Methods of studying algebraic material in the initial course of mathematics. Elements of algebra in elementary school

9.3.1. The method of introducing the concept of "Monomial" and the formation of the ability to find its numerical value.

The basic knowledge includes the concepts of an algebraic expression, the product of algebraic expressions, a multiplier (numerical and alphabetic); to skills - writing an algebraic expression by its elements, highlighting the elements of a given algebraic expression.

Updating knowledge is carried out through exercises.

1. From this set, select such algebraic expressions that are products of several factors: a) 5 a 2 b; b) (7 ab 2 + since 2):(5m 2 n); at 8; d) 5 a 6 bb 4 a; e) ; f) g)

The specified condition is satisfied by algebraic expressions: 5 a 2 b; 8; 5a 6 bb 4 a; ; Most likely, students will not name 8 among the required algebraic expressions; ; although some may guess what can be represented as s. Having taken several algebraic expressions, one should practice in isolating their numerical factor, literal factors, in writing new expressions according to the given algebraic expressions.

2. Write a new algebraic expression using expressions 3 a 2 b and a. Possible student responses: 3 a 2 b+ a; 3a 2 b– a; 3a 2 b a; 3a 2 b: a.

3. Which of the following expressions are monomials: a) 5 a 3 bcb 4; b) a; c) d) 3 4 e) 7 ab 2:n; e) - 5 a 6 b c 2; e) - a 3; g) h) - mnx. Name the numerical and alphabetic multipliers of monomials.

4. Write down several algebraic expressions that are monomials.

5. Write down several monomials that differ only in numerical coefficient.

6. Fill in the gaps: a) 12 a 3 b 4= 2a… b 2; b) - 24 m 2 b 7 p 6= 24bp …

7. Instead of a verbal formulation, write algebraic expressions: a) double product of numbers a and b; b) triple the product of the square of a number a and numbers b.

8. Explain the expressions: a) 2 a b; b) a 5b.

For example, the expression a 5b can be explained as: 1) the product of numbers a, 5 and b;2) product of numbers a and 5 b;3) area of a rectangle with sides a and 5 b.

Exercises of type 7 and 8 also contribute to mastering the method of solving text problems using equations, since the translation of verbal formulations into the language of numbers and letters and the verbal interpretation of algebraic expressions are important components of the method of solving problems using equations.

9. Find the numerical value of the monomial: 1) 5 mnx at m= 3, n= ; x=8; 2) (– 0,25)a b at a=12; b=8. When performing such exercises, special students should be pointed out to the need to use the properties and laws of arithmetic operations to rationalize calculations.

The organization of the exercises can be different: a solution at the blackboard, an independent solution, a commented solution, simultaneous execution of exercises on the blackboard with the involvement of weak students and independent work of strong students, etc.

For homework, you can use exercises to write numbers in a standard form, which will serve as a motive for introducing the concept of a standard form of a monomial in the next lesson.

9.3.2. Generalization and systematization of knowledge on the topic: "Progressions".

Reproduction and correction of basic knowledge can be done through exercises to fill in the table, followed by a discussion of the results.

Note that arithmetic and geometric progressions provide an example of studying material in similar situations, so the methods of opposition and comparison should occupy an important place in the systematization of knowledge about progressions. The discussion of key issues is based on the clarification of the reasons for the difference and common in progressions.

Issues for discussion.

BUT). Name the common and different in the structure of the definition of arithmetic and geometric progressions.

B). Define an infinitely decreasing geometric progression.

AT). What is the sum of an infinitely decreasing geometric progression? Write down its formula.

G). How to prove that a given sequence is an arithmetic (geometric) progression?

D). Use the arrows to show the links between the indicated definitions, formulas (Fig. 7):

a	a n = a n -1 + d	a 1 , a 2 , … …	a n \u003d a l + d (n-1)
		a n , d

	a n = (a n -1 + a n +1)	Sign of an arithmetic progression	S n = (a 1 + a 2) n

3. Write down all the definitions, formulas on the topic "Geometric progression" and indicate the dependencies between them.

Exercises 2 and 3 can be offered to students to complete on their own, followed by a discussion of the results with all students in the class. Exercise 2 can be done collectively, and exercise 3 can be offered as independent work.

The next stages of the generalizing lesson are implemented with the help of exercises, the implementation of which requires the analysis and use of basic facts, leading to new connections and relationships between the concepts and theorems studied.

4. Between the numbers 4 and 9, insert a positive number so that you get three consecutive members of a geometric progression. Formulate and solve a similar problem in relation to an arithmetic progression.

5. Determine the numbers a 1 , a 2 , a 3 and a 4, if a 1 , a 2 , a 3 are successive members of a geometric progression, and a 1 , a 3 and a 4– arithmetic progression and a 1 + a 4= 14, a 2 + a 3 = 12.

7. Can three positive numbers be simultaneously three consecutive members of an arithmetic and geometric progression?

8. Is it possible to assert that arithmetic and geometric progressions are functions? If so, what types of functions do they belong to?

9. It is known that a n = 2n+1 is an arithmetic progression. What is common and different in the graphs of this progression and a linear function f(X) = 2x+1?

10. Is it possible to specify sequences that are
both arithmetic and geometric progressions?

The forms of performing exercises can be different: performing exercises at the blackboard, commented on the solution, etc. Some of the above exercises can be performed by students on their own, and their implementation can be carried out depending on the capabilities of the students using cards containing missing lines or instructions for their implementation. Obviously, the lower the student's abilities, the more extensive the set of recommendations (instructions for implementation) should be for him.

9.3.3. Testing, evaluation and correction of knowledge, skills and abilities on the topic: "Multiplication and division of rational numbers".

Checking students' knowledge of the factual material, the ability to explain the essence of the basic concepts is carried out in the process of conversation, followed by exercises.

Questions for conversation

1. Formulate a rule for multiplying two numbers with the same signs. Give examples.

2. Formulate a rule for multiplying two numbers with different signs. Give examples.

3. What is the product of several numbers if one of them is zero? Under what conditions a b= 0?

4. What is the product a(-one)? Give examples.

5. How will the product change when the sign of one of the factors changes?

6. Formulate the commutative law of multiplication.

7. How is the associative law of multiplication formulated?

8. Write down, using letters, the commutative and associative laws of multiplication.

9. How to find the product of three, four rational numbers?

10. The student, performing the exercise to find the product 0.25 15 15 (–4), used the following sequence of actions: (0.25 (–4)) 15 15 = (–1) 15 15 = –15 15. What laws does he used?

11. What factor of an algebraic expression is called a coefficient?

12. How to find the coefficient of a product, in which there are several alphabetic and numerical factors?

13. What is the coefficient of the expression: a; – a; ab; – ab?

14. Formulate the distributive law of multiplication. Write it down with letters.

15. What terms algebraic sum called similar?

16. Explain what it means to bring like terms.

17. Explain with the help of what laws the reduction of similar terms in the expression 5.2 is carried out y- 8a - 4,8y- 2a.

18. What is the rule for dividing rational numbers with the same signs?

19. What is the rule for dividing rational numbers with different signs?

20. When is the quotient of two rational numbers equal to zero?

21. In what order are joint actions performed with rational numbers?

Some questions can be the subject of collective discussion, others - sheets of mutual control of students, it is possible to conduct a mathematical dictation on the basis of some questions, etc.

The subsequent series of exercises is aimed at monitoring, evaluating, and correcting students' skills. Various forms of performing exercises are possible: an independent solution, accompanied by self-control of students, a commented solution, performing exercises on the board, an oral survey, etc. This series covers two groups of exercises. The first group does not require a reconstructive nature to perform mental activity, the implementation of the second group involves the reconstruction of knowledge and skills on the topic being studied.

1. Which of the following equalities are true:

1) (–9) (–8) = –72; 2) (–1,4) 0,5 = – 0,7;

3) 12 (–0,2) = –0,24; 4) (–3,2) (–2,1) = 6,72?

Choose the correct answer.

Answer: 1); 2); 3); four); there are no true equalities.

2. Without performing calculations, determine which product is positive:

1) 0,26 (–17) (–52) (–34); 2) (–1) (–8) 0,4 (–3,4);

3) (–16) (–0,87) (– ) (–5); 4) 5 (–3,2) 0 (0,7).

