The article examines the topical issue at this stage of the implementation of the principle of continuity at different stages of mathematical education in the study of natural numbers and ordinary fractions. The authors put forward as key positions the scientific approach in the presentation of the indicated issues, the formation of a theoretical type of thinking, an orientation towards the developing principles of teaching mathematics.
Mathematical education, continuity, scientific character, theoretical thinking, natural numbers
The implementation of the idea of continuity at different levels of mathematics education is today an urgent problem, the solution of which is aimed at the research of many teachers, mathematicians, methodologists. The modernization of the school mathematics course meets the modern requirements of the development of science, the introduction of mathematics in the most diverse areas of knowledge. At present, not only in our country, but also in many countries of the world, intensive searches are being carried out for ways to improve school teaching in mathematics, which would bring it closer to the modern level of development of mathematical science. The task of such teaching mathematics is:
– not the transfer of ready-made knowledge, but in the organization of students' activities to master the system of knowledge, methods of analysis and generalization of educational material based on practical experience, knowledge gained during the period of informal learning (from everyday life, communication with adults, programs of additional educational institutions);
– systematic assimilation of knowledge, skills, skills, subject competences for the use of mathematical representations and concepts in the process of presenting information about the features of surrounding objects, the course of development of any phenomenon in quantitative and spatial terms;
– creative use of the acquired knowledge in solving theoretical and practical problems;
– "the formation of the ability for continuous mental activity, the foundations of logical thinking, spatial imagination, mathematical speech and argumentation, the ability to distinguish between justified and unfounded judgments" [1, p. 28].
In accordance with the Model Program in Mathematics that meets the requirements of the new standards, you attach particular importance to mathematics as a subject in the mental development of the subject of study. The purpose of mathematical development is "the formation of the ability for intellectual activity (logical and sign-symbolic thinking), spatial imagination, mathematical speech; the ability to build reasoning, choose argumentation, search for information; understand the meaning of quantities and how to measure them; the development of interest in mathematics, the desire to use mathematical knowledge in everyday life"[6].
The solution of the indicated tasks and goals of teaching mathematics lies in:
– difficulties associated with the specifics of the subject of mathematics, its logical and psychological foundations of design, which requires the cognitive side of education, which is a powerful means of developing younger schoolchildren and a strong, faithful internal support of the present, original, not copied mastery of knowledge, skills, and abilities;
– on the peculiarities of perception, thinking, memory of schoolchildren (not only their acquisition of knowledge, but also the mastery of techniques and methods of obtaining them, requiring mental effort and adequate application);
– on the diversity of the current state of pedagogical science and practice, characterized by the variability of concepts, technologies, methodological and methodological systems and approaches;
– implementation of the idea of continuity at different levels of mathematics education.
How can a teacher understand the merits and demerits of this or that, often compulsory for the leadership of an educational institution, training program, how to learn to consider all issues of the program of mathematical education in interrelation and continuity?
Of course, first of all, we are witnessing a new stage in the development of mathematical education, which is determined by the rejection of a uniform, unitary system of schooling. However, already between elementary and basic school, new contradictions arise more and more often, connected with the problem of continuity and continuity in teaching mathematics. The essence of these contradictions is as follows. As you know, fundamental scientific research on the problem of the relationship between learning and development was carried out in dignitaries on younger students. This led to the fact that the primary school has a fundamental psychological and pedagogical base for the implementation of the ideas of developing education at the methodological level. However, these ideas have not yet received proper development in the mathematics course of 5-6 classes. Secondary school methodologists see the solution to the problem of continuity between primary and secondary schools in the creation of a unified concept of mathematical education, the main principle of which is the priority of the developmental function, based on the scientific presentation of all formed mathematical concepts from the initial stages of education. However, declaring the priority of the developmental function of teaching, one must not forget that it is in the primary grades that the ability to learn should be formed, on the basis of which all subsequent mathematical education will be built.
Thus, the Concept for the Development of Mathematical Education in the Russian Federation states that "the initial stage of teaching mathematics has two main goals: internal (didactic) - preparing students for continuing education - and external (pragmatic) - the formation of mathematical literacy" [5]. Some authors, highlighting the external goal as a priority, argue that the content of elementary mathematics education should be "relatively closed" [2, p. 31]. In other words, the pragmatic goal - the formation of mathematical literacy - corresponds to the thesis about the sufficiency of primary education for a person's everyday life.
In this concept, teaching in grades 5-6 is also assigned a preparatory role, where it is already a question of the functional literacy of students. The closest to this concept is the statement of M.V. Lomonosov that "mathematics should be studied for that reason, that it puts the mind in order."
