to learn any language is to apply it. And it’s best to start at an early age.

About math “smartly.”
Usually idealized properties of objects and processes under study are formulated as axioms, then by strict rules of logical deduction other true properties (theorems) are derived from them. This theory in aggregate forms a mathematical model of the object under study. Thus, initially starting from spatial and quantitative relations, mathematics obtains more abstract relations, the study of which is also the subject of modern mathematics.

Traditionally, mathematics is divided into theoretical mathematics, which performs in-depth analysis of intramathematical structures, and applied mathematics, which provides its models to other sciences and engineering disciplines, some of which occupy a borderline position to mathematics. In particular, formal logic can be considered both part of philosophical sciences and part of mathematical sciences; mechanics is both physics and mathematics; computer science, computer technology, and algorithms belong both to engineering and mathematical sciences, etc. There are many different definitions of mathematics in the literature.

Mathematics studies imaginary, ideal objects and the relations between them, using a formal language. In general, mathematical concepts and theorems do not necessarily correspond to anything in the physical world. The main task of the applied mathematician is to create a mathematical model that is sufficiently adequate to the real object under study. The task of the theoretical mathematician is to provide a sufficient set of convenient means to achieve this goal.

The content of mathematics can be defined as a system of mathematical models and tools for their creation. A model of an object does not take into account all of its features, but only the most necessary for the purposes of study (idealized). For example, when studying the physical properties of an orange, we can abstract from its color and taste and imagine it (albeit not perfectly accurately) as a sphere. If we need to understand how many oranges we will get if we put two and three together, we can also abstract away from shape, leaving the model with only one characteristic – quantity. Abstraction and making connections between objects in the most general form is one of the main directions of mathematical creativity.

Another direction, along with abstraction, is generalization. For example, generalizing the concept of “space” to the space of n-dimensions. The space Rn, for n>3 is a mathematical fiction. A very ingenious fiction, though, which helps mathematically understand complex phenomena.

The study of intramathematical objects, as a rule, is carried out using axiomatic method: first a list of basic notions and axioms is formulated for the studied objects and then substantive theorems, together forming a mathematical model, are obtained from the axioms by means of deduction rules.