Answer: 1); 2); 3); four).

3. Specify expressions that have equal coefficients:

1) 9ace and 3 x(4y); 2) (–3) (–8cb) and 4 X 6y;

3) abs and 2.75 xy; 4) 3,15abs and 0.001 abs.

4. Which of the expressions contains similar terms:

1) 7a– 12ab+ 14; 2) – 0,5xy + 2,7kh - 0,5;

3) 3With – 2,7hus – ;4) 72ab- ab + 241?

Specify the correct answer.

Answer: 1); 2); four); there are no expressions containing similar terms.

5. Indicate the correct equalities: : (–18.2

3. Choose the largest and smallest number among the numbers
a,a 2 ,a 3 ,a 4 , a 5 , a 6 , a 7 at a = – 5, a = 3.

4. Simplify the expression:

1) – X(y - 4) – 2(hu– 3) – 3X; 2) a(b+ 3) – 3(2 – ab) + a.

The above set of tasks and their sequence cover all levels of knowledge acquisition. The fulfillment of the entire set of tasks corresponds to the qualitative assimilation of knowledge and skills and can be rated as "excellent". The exercises of the first group correspond to the assimilation of knowledge and skills at the level of their application in situations that do not require the reconstruction of knowledge and skills. Correct answers to the questions characterize the assimilation of knowledge at the level of reproduction. A grade of "satisfactory" can be given to a student who has completed most of the exercises in the first group. The “good” rating corresponds to the correctly performed majority of the exercises of the first and second groups.

Tasks

1. Choose a specific topic for the correctional and developmental course in algebra at the main school. Study the relevant sections of the program and textbook. Identify the methodological features of the study of the topic. Develop fragments of a methodology for teaching a topic. Prepare a set of cards to correct students' knowledge.

2. Attend several algebra lessons at one of the special (correctional) institutions of type VII in your region. Analyze one lesson from the point of view of its educational, correctional-developing, educational and practical orientation.

3. One of the goals of teaching mathematics is the formation of a mathematical culture. Computational culture is one of the components of mathematical culture. Suggest your interpretation of the concept of "computational culture". At what stages of teaching mathematics to special students, when teaching what content is it possible and appropriate to set the goal of “forming a computing culture”? Lead specific example with the corresponding task system. Make a list of literature on the development of the concept of number for extracurricular reading of special students. Specify in which classes it can be used.

CHAPTER 10.

(8 ocloc'k)

Plan:

1. Goals of studying algebraic material in primary school.

2. Properties of arithmetic operations studied in elementary grades.

3. Learning numerical expressions and rules for the order in which actions are performed:

One order without brackets;

One order with brackets;

Expressions without brackets, including 4 arithmetic operations, with brackets.

4. Analysis of numerical equalities and inequalities studied in elementary grades (comparison of two numbers, a number and a numerical expression, two numerical expressions).

5. The introduction of alphabetic symbols with a variable.

6. Methodology for studying equations:

a) give a definition of the equation (from lectures on mathematics and from a mathematics textbook for elementary school),

b) highlight the scope and content of the concept,

c) what method (abstract-deductive or concrete-inductive) will you introduce this concept? Describe the main steps in working on an equation.

Complete the tasks:

1. Explain the expediency of using inequalities with a variable in the initial classes.

2. Prepare a message for the lesson on the possibility of developing functional propaedeutics in students (through the game, through the study of inequalities).

3. Select tasks for students to fulfill the essential and non-essential properties of the concept of "equation".

1. Abramova O.A., Moro M.I. Solving Equations // Elementary School. - 1983. - No. 3. - S. 78-79.

2. Ymanbekova P. Means of visibility in the formation of the concept of "equality" and "inequality" // Elementary school. - 1978. - No. 11. - S. 38-40.

3. Shchadrova I.V. On the order of actions in an arithmetic expression // Elementary school. - 2000. - No. 2. - S. 105-107.

4. Shikhaliev Kh.Sh. Unified approach to solving equations and inequalities // Elementary school. - 1989. - No. 8. - S. 83-86.

5. Nazarova I.N. Familiarization with functional dependence in teaching problem solving // Elementary school. - 1989. - No. 1. - S. 42-46.

6. Kuznetsova V.I. About some common mistakes students associated with issues of algebraic propaedeutics // Elementary school. - 1974. - No. 2. – S. 31.

General characteristics of the study methodology

algebraic material

The introduction of algebraic material into the elementary course of mathematics makes it possible to prepare students for the study of the basic concepts of modern mathematics, for example, such as "variable", "equation", "inequality", etc., and contributes to the development of functional thinking in children.

The main concepts of the topic are “expression”, “equality”, “inequality”, “equation”.

The term "equation" is introduced when studying the topic "Thousand", but the preparatory work for familiarizing students with equations begins from grade 1. The terms "expression", "expression value", "equality", "inequality" are included in the vocabulary of students starting from grade 2. The concept of “solve inequality” is not introduced in primary grades.

Numeric expressions

In mathematics, an expression is understood as a sequence that is constant according to certain rules mathematical symbols denoting numbers and operations on them. Expression examples: 7; 5+4; 5 (3+ in); 40: 5 + 6, etc.

Expressions of the form 7; 5+4; 10:5+6; (5 + 3) 10 are called numerical expressions, in contrast to expressions of the form 8 - a; (3 + in); 50: to, called literal or variable expressions.

The tasks of studying the topic

2. To acquaint students with the rules for the order of performing operations on numbers and, in accordance with them, develop the ability to find numerical values expressions.

3. To acquaint students with identical transformations of expressions based on arithmetic operations.

In the familiarization method junior schoolchildren with the concept of a numerical expression, three stages can be distinguished, involving familiarization with expressions containing:

One arithmetic operation (stage I);

Two or more arithmetic operations of one stage (stage II);

Two or more arithmetic operations of different levels (stage III).

With the simplest expressions - sum and difference - students are introduced in grade I (when studying addition and subtraction within 10); with the product and the quotient of two numbers - in the II class.

Already when studying the topic “Ten”, the names of arithmetic operations, the terms “term”, “sum”, “reduced”, “subtracted”, “difference” are introduced into the vocabulary of students. In addition to terminology, they must also learn some elements of mathematical symbolism, in particular, action signs (plus, minus); they must learn to read and write simple mathematical expressions like 5 + 4 (the sum of the numbers "five" and "four"); 7 - 2 (the difference between the numbers "seven" and "two").

First, students are introduced to the term "sum" in the meaning of the number that is the result of the action of addition, and then in the meaning of the expression. Reception of subtraction of the form 10 - 7, 9 - 6, etc. based on knowledge of the relationship between addition and subtraction. Therefore, it is necessary to teach children to represent a number (reduced) as the sum of two terms (10 is the sum of the numbers 7 and 3; 9 is the sum of the numbers 6 and 3).

With expressions containing two or more arithmetic operations, children get acquainted in the first year of study with the assimilation of computational techniques ± 2, ± 3, ± 1. They solve examples of the form 3 + 1 + 1, 6 - 1 - 1, 2 + 2 + 2, etc. Calculating, for example, the value of the first expression, the student explains: “Add one to three, you get four, add one to four, you get five.” The solution of examples of the form 6 - 1 - 1, etc. is explained in a similar way. Thus, first-graders are gradually preparing to derive a rule on the order in which actions are performed in expressions containing actions of one stage, which is generalized in grade II.

In grade I, children will practically master another rule for the order of performing actions, namely, performing actions in expressions of the form 8 - (4 + 2); (6 - 2) + 3, etc.

Students' knowledge of the rules for the order in which actions are performed is summarized and another rule is introduced about the order in which actions are performed in expressions that do not have brackets and contain arithmetic operations of different levels: addition, subtraction, multiplication and division.

When familiarizing yourself with the new rule on the order of actions, work can be organized in different ways. You can invite children to read the rule from the textbook and apply it when calculating the values of the corresponding expressions. You can also invite students to calculate, for example, the value of the expression 40 - 10: 2. The answers may turn out to be different: for some, the value of the expression will be equal to 15, for others 35.