Summarizing the experience of teachers, as well as their own experience in mathematical training at different levels of education, we note that in order to characterize the concept of continuity in the framework of mathematical education, training must be considered as a process of the formation of a person's personality through mastering the basics of mathematical knowledge, skills, experience of mathematical activity. To characterize the continuity in teaching, an integrated, systematic approach is needed, which reflects: the logic of constructing the main content and methodological lines of the course, taking into account the relationship and development of the concepts studied by schoolchildren.
We will offer key positions for the implementation of continuity in the study of natural numbers and fractions at the initial and basic stages of training. In modern programs and textbooks, the method of studying natural numbers and fractions is carried out within the framework of the concept presented by N.B. Istomina [4], the main goal of which is to develop students' thinking in the process of assimilating mathematical content. The fundamental key of this concept is also in the study of natural numbers and fractions in grade 5, as well as rational numbers in grade 6. Thus, the main directions of the methodology for studying natural numbers in primary grades received their further development in the study of natural numbers and fractions in grade 5.
Let us characterize the teacher's activities aimed at the implementation of continuity in the mathematical education of schoolchildren.
An elementary school teacher, realizing that the cognitive development of a junior schoolchild takes place in the joint educational activity of him and his students, should be guided not by the type of thinking already mastered by the child, but by the new emerging type - by conceptual, theoretical thinking, because only in this case, learning will be truly developmental.
The traditional elementary course of mathematics with all its content is aimed at the intellectual development of younger students and has great opportunities for the successful organization of developmental education. Like most other initial courses in mathematics, it is largely based on the use of a number system built on the basis of the set-theoretic approach [3, p. 59]. Therefore, the teacher, in order to focus on the formation of theoretical thinking in junior schoolchildren in the process of teaching the elementary course of mathematics, it is necessary to know well the theoretical foundations of this number system, to see and use the possibilities of their application in the further educational process.
There are two independent approaches to constructing the set of non-negative integers - axiomatic and set-theoretic.
In a high school mathematics course, with the axiomatic construction of a set of non-negative integers, the concept of relations "more", "less" is introduced on the basis of the addition operation, already well studied by this time [2, p. 81]. As you know, in the traditional initial course of mathematics, this relation is considered earlier than addition. Therefore, consider the second approach to constructing a set of non-negative integers.
In the set-theoretic construction of the set of non-negative integers, the concepts "more", "less" are defined on the basis of a comparison of the sets representing the numbers under consideration. If these sets are not of equal cardinality, then the numbers under consideration are not equal, since they define different classes of equipotent finite sets. This requires the introduction of new concepts of comparison on the set of non-negative integers - the concepts of "more", "less" [4, p. 81].
Let us give a brief statement of the quantitative theory of non-negative integers, on the set-theoretical basis of its construction, which is the theoretical basis of the relations "more", "less".
Definition. A correspondence between elements of non-empty sets X and Y is a subset of the Cartesian product of the sets X and Y.
Definition. Let X and Y be non-empty sets. A bijection from a set X to a set Y is a correspondence f between elements of the sets X and Y, which has the following properties: a) to each element of the set X there corresponds some element of the set Y; b) different elements of the set X correspond to different elements of the set Y; c) each element of the set Y corresponds to some element of the set X; d) different elements of the set Y correspond to different elements of the set X. Designation: f: X 
Y.
Definition. Non-empty sets X and Y are called equipotent if there is a bijection from the set X to the set Y. Notation: |X| = |Y|.
Definition. Let X and Y be finite sets. There are as many elements in the set X as in the set Y, if X is equal to Y. The set X has fewer elements than in the set Y (or in the set Y there are more elements than in the set X), if X is equivalent to some sub-set of the set Y, provided that this subset is not equal to the set Y itself.
If on the elements of the set of finite sets M we consider the relation of equal cardinality ~ (tilde): ~ 
and ~ = {(x, y)| |x| = |y|}, and X, then, being an equivalence, it splits the set M into classes of equally powerful sets. Each class is assigned a name and a symbol to record it. For example, a class containing an empty set is assigned the name "zero" and the character to write it "0"; the class of equally powerful sets containing a set consisting only of the element d - the name "one" and the symbol 1, and so on.
Definition. Let M be a set whose elements are finite sets. On M, the ratio of equal power "set X is equal to set Y" is given.
A non-negative integer is the name of the class of the partition of the set M by the equivalence relation. A natural number is a non-zero non-negative integer.