After that, the teacher explains: “To find the value of an expression that does not have brackets and contains the operations of addition, subtraction, multiplication and division, one must perform in order (from left to right) first the operations of multiplication and division, and then (also from left to right) addition and subtraction. In this expression, you must first divide 10 by 2, and then subtract the result 5 from 40. The value of the expression is 35.

Primary school students actually get acquainted with the identical transformations of expressions.

The identical transformation of expressions is the replacement of a given expression with another, the value of which is equal to the value of the given one (the term and definition are not given to primary school students).

With the transformation of expressions, students meet from grade 1 in connection with the study of the properties of arithmetic operations. For example, when solving examples like 10 + (50 + 3) convenient way children reason like this: “It’s more convenient to add tens to tens and add 3 units to the result of 60. I will write down: 10 (50 + 3) \u003d (10 + 50) + 3 \u003d 63.

Performing a task in which it is necessary to complete the entry: (10 + 7) 3 = 10 3 + 7 3 ..., the children explain: “On the left, the sum of the numbers 10 and 7 is multiplied by the number 3, on the right, the first term 10 of this sum is multiplied by the number 3; in order to preserve the “equal” sign, the second term 7 must also be multiplied by the number 3 and the resulting products added. I will write it down like this: (10 + 7) 3 = 10 3 + 7 3.

When transforming expressions, students sometimes make errors of the form (10 + 4) 3 = - 10 3 + 4. The reason for this kind of errors is associated with the incorrect use of previously acquired knowledge (in this case, using the rule of adding a number to the sum when solving an example, in which the sum must be multiplied by the number). To prevent such errors, you can offer students the following tasks:

a) Compare the expressions written on the left side of the equalities. How are they similar, how are they different? Explain how you calculated their values:

(10 + 4) + 3 = 10 + (4 + 3) = 10 + 7 = 17

(10 + 4) 3 = 10 3 + 4 3 = 30 + 12 = 42

b) Fill in the gaps and find the result:

(20 + 3) + 5 = 20 + (3 + ð); (20 + 3) 5 = 20 ð + 3 ð.

c) Compare the expressions and put a > sign between them,< или =:

(30 + 4) + 2 ... 30 + (4 + 2); (30 + 4) + 2 ... 30 2 + 4 2.

d) Check by calculation whether the following equalities are true:

8 3 + 7 3 = (8 + 7) 3; 30 + (5 + 7) = 30 + 7.

Literal expressions

In the elementary grades, it is planned to carry out - in close connection with the study of numbering and arithmetic operations - preparatory work to reveal the meaning of the variable. To this end, mathematics textbooks include exercises in which the variable is denoted by a “window”. For example, ð< 3, 6 < ð, ð + 2 = 5 и др.

Here it is important to encourage students to try to substitute in the “window” not one, but several numbers in turn, checking each time whether the entry is correct.

Thus, in the case of ð< 3 в «окошко» можно подставить числа 0, 1, 2,; в случае 6 < ð - числа 7, 8, 9, 10, 20 и др.; в случае ð + 2 = 5 можно подставить только число 3.

In order to simplify the mathematics curriculum for elementary grades and ensure its accessibility, letter symbols as a means of generalizing arithmetic knowledge are not used. All letter designations are replaced by verbal formulations.

For example, instead of setting

A task is proposed in the following form: “Increase the number 3 by 4 times; 5 times; 6 times; ... ".

Equalities and inequalities

Familiarization of primary school students with equalities and inequalities is associated with the solution of the following tasks:

To teach how to establish the relationship "greater than", "less than" or "equal to" between expressions and write the comparison results using a sign;

The methodology for the formation of ideas about numerical equalities and inequalities among younger schoolchildren provides for the following stages of work.

At the first stage, first of all, the school week, first-graders perform exercises to compare sets of objects. Here it is most expedient to use the method of establishing a one-to-one correspondence. At this stage, the results of the comparison are not yet written using the appropriate ratio signs.

At stage II, students compare numbers, first relying on object visibility, and then on the property of numbers in the natural series, according to which, of two different numbers, the number is greater, which is called later when counting, and the number is smaller, which is called earlier. The relations established in this way are recorded by the children with the help of appropriate signs. For example, 3 > 2, 2< 3. В дальнейшем при изучении нумерации (в концентрах «Сотня», «Тысяча», «Многозначные числа») для сравнения чисел полезно применять два способа, а именно устанавливать отношения между числами: 1) по месту их расположения в натуральном ряду; 2) на основе сравнения соответствующих разрядных чисел, начиная с высших разрядов. Например, 826 < 829, так как сотен и десятков в этих числах поровну, а единиц в первом числе меньше, чем во втором.

You can also compare the values: 4 dm 5 cm > 4 dm 3 cm, since there are more decimeters than in the second. In addition, the values can be first expressed in units of one measurement and only after that they can be compared: 45 cm > 43 cm.

Similar exercises are already introduced when studying addition and subtraction within 10. It is useful to perform them based on clarity, for example: students lay out four circles on their desks on the left, and four triangles on the right. It turns out that the figures are equally divided - four each. They write down the equality: 4 \u003d 4. Then the children add one circle to the figures on the left and write down the sum 4 + 1. There are more figures on the left than on the right, which means 4 + 1\u003e 4.

Using the technique of the equation, students move from inequality to equality. For example, 3 mushrooms and 4 squirrels are placed on a typesetting canvas. To make mushrooms and squirrels equally, you can: 1) add one mushroom (then there will be 3 mushrooms and 3 squirrels).

There are 5 cars and 5 trucks on the typesetting canvas. In order to have more cars than others, you can: 1) remove one (two, three) cars (cars or trucks) or 2) add one (two, three) cars.

Gradually, when comparing expressions, children move from relying on visualization to comparing their meanings. This method is the main one in elementary grades. When comparing expressions, students can also rely on knowledge: a) the relationship between the components and the result of an arithmetic operation: 20 + 5 * 20 + 6 (the sum of the numbers 20 and 5 is written on the left, the sum of the numbers 20 and 6 on the right. The first terms of these sums are the same , the second summand on the left is less than the second summand on the right, so the sum on the left is less than the sum on the right: 20 + 5< 20 + 6); б) отношение между результатами и компонентами арифметических действий: 15 + 2 * 15 (слева и справа сначала было поровну – по 15. Затем к 15 прибавили 2, стало больше, чем 15); в) смысла действия умножения: 5 + 5 + 5 + 5 + 5 * 5 · 3 (слева число 5 взяли слагаемым 5 раз, справа число 5 взяли слагаемым 3 раза, значит, сумма слева будет больше, чем справа: 5 + 5 + 5 + 5 + 5 >5 + 5 + 5); d) properties of arithmetic operations: (5 + 2) 3 * 5 3 + 2 3 (on the left, the sum of the numbers 5 and 2 is multiplied by the number 3, on the right, the products of each term by the number 3 are found and added. So, instead of an asterisk, you can put an equal sign: (5 + 2) 3 = 5 3 + 2 3).

In these cases, the evaluation of the values of expressions is used to check the correctness of the sign. To write inequalities with a variable in elementary grades, a “window” is used: 2 > ð, ð = 5, ð > 3.

It is useful to perform the first exercises of this type based on a number series, referring to which students notice that the number 2 is greater than one and zero, therefore, the numbers 0 and 1 can be substituted into the “window” (2> ð) (2> 0, 2> 1 ).

Other exercises with a window are performed similarly.

The main way when considering inequalities with a variable is the selection method.

To facilitate the values of the variable in the inequalities, it is proposed to choose them from a specific series of numbers. For example, you can suggest writing out those numbers from the series 7, 8, 9, 10, 11, 12, 13, 14, for which the record ð - 7 is correct< 5.

When completing this task, the student can reason like this: “Let's substitute the number 7 in the “window”: 7 minus 7 will be 0, 0 is less than 5, so the number 7 is suitable. Substitute the number 8:8 minus 7 into the “window” will be 1, 1 is less than 5, which means that the number 8 is also suitable ... Substitute the number 12 into the “window”: 12 minus 7 will be 5, 5 less than 5 is incorrect, then the number 12 is not suitable . To write ð - 7< 5 была верной, в «окошко» можно подставить любое из чисел 7, 8, 9, 10, 11».

Equations

At the end of grade 3, children get acquainted with the simplest equations of the form: X+8 =15; 5+X=12; X–9 =4; 13–X=6; X 7 \u003d 42; four· X=12; X:8 =7; 72:X=12.

The child should be able to solve equations in two ways:

1) selection method (in the simplest cases); 2) in a way based on the application of the rules for finding unknown components of arithmetic operations. Here is an example of writing a solution to an equation along with a check and the child's reasoning when solving it:

X – 9 = 4 X = 4 + 9 X = 13

13 – 9 = 4 4 = 4

"In the equation X– 9 = 4 x stands in the place of the reduced one. To find the unknown minuend, you need to add the subtrahend to the difference ( X\u003d 4 + 9.) Let's check: we subtract 9 from 13, we get 4. We got the correct equality 4 \u003d 4, which means the equation was solved correctly.

In the 4th grade, the child can be introduced to the solution simple tasks how to write an equation.

Introduction ................................................ ................................................. ....... 2

Chapter I. General theoretical aspects of the study of algebraic material in elementary school .............................................................. ................................................. .... 7

1.1 Experience of introducing elements of algebra in elementary school .............................. 7

1.2 Psychological foundations for the introduction of algebraic concepts

in primary school............................................... ................................ 12

1.3 The problem of the origin of algebraic concepts and its significance

for the construction of a subject .................................................................. ....... twenty

2.1 Education in primary school in terms of needs

high school ................................................................ ...................................... 33

2.1 Comparison (opposition) of concepts in mathematics lessons .... 38

2.3 Joint study of addition and subtraction, multiplication and division 48

Chapter III. The practice of studying algebraic material in mathematics lessons in the primary grades of secondary school No. 4 in Rylsk .................................................. ... 55

3.1 Rationale for use innovative technologies(technology

enlargement of didactic units) .............................................. ...... 55

3.2 About the experience of getting acquainted with algebraic concepts in grade I .... 61

3.3 Learning to solve problems related to the movement of bodies .............................. 72

Conclusion................................................. ................................................. .76

Bibliographic list.......................................................................... 79

In any modern system of general education, mathematics occupies one of the central places, which undoubtedly speaks of the uniqueness of this field of knowledge.

What is modern mathematics? Why is she needed? These and similar questions are often asked to teachers by children. And each time the answer will be different depending on the level of development of the child and his educational needs.

It is often said that mathematics is the language of modern science. However, this statement seems to have a significant flaw. The language of mathematics is so widespread and so often effective precisely because mathematics is not reducible to it.

An outstanding Russian mathematician A.N. Kolmogorov wrote: “Mathematics is not just one of the languages. Mathematics is a language plus reasoning, it’s like a language and logic together. Mathematics is a tool for thinking. It concentrates the results of the exact thinking of many people. With the help of mathematics, one reasoning can be connected with another ... The obvious complexities of nature with its strange laws and rules, each of which allows a separate very detailed explanation are in fact closely related. However, if you do not want to use mathematics, then in this huge variety of facts you will not see that logic allows you to move from one to another "(, p. 44).

Thus, mathematics allows us to form certain forms of thinking necessary for studying the world around us.

At present, the disproportion between the degree of our knowledge of nature and the understanding of man, his psyche, and thought processes is becoming more and more noticeable. W. W. Sawyer in his book "Prelude to Mathematics" (, p. 7) notes: "You can teach students to solve quite a lot of types of problems, but true satisfaction will come only when we are able to transfer to our pupils not just knowledge, but flexibility of mind which would give them the opportunity in the future not only to solve independently, but also to set new tasks for themselves.

Of course, there are certain limits here, which should not be forgotten: a lot is determined by innate abilities, talent. However, it is possible to note a whole set of factors depending on education and upbringing. This makes it extremely important to make a correct assessment of the vast untapped possibilities of education in general and mathematics education in particular.

In recent years, there has been a steady trend of the penetration of mathematical methods into such sciences as history, philology, not to mention linguistics and psychology. Therefore, the circle of persons who, in their subsequent professional activity perhaps will apply mathematics, expands.

Our education system is arranged in such a way that for many, school provides the only opportunity in life to join the mathematical culture, to master the values contained in mathematics.

What is the influence of mathematics in general and school mathematics in particular on the upbringing of a creative person? Teaching the art of problem solving in mathematics classes provides us with an exceptionally favorable opportunity for the formation of a certain mindset in students. The need for research develops interest in patterns, teaches to see the beauty and harmony of human thought. All this is on our mind essential element general culture. An important influence is exerted by the course of mathematics on the formation various forms thinking: logical, spatial-geometric, algorithmic. Any creative process begins with the formulation of a hypothesis. Mathematics, with the appropriate organization of education, being a good school for constructing and testing hypotheses, teaches us to compare various hypotheses, find the best option, set new tasks, and look for ways to solve them. Among other things, she also develops the habit of methodical work, without which no creative process is conceivable. Maximizing the possibilities of human thinking, mathematics is its highest achievement. It helps a person in self-awareness and the formation of his character.

This is just a small part of a large list of reasons why mathematical knowledge should become an integral part of the general culture and an indispensable element in the upbringing and education of a child.

The course of mathematics (without geometry) in our 10-year school is actually divided into three main parts: arithmetic (grades I-V), algebra (grades VI-VIII) and elements of analysis (grades IX-X). What is the basis for such a subdivision?

Of course, each of these parts has its own special "technology". So, in arithmetic, for example, it is associated with calculations performed on multi-valued numbers, in algebra - with identical transformations, logarithm, in analysis - with differentiation, etc. But what are the deeper foundations associated with the conceptual content of each part?

The next question concerns the grounds for distinguishing between school arithmetic and algebra (i.e., the first and second parts of the course). Arithmetic includes the study of natural numbers (positive integers) and fractions (prime and decimal). However, a special analysis shows that the combination of these types of numbers in one school subject is illegal.

The fact is that these numbers have different functions: the first ones are associated with account items, the second - with measurement. This circumstance is very important for understanding the fact that fractional (rational) numbers are only a special case of real numbers.

From the point of view of measuring quantities, as noted by A.N. Kolmogorov, "there is no such deep difference between rational and irrational real numbers. For pedagogical reasons, they linger on rational numbers for a long time, since they are easy to write in the form of fractions; however, the use that is given to them from the very beginning should have immediately led to real numbers in all their generality" (), p. 9).

A.N. Kolmogorov considered justified both from the point of view of the history of the development of mathematics, and in essence, A. Lebesgue's proposal to move in teaching after natural numbers immediately to the origin and logical nature of real numbers. At the same time, as noted by A.N. Kolmogorov, "the approach to the construction of rational and real numbers from the point of view of measuring quantities is no less scientific than, for example, the introduction of rational numbers in the form of "pairs". For the school, however, it has an undeniable advantage" (, p. 10).

Thus, on the basis of natural (integer) numbers, there is a real possibility to immediately form the "most general concept of a number" (in the terminology of A. Lebesgue), the concept of a real number. But from the point of view of the construction of the program, this means no more, no less than the elimination of the arithmetic of fractions in its school interpretation. The transition from integers to real numbers is a transition from arithmetic to "algebra", to the creation of a foundation for analysis.

These ideas, expressed more than 20 years ago, are still relevant today. Is it possible to change the structure of teaching mathematics in elementary school in this direction? What are the advantages and disadvantages of "algebraizing" the primary teaching of mathematics? The purpose of this work is to try to answer the questions posed.

Achieving this goal requires solving the following tasks:

Consideration of the general theoretical aspects of the introduction in elementary school of the algebraic concepts of magnitude and number. This task is posed in the first chapter of the work;

The study of a specific methodology for teaching these concepts in elementary school. Here, in particular, it is supposed to consider the so-called theory of enlargement of didactic units (UDE), which will be discussed below;

Show the practical applicability of the provisions under consideration on school lessons mathematics in elementary school (lessons were taught by the author in high school No. 4 of Rylsk). This is the subject of the third chapter of the work.

With regard to the bibliography devoted to this issue, the following can be noted. Despite the fact that lately total published methodological literature in mathematics is extremely small, there was no lack of information when writing the work. Indeed, from 1960 (the time the problem was posed) to 1990. In our country, a huge amount of educational, scientific and methodological literature has been published, to one degree or another affecting the problem of introducing algebraic concepts in the course of mathematics for elementary school. In addition, these issues are regularly covered in specialized periodicals. So, when writing the work, publications in the journals Pedagogy, Teaching Mathematics at School and Primary School were used to a large extent.