From the last definition it immediately follows that N0 = N
{0} (N – is the designation of the set of natural numbers).
Definition. A representative of a non-negative integer p is a set that is an element of the class p of the partition of the set M by the equivalence relation (M/~). Designation: capital letter of the Latin alphabet with subscript p.
Definition. A non-negative integer p is called the number of elements of the set X if the set X is a representative of the number p. Designation: n(X).
It's obvious, that n(X) = p 
X
p.
Definition. A non-negative integer m is less than a non-negative integer p (or a number p is greater than m) if there is a representative of the number m, which is a subset of some representative of the number p and is unequal to it. Designation: m < p or p > m. 
, 
.
It is obvious from the definition, that m < p p > m.
Definition. The set of all natural numbers ordered by the ratio "less" is called a natural series.
Designation: (N, <). For example, (N, <) = {1, 2, 3, 4, 5, 6, …}.
Definition. Let p be an arbitrary natural number. A p-segment of a natural number series is a set of all natural numbers, ordered by the relation "less than", each of which is less than or equal to p.
Designation: Np. Np = {1, 2, 3, 4, …, p}.
For example: N1 = {1}, N2 = {1, 2}, N7 = {1, 2, 3, 4, 5, 6, 7}.
Definition. A count (or numbering) of elements of a finite set A is a bijection of a set A onto some p-segment of a natural series; in this case, the image of an element is called its number.
Example. Count the elements of a set X = {a, b, c, d, e}.
Solution. Consider f = {(a, 1); (b, 2); (c, 3); (d, 4); (e, 5)}.
Since 
, then f – numbering of elements X by definition.
For convenience, it is customary to write the number with a subscript in the numbering. For example, (d, 4) d4 and f = {a1, b2, c3, d4, e5}.
Note that the content of the elementary course in MEC "School of Russia" is based on the use of mainly a numerical system, in the construction of which a set-theoretic approach is used. Therefore, it is quite understandable that the method of studying the relationship "less", "more" on the set of non-negative integers in the first grade of traditional education uses a set-theoretic interpretation.
In accordance with this interpretation, the basis of the concept of relations "more", "less" on the set of non-negative integers is the ratio of equal power of finite sets. Let's remind him. One number is less than the other, if there is a representative of the first number, which is a subset of some representative of the second number, and the representatives of the numbers are unequal sets.
Preparatory work for the formation in students of the concept of the relationship "more", "less" (and at the same time the relationship "equal") on the basis of such an interpretation begins already in the pre-number period, at the very beginning of their study of the initial course of mathematics. This work is carried out on the basis of practical actions with various groups of objects. This approach makes it possible to use the experience previously accumulated by students and from the very beginning to teach in close connection with life and in the implementation of the principle of continuity.
Note that the ability to compare two numbers based on this method, that is, characterizing the comparison results with the words "more", "less", "equal", is one of the main requirements for the mathematical training of students graduating from the first grade. Using the example of studying the numbers of the first ten, younger students get acquainted with various ways of comparing numbers, one of which is based on comparing the corresponding groups of subjects. The main task of these lessons is "to teach junior schoolchildren to establish a mutually unambiguous correspondence between the objects of two sets". Its solution is closely related to the formation of an idea of quantity among junior schoolchildren. It is not excluded that the majority of students will turn to the counting of subjects, but the focus of children should be various methods of establishing a one-to-one correspondence [7, p. 19].
When organizing work on the formation of the concepts under consideration, subject, graphic and symbolic models are used. Therefore, in the process of this work, teachers' instructions for the most part will have to reflect precisely the stages of the modeling process and rely on the theoretical foundations of these concepts. Obviously, in this case, in order to formulate clear, concise and correct instructions, the teacher needs to know well the relevant theoretical material of the university course in mathematics.
Naturally, the instructions and instructions of the teacher for younger students cannot contain the terms of the university course in mathematics. But the teacher should be able to express their essence correctly in a language accessible to younger schoolchildren. He can do this only if he knows the theoretical basis of the concepts used well enough.
The basis of the concepts "more", "less" is the ratio of equal power of finite sets. And the concept "less" for non-negative integers can be expressed as follows: one number is less than the other, if in the set, the number of elements of which is equal to the first number, there are so many elements in the part of the set, the number of elements of which is equal to another number. For example, for a set A (a set of circles) and a set B (a set of rectangles), shown in Figure 1, it is true that sets A and B are not equal in power and, since from set B one can select a part in which there are the same number of rectangles, how many circles are in set A, then the number 4 (the number of circles) is less than the number 5 (the number of rectangles).
|
B |
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