1.1. General questions of methods of studying algebraic material.

1.2. Methodology for studying numerical expressions.

1.3. The study of literal expressions.

1.4. The study of numerical equalities and inequalities.

1.5. Technique for studying equations.

1.6. Solve simple arithmetic problems by writing equations.

1.1. General questions of the methodology for studying algebraic material

The introduction of algebraic material into the initial course of mathematics makes it possible to prepare students for the study of the basic concepts of modern mathematics (variable, equation, equality, inequality, etc.), contributes to the generalization of arithmetic knowledge, and the formation of functional thinking in children.

Primary school students should receive initial information about mathematical expressions, numerical equalities and inequalities, learn to solve the equations provided for by the curriculum and simple arithmetic problems by drawing up an equation (the theoretical basis for choosing an arithmetic operation in which the relationship between the components and the result of the corresponding arithmetic operation0.

The study of algebraic material is carried out in close connection with arithmetic material.

1.2. Methodology for studying numerical expressions

In mathematics, an expression is understood as a sequence of mathematical symbols built according to certain rules, denoting numbers and operations on them.

Expressions like: 6; 3+2; 8:4+(7-3) - numerical expressions; type: 8-a; 30:in; 5+(3+s) - literal expressions (expressions with a variable).

The tasks of studying the topic

2) To acquaint students with the rules for the order of performing arithmetic operations.

3) Learn to find the numerical values of expressions.

4) Familiarize yourself with identical transformations of expressions based on the properties of arithmetic operations.

The solution of the tasks set is carried out throughout all the years of education in the primary grades, starting from the first days of the child's stay at school.

The methodology for working on numerical expressions provides for three stages: at the first stage - the formation of concepts about the simplest expressions (sum, difference, product, quotient of two numbers); at the second stage - about expressions containing two or more arithmetic operations of one stage; at the third stage - about expressions containing two or more arithmetic operations of different levels.

With the simplest expressions - sum and difference - students are introduced in the first grade (according to the program 1-4) with the product and private - in the second grade (with the term "product" - in grade 2, with the term "private" - in the third grade).

Consider the method of studying numerical expressions.

Performing operations on sets, children, first of all, learn the specific meaning of addition and subtraction, therefore, in records of the form 3 + 2, 7-1, the signs of actions are perceived by them as a short designation of the words “add”, “subtract” (add 2 to 3). In the future, the concepts of actions deepen: students learn that by adding (subtracting) a few units, we increase (decrease) the number by the same number of units (reading: 3 increase by 2), then the children will learn the name of the plus signs (reading: 3 plus 2), "minus".

In the topic “Addition and subtraction within 20”, children are introduced to the concepts of “sum”, “difference” as the names of mathematical expressions and as the name of the result of the arithmetic operations of addition and subtraction.

Consider a fragment of the lesson (grade 2).

Attach 4 red and 3 yellow circles to the board using water:

OOOO OOO

How many red circles? (Write down the number 4.)

How many yellow circles? (Write down the number 3.)

What action should be performed on the written numbers 3 and 4 in order to find out how many red and how many yellow circles are together? (record appears: 4+3).

Tell me, without counting how many circles are there?

Such an expression in mathematics, when there is a “+” sign between the numbers, is called the sum (Let's say together: sum) and read like this: the sum of four and three.

Now let's find out what the sum of the numbers 4 and 3 is equal to (we give a complete answer).

Likewise for the difference.

When studying addition and subtraction within 10, expressions consisting of 3 or more numbers connected by the same and different signs of arithmetic operations are included: 3+1+2, 4-1-1, 7-4+3, etc. By revealing the meaning of such expressions, the teacher shows how to read them. Calculating the values of these expressions, children practically master the rule about the order of arithmetic operations in expressions without brackets, although they do not formulate it: 10-3+2=7+2=9. Such records are the first step in performing identical transformations.

The methodology for familiarizing yourself with expressions with brackets can be different (Describe a fragment of the lesson in your notebook, prepare for practical exercises).

The ability to compose and find the meaning of an expression is used by children in solving arithmetic problems, at the same time, here the concept of “expression” is further mastered, the specific meaning of expressions in the records of solving problems is assimilated.

Of interest is the type of work proposed by the Latvian methodologist Ya.Ya. Mentzis.

A text is given, for example, like this: “The boy had 24 rubles, a cake costs 6 rubles, a candy 2 rubles”, it is proposed:

a) make all kinds of expressions on this text and explain what they show;

b) explain what the expressions show:

2 cells 3 cells

24-2 24-(6+2) 24:6 24-6 3

In grade 3, along with the expressions discussed earlier, they include expressions consisting of two simple expressions (37+6) - (42+1), as well as consisting of a number and a product or a quotient of two numbers. For example: 75-50:25+2. Where the order in which actions are performed does not match the order in which they are written, brackets are used: 16-6:(8-5). Children must learn to read and write these expressions correctly, to find their meanings.

The terms "expression", "expression value" are introduced without definitions. In order to make it easier for children to read and find the meaning of complex expressions, methodologists recommend using a scheme that is compiled collectively and used when reading expressions:

1) I will establish which action is performed last.

2) I'll think about how the numbers are called when performing this action.

3) I will read how these numbers are expressed.

The rules for the order of actions in complex expressions are studied in the 3rd grade, but children practically use some of them in the first and second grades.

The first is the rule on the order of performing actions in expressions without brackets, when numbers are either only addition and subtraction, or multiplication and division (3 cl.). The purpose of the work at this stage is, based on the practical skills of students acquired earlier, to pay attention to the order in which actions are performed in such expressions and formulate a rule.

Leading children to the formulation of the rule, understanding it can be different. The main reliance on existing experience, the maximum possible independence, the creation of a situation of search and discovery, evidence.

You can use the methodical technique of Sh.A. Amonashvili "teacher's mistake".

For example. The teacher reports that when finding the meaning of the following expressions, he got answers, in the correctness of which he is sure (answers are closed).

36:2 6=6 etc.

Invites the children to find the meanings of the expressions themselves, and then compare the answers with the answers received by the teacher (at this point, the results of arithmetic operations are revealed). Children prove that the teacher made mistakes and, based on the study of particular facts, formulate a rule (see mathematics textbook, grade 3).

Similarly, you can introduce the rest of the rules for the order of actions: when expressions without brackets contain actions of the 1st and 2nd stage, in expressions with brackets. It is important that children realize that changing the order of performing arithmetic operations leads to a change in the result, in connection with which mathematicians decided to agree and formulate rules that must be strictly observed.

Expression conversion is the replacement of a given expression with another one with the same numerical value. Students perform such transformations of expressions, based on the properties of arithmetic operations and the consequences of them (, pp. 249-250).

When studying each property, students are convinced that in expressions of a certain type, actions can be performed in different ways, but the value of the expression does not change. In the future, students apply knowledge of the properties of actions to transform given expressions into identical expressions. For example, tasks of the form are offered: continue recording so that the “=” sign is preserved:

76-(20 + 4) =76-20... (10 + 7) -5= 10-5...

60: (2 10) =60:10...

When completing the first task, the students reason as follows: on the left, the sum of the numbers 20 and 4 is subtracted from 76 , on the right, 20 was subtracted from 76; in order to get the same amount on the right as on the left, it is necessary to subtract 4 more on the right. Other expressions are similarly transformed, that is, after reading the expression, the student remembers the corresponding rule. And, performing actions according to the rule, it receives the transformed expression. To make sure the conversion is correct, the children calculate the values of the given and converted expressions and compare them.

Applying knowledge of the properties of actions to substantiate calculation methods, students of grades I-IV perform transformations of expressions of the form:

72:3= (60+12):3 = 60:3+12:3 = 24 1830= 18(310) = (183) 10=540

It is also necessary here that students not only explain on the basis of what they receive each subsequent expression, but also understand that all these expressions are connected by the “=” sign, because they have the same meaning. To do this, occasionally you should offer children to calculate the values of expressions and compare them. This prevents errors like: 75 - 30 = 70 - 30 = 40+5 = 45, 24 12= (10 + 2) =24 10+24 2 = 288.

Students of grades II-IV perform the transformation of expressions not only on the basis of the properties of the action, but also on the basis of their specific meaning. For example, the sum of identical terms is replaced by the product: (6+ 6 + 6 = 6 3, and vice versa: 9 4 = = 9 + 9 + 9 + 9). Based also on the meaning of the action of multiplication, more complex expressions are converted: 8 4 + 8 = 8 5, 7 6-7 = 7 5.

On the basis of calculations and analysis of specially selected expressions, students of grade IV are led to the conclusion that if brackets in expressions with brackets do not affect the order of actions, then they can be omitted. In the future, using the learned properties of actions and the rules for the order of actions, students practice converting expressions with brackets into expressions that are identical to them without brackets. For example, it is proposed to write these expressions without brackets so that their values do not change:

(65 + 30)-20 (20 + 4) 3

96 - (16 + 30) (40 + 24): 4

So, the children replace the first of the given expressions with the expressions: 65 + 30-20, 65-20 + 30, explaining the order of performing actions in them. In this way, students make sure that the meaning of the expression does not change when changing the order of actions only if the properties of the actions are applied in this case.

Introduction ................................................ ................................................. ....... 2

1.1 Experience of introducing elements of algebra in elementary school .............................. 7

1.2 Psychological foundations for the introduction of algebraic concepts

in primary school............................................... ................................ 12

1.3 The problem of the origin of algebraic concepts and its significance

for the construction of a subject .................................................................. ....... twenty

2.1 Education in primary school in terms of needs

high school ................................................................ ...................................... 33

2.1 Comparison (opposition) of concepts in mathematics lessons .... 38

2.3 Joint study of addition and subtraction, multiplication and division 48

Chapter III. The practice of studying algebraic material in mathematics lessons in the primary grades of secondary school No. 4 in Rylsk .................................................. ... 55

3.1 Rationale for the use of innovative technologies (technologies

enlargement of didactic units) .............................................. ...... 55

3.2 About the experience of getting acquainted with algebraic concepts in grade I .... 61

3.3 Learning to solve problems related to the movement of bodies .............................. 72

Conclusion................................................. ................................................. .76

Bibliographic list .................................................................. ......................... 79

Introduction

In any modern system of general education, mathematics occupies one of the central places, which undoubtedly speaks of the uniqueness of this field of knowledge.

An outstanding Russian mathematician A.N. Kolmogorov wrote: “Mathematics is not just one of the languages. Mathematics is a language plus reasoning, it’s like a language and logic together. Mathematics is a tool for thinking. It concentrates the results of the exact thinking of many people. With the help of mathematics, one reasoning can be connected with another ... The obvious complexities of nature, with its strange laws and rules, each of which allows a separate very detailed explanation, are in fact closely connected. However, if you do not want to use mathematics, then in this huge variety of facts you will not see that logic allows you to go over from one to another "(, p. 44).

Thus, mathematics allows us to form certain forms of thinking necessary for studying the world around us.

In recent years, there has been a steady trend of the penetration of mathematical methods into such sciences as history, philology, not to mention linguistics and psychology. Therefore, the circle of people who, in their subsequent professional activities, will probably apply mathematics, is expanding.

Our education system is arranged in such a way that for many, school provides the only opportunity in life to join the mathematical culture, to master the values contained in mathematics.

What is the influence of mathematics in general and school mathematics in particular on the upbringing of a creative person? Teaching the art of problem solving in mathematics classes provides us with an exceptionally favorable opportunity for the formation of a certain mindset in students. The need for research develops interest in patterns, teaches to see the beauty and harmony of human thought. All this is in our opinion the most important element of a common culture. An important influence is exerted by the course of mathematics on the formation of various forms of thinking: logical, spatial-geometric, algorithmic. Any creative process begins with the formulation of a hypothesis. Mathematics, with the appropriate organization of education, being a good school for constructing and testing hypotheses, teaches us to compare various hypotheses, find the best option, set new tasks, and look for ways to solve them. Among other things, she also develops the habit of methodical work, without which no creative process is conceivable. Maximizing the possibilities of human thinking, mathematics is its highest achievement. It helps a person in self-awareness and the formation of his character.

The fact is that these numbers have different functions: the former are associated with the counting of objects, the latter with the measurement of quantities. This circumstance is very important for understanding the fact that fractional (rational) numbers are only a special case of real numbers.

Achieving this goal requires solving the following tasks:

Consideration of the general theoretical aspects of the introduction in elementary school of the algebraic concepts of magnitude and number. This task is posed in the first chapter of the work;

Show the practical applicability of the provisions under consideration in school mathematics lessons in elementary school (the lessons were conducted by the author in secondary school No. 4 in Rylsk). This is the subject of the third chapter of the work.

With regard to the bibliography devoted to this issue, the following can be noted. Despite the fact that recently the total amount of published methodological literature in mathematics is extremely small, there was no shortage of information when writing the work. Indeed, from 1960 (the time the problem was posed) to 1990. In our country, a huge amount of educational, scientific and methodological literature has been published, to one degree or another affecting the problem of introducing algebraic concepts in the course of mathematics for elementary school. In addition, these issues are regularly covered in specialized periodicals. So, when writing the work, publications in the journals Pedagogy, Teaching Mathematics at School and Primary School were used to a large extent.

Chapter I. General theoretical aspects of studying algebraic material in elementary school 1.1 Experience in introducing elements of algebra in elementary school

The content of a subject, as you know, depends on many factors - on the requirements of life to the knowledge of students, on the level of relevant sciences, on the mental and physical age capabilities of children, etc. The correct consideration of these factors is an essential condition for the most effective learning schoolchildren, expanding their cognitive abilities. But sometimes this condition is not met for one reason or another. In this case, teaching does not give the desired effect both in relation to the assimilation of the range of necessary knowledge by children, and in relation to the development of their intellect.

It seems that at present the programs of teaching some subjects, in particular mathematics, do not meet the new requirements of life, the level of development modern sciences(for example, mathematics) and new data developmental psychology and logic. This circumstance dictates the need for a comprehensive theoretical and experimental verification of possible projects for the new content of educational subjects.

Foundation mathematical knowledge established in elementary school. But, unfortunately, both mathematicians themselves and methodologists and psychologists pay very little attention to the content of elementary mathematics. Suffice it to say that the curriculum in mathematics in elementary school (I-IV grades) in its main features took shape 50-60 years ago and naturally reflects the system of mathematical, methodological and psychological ideas of that time.

Consider the characteristic features of the state standard for mathematics in elementary school. Its main content is integers and operations on them, studied in a certain sequence. First, four actions are studied in the limit of 10 and 20, then - oral calculations in the limit of 100, oral and written calculations in the limit of 1000 and, finally, in the limit of millions and billions. In grade IV, some relationships between data and the results of arithmetic operations, as well as simple fractions, are studied. Along with this, the program involves the study of metric measures and measures of time, mastering the ability to use them for measurement, knowledge of some elements of visual geometry - drawing a rectangle and a square, measuring segments, areas of a rectangle and a square, calculating volumes.

Students should apply the acquired knowledge and skills to solving problems and performing simple calculations. Throughout the course, problem solving is carried out in parallel with the study of numbers and actions - half of the corresponding time is allocated for this. Solving problems helps students understand the specific meaning of actions, understand the various cases of their application, establish the relationship between quantities, and gain elementary skills in analysis and synthesis. From grades I to IV, children solve the following main types of problems (simple and compound): finding the sum and remainder, product and quotient, increasing and decreasing these numbers, difference and multiple comparison, simple triple rule, proportional division, finding the unknown by two differences, the calculation of the arithmetic mean and some other types of tasks.

Children encounter different types of dependencies of quantities when solving problems. But it is very characteristic - students start tasks after and as they study numbers; the main thing that is required when solving is to find a numerical answer. Children with great difficulty reveal the properties of quantitative relations in specific, private situations, which are commonly considered arithmetic problems. Practice shows that the manipulation of numbers often replaces the actual analysis of the conditions of the problem from the point of view of the dependences of real quantities. The tasks introduced into the textbooks do not, moreover, represent a system in which more "complex" situations would be connected with "deeper" layers of quantitative relations. Problems of the same difficulty can be found both at the beginning and at the end of the textbook. They change from section to section and from class to class according to the complexity of the plot (the number of actions increases), according to the rank of numbers (from ten to a billion), according to the complexity of physical dependencies (from distribution problems to problems on movement) and other parameters. Only one parameter - deepening into the system of proper mathematical laws - is manifested in them weakly, indistinctly. Therefore, it is very difficult to establish a criterion for the mathematical difficulty of a particular problem. Why are the tasks of finding the unknown by two differences and finding the arithmetic mean (grade III) more difficult than the tasks of difference and multiple comparisons (grade II)? The methodology does not give a convincing and logical answer to this question.

Thus, primary school students do not receive adequate, full-fledged knowledge about the dependencies of quantities and the general properties of quantity, either when studying the elements of number theory, because they are mainly associated with the technique of calculations in the school course, or when solving problems, because the latter do not have the appropriate form and do not have the required system. Attempts by methodologists to improve teaching methods, although they lead to partial success, however, do not change the general state of affairs, since they are limited in advance by the framework of the accepted content.

It seems that the critical analysis of the adopted program in arithmetic should be based on the following provisions:

The concept of number is not identical with the concept of the quantitative characteristics of objects;

The number is not the original form of expressing quantitative relations.

We present the rationale for these provisions.

It is well known that modern mathematics (in particular, algebra) studies such moments of quantitative relations that do not have a numerical shell. It is also well known that some quantitative relations are quite expressible without numbers and before numbers, for example, in segments, volumes, etc. (relation "greater than", "less than", "equal to"). The presentation of the initial general mathematical concepts in modern manuals is carried out in such a symbolism that does not imply the obligatory expression of objects by numbers. So, in the book of E.G. Gonin "Theoretical Arithmetic", the main mathematical objects from the very beginning are denoted by letters and special signs (, pp. 12 - 15). It is characteristic that certain types of numbers and numerical dependencies are given only as examples, illustrations of the properties of sets, and not as their only possible and unique existing form expressions. Further, it is noteworthy that many illustrations of individual mathematical definitions are given in graphical form, through the ratio of segments, areas (, pp. 14-19). All the basic properties of sets and quantities can be derived and substantiated without involving numerical systems; moreover, the latter themselves receive justification on the basis of general mathematical concepts.

In turn, numerous observations by psychologists and educators show that quantitative representations arise in children long before they acquire knowledge about numbers and methods of operating with them. True, there is a tendency to refer these representations to the category of "pre-mathematical formations" (which is quite natural for traditional methods that identify the quantitative characteristic of an object with a number), but this does not change their essential function in the child's general orientation in the properties of things. And sometimes it happens that the depth of these supposedly "pre-mathematical formations" is more essential for the development of a child's own mathematical thinking than knowledge of the intricacies of computer technology and the ability to find purely numerical dependencies. It is noteworthy that acad. A.N. Kolmogorov, characterizing the features of mathematical creativity, specifically notes the following circumstance: “The majority of mathematical discoveries are based on some simple idea: a visual geometric construction, a new elementary inequality, etc. It is only necessary to properly apply this simple idea to solving a problem that seems inaccessible at first sight" (, p. 17).

A wide variety of ideas regarding the structure and methods of constructing a new program are currently expedient. It is necessary to involve mathematicians, psychologists, logicians, and methodologists in the work on its construction. But in all its specific variants, it seems to have to satisfy the following basic requirements:

Bridge the existing gap between the content of mathematics in primary and secondary schools;

To give a system of knowledge about the basic regularities of the quantitative relations of the objective world; at the same time, the properties of numbers, as a special form of expressing quantity, should become a special, but not the main section of the program;

To instill in children the techniques of mathematical thinking, and not just calculation skills: this involves the construction of such a system of tasks, which is based on deepening into the sphere of dependencies of real quantities (the connection of mathematics with physics, chemistry, biology and other sciences that study specific quantities);

Resolutely simplify the entire technique of calculation, reducing to a minimum the work that cannot be done without appropriate tables, reference books and other auxiliary (in particular, electronic) means.

The meaning of these requirements is clear: in elementary school it is quite possible to teach mathematics as a science about the regularities of quantitative relations, about the dependencies of quantities; computational techniques and elements of number theory should become a special and private section of the program.

The experience of constructing a new program in mathematics and its experimental verification, carried out since the late 1960s, already at the present time allow us to talk about the possibility of introducing a systematic course of mathematics into school, starting from the 1st grade, providing knowledge about quantitative relationships and dependencies of quantities in algebraic form. .

1.2 Psychological foundations for the introduction of algebraic concepts in elementary school

Recently, in the modernization of curricula, special attention has been given to bringing a set-theoretical foundation to the school curriculum (this trend is clearly manifested both in our country and abroad). The implementation of this trend in teaching (especially in the primary grades, as is observed, for example, in American school) will inevitably pose a number of difficult questions for child and educational psychology and for didactics, because now there are almost no studies that reveal the features of the child's assimilation of the meaning of the concept of set (in contrast to the assimilation of counting and number, which has been studied very many-sidedly).

Logical and psychological research in recent years (especially the work of J. Piaget) has revealed the connection between certain "mechanisms" of children's thinking and general mathematical concepts. Below, the features of this connection and their significance for the construction of mathematics as an academic subject are specially considered (in this case, we will focus on the theoretical side of the matter, and not on any particular version of the program).

The natural number has been a fundamental concept of mathematics throughout its history; it plays a very significant role in all areas of production, technology, and everyday life. This allows theoretical mathematicians to give it a special place among other concepts of mathematics. In various forms, statements are made that the concept of a natural number is the initial stage of mathematical abstraction, that it is the basis for the construction of most mathematical disciplines.

The choice of the initial elements of mathematics as an academic subject essentially implements these general provisions. At the same time, it is assumed that, while getting acquainted with the number, the child simultaneously reveals for himself the initial features of quantitative relations. Counting and number are the basis of all subsequent learning of mathematics at school.

However, there is reason to believe that these provisions, while rightly highlighting the special and fundamental meaning of the number, at the same time inadequately express its connection with other mathematical concepts, inaccurately assess the place and role of the number in the process of mastering mathematics. Because of this circumstance, in particular, some significant shortcomings of the adopted programs, methods and textbooks in mathematics result. It is necessary to specially consider the actual connection of the concept of number with other concepts.

Many general mathematical concepts, and in particular the concepts of equivalence and order relations, are systematically considered in mathematics, regardless of the numerical form. These concepts do not lose their independent character; on their basis, one can describe and study a particular subject - various numerical systems, the concepts of which by themselves do not cover the meaning and meaning of the original definitions. Moreover, in the history of mathematical science, general concepts have developed precisely to the extent that "algebraic operations", famous example which are delivered by four operations of arithmetic, began to be applied to elements of a completely non-"numerical" character.

Recently, attempts have been made to develop in teaching the stage of introducing the child to mathematics. This trend finds expression in methodological manuals, as well as in some experimental textbooks. So, in one American textbook intended for teaching children 6-7 years old (), tasks and exercises are introduced on the first pages that specifically train children in establishing the identity of subject groups. The children are shown the technique of connecting sets, - at the same time, the corresponding mathematical symbolism is introduced. Working with numbers is based on elementary information about sets.

It is possible to assess the content of specific attempts to implement this trend in different ways, but it, in our opinion, is quite legitimate and promising.

At first glance, the concepts of "relationship", "structure", "laws of composition", etc., which have complex mathematical definitions, cannot be associated with the formation of mathematical representations in young children. Of course, the entire true and abstract meaning of these concepts and their place in the axiomatic construction of mathematics as a science is an object of assimilation by a head that is already well developed and "trained" in mathematics. However, certain properties of things fixed by these concepts, in one way or another, appear for the child already relatively early: there are concrete psychological data for this.

First of all, it should be borne in mind that from the moment of birth to 7 - 10 years, the child develops and forms the most complex systems of general ideas about the world around him and the foundation of content-objective thinking is laid. Moreover, on the basis of relatively narrow empirical material, children identify general schemes of orientation in the spatio-temporal and cause-and-effect relationships of things. These schemes serve as a kind of framework for the "coordinate system" within which the child begins to more and more deeply master the various properties of the diverse world. Of course, these general schemes are little realized and to a small extent can be expressed by the child himself in the form of an abstract judgment. Figuratively speaking, they are an intuitive form of organization of the child's behavior (although, of course, they are more and more reflected in judgments as well).

In recent decades, the questions of the formation of the intellect of children and the emergence of their general ideas about reality, time and space have been studied especially intensively by the famous Swiss psychologist J. Piaget and his collaborators. Some of his works are directly related to the problems of the development of a child's mathematical thinking, and therefore it is important for us to consider them in relation to design issues. curriculum.

In one of his recent books() J. Piaget gives experimental data on the genesis and formation in children (up to 12 - 14 years old) of such elementary logical structures as classification and seriation. Classification involves the implementation of an inclusion operation (for example, A + A "= B) and an operation inverse to it (B - A" = A). Seriation is the ordering of objects in systematic rows (for example, sticks of different lengths can be arranged in a row, each member of which is greater than all previous ones and less than all subsequent ones).

Analyzing the formation of classification, J. Piaget shows how, from its original form, from the creation of a "figured set" based only on the spatial proximity of objects, children move on to a classification based on the relation of similarity ("non-shaped sets"), and then to the classification itself. complex form - to the inclusion of classes, due to the relationship between the volume and content of the concept. The author specifically considers the issue of forming a classification not only according to one, but also according to two or three signs, about the formation in children of the ability to change the basis of the classification when new elements are added. The authors also find similar stages in the process of the development of seriation.

These studies were aimed at definite purpose- to reveal the patterns of formation of the operator structures of the mind and, above all, their constitutive property as reversibility, i.e. the ability of the mind to move forward and backward. Reversibility occurs when "operations and actions can unfold in two directions, and the understanding of one of these directions causes ipso facto [by the very fact] the understanding of the other" (, p. 15).

Reversibility, according to J. Piaget, represents the fundamental law of composition inherent in the mind. It has two complementary and irreducible forms: reversal (inversion or negation) and reciprocity. The inversion takes place, for example, in the case when the spatial displacement of an object from A to B can be canceled by transferring the object back from B to A, which is ultimately equivalent to a zero transformation (the product of an operation on the inverse is an identical operation, or a zero transformation).

Reciprocity (or compensation) implies the case when, for example, when an object is moved from A to B, the object remains in B, but the child himself moves from A to B and reproduces the initial position when the object was against his body. The movement of the object is not canceled here, but it was compensated by appropriate displacement own body- and this is already a different form of transformation than conversion (, p. 16).

In his works, J. Piaget showed that these transformations first appear in the form of sensory-motor circuits (from 10 to 12 months). Gradual coordination of sensory-motor schemes, functional symbolism and linguistic display lead to the fact that through a number of stages, conversion and reciprocity become properties of intellectual actions (operations) and are synthesized in a single operator structure (in the period from 7 to 11 and from 12 to 15 years old) . Now the child can coordinate all movements into one in two frames of reference at once - one is mobile, the other is stationary.

J. Piaget believes that a psychological study of the development of arithmetic and geometric operations in the mind of a child (especially those logical operations that carry out preliminary conditions in them) makes it possible to accurately correlate the operator structures of thinking with algebraic structures, order structures and topological structures (, p. 13). Thus, the algebraic structure ("group") corresponds to the operator mechanisms of the mind, subject to one of the forms of reversibility - inversion (negation). A group has four elementary properties: the product of two group elements also gives a group element; direct operation corresponds to one and only one reverse; there is an identity operation; successive compositions are associative. In the language of intellectual action, this means:

The coordination of the two systems of action is new scheme, attached to the previous ones;

The operation can develop in two directions;

When we return to the starting point, we find it unchanged;

One and the same point can be reached in different ways, and the point itself remains unchanged.

The facts of the “independent” development of the child (i.e. development independent of the direct influence schooling) show a discrepancy between the order of the stages of geometry and the stages of the formation of geometric concepts in a child. The latter approach the order of succession of the main groups, where the topology comes first. According to Piaget, the child first develops a topological intuition, and then he orients himself in the direction of projective and metric structures. Therefore, in particular, as J. Piaget notes, at the first attempts at drawing, the child does not distinguish between squares, circles, triangles and other metric figures, but perfectly distinguishes between open and closed figures, the position "outside" or "inside" in relation to the border, separation and neighborhood (for the time being, not distinguishing distances), etc. (, p. 23).

Let us consider the main provisions formulated by J. Piaget in relation to the issues of constructing a curriculum. First of all, Piaget's research shows that during preschool and school childhood, a child develops such operator structures of thinking that allow him to evaluate the fundamental characteristics of classes of objects and their relationships. Moreover, already at the stage of concrete operations (from the age of 7-8), the child's intellect acquires the property of reversibility, which is extremely important for understanding the theoretical content of educational subjects, in particular mathematics.

These data indicate that traditional psychology and pedagogy did not sufficiently take into account the complex and capacious nature of those stages of a child's mental development that are associated with the period from 2 to 7 and from 7 to 11 years.

Consideration of the results obtained by J. Piaget allows us to draw a number of significant conclusions in relation to the design of a curriculum in mathematics. First of all, the actual data on the formation of the child's intellect from 2 to 11 years old show that at this time not only are the properties of objects described by means of the mathematical concepts of "relationship - structure" not "alien" to him, but the latter themselves organically enter into the child's thinking.

Traditional programs do not take this circumstance into account. Therefore, they do not realize many of the possibilities lurking in the process of the child's intellectual development.

The materials available in modern child psychology make it possible to positively evaluate the general idea of constructing such an educational subject, which would be based on the concepts of initial mathematical structures. Of course, great difficulties arise along this path, since there is still no experience in constructing such a subject. In particular, one of them is related to the definition of the age "threshold" from which training under the new program is feasible. If we follow the logic of J. Piaget, then, apparently, these programs can be taught only when the operator structures have already fully formed in children (from the age of 14-15). But if we assume that the real mathematical thinking of a child is formed precisely within the process that is designated by J. Piaget as the process of folding operator structures, then these programs can be introduced much earlier (for example, from 7 - 8 years), when children begin to form specific operations with the highest level of reversibility. In "natural" conditions, when studying according to traditional programs, formal operations, perhaps, only take shape by the age of 13-15. But is it possible to "accelerate" their formation by earlier introduction of such educational material, the assimilation of which requires a direct analysis of mathematical structures?

It seems that such possibilities exist. By the age of 7 - 8, children have already developed a plan of mental actions to a sufficient extent, and by teaching according to the appropriate program, in which the properties of mathematical structures are given "explicitly" and the children are given the means of analyzing them, it is possible to quickly bring children to the level of "formal" operations, than in the terms in which it is carried out with the "independent" discovery of these properties.

In this case, it is important to take into account the following circumstance. There is reason to believe that the peculiarities of thinking at the level of specific operations, dated by J. Piaget to 7-11 years old, are themselves inextricably linked with the forms of organization of education characteristic of the traditional elementary school. This training (both in our country and abroad) is conducted on the basis of an extremely empirical content, often not connected at all with the conceptual (theoretical) attitude to the object. Such training supports and consolidates in children thinking based on external, perceptible signs of things by direct perception.

Thus, at the present time there are factual data showing a close connection between the structures of children's thinking and general algebraic structures, although the "mechanism" of this connection is far from clear and has hardly been studied. The presence of this connection opens up fundamental possibilities (so far only possibilities!) for constructing an educational subject that develops according to the scheme "from simple structures to their complex combinations." One of the conditions for the realization of these possibilities is the study of the transition to mediated thinking and its age standards. This method of constructing mathematics as an academic subject can itself be a powerful lever for the formation of such thinking in children, which is based on a fairly solid conceptual foundation.

1.3 The problem of the origin of algebraic concepts and its significance for the construction of a subject

The division of the school course of mathematics into algebra and arithmetic, of course, is conditional. The transition from one to the other is gradual. In school practice, the meaning of this transition is masked by the fact that the study of fractions actually takes place without a detailed reliance on the measurement of quantities - fractions are given as ratios of pairs of numbers (although formally the importance of measuring quantities is recognized in methodological manuals). An expanded introduction of fractional numbers based on the measurement of quantities inevitably leads to the concept of a real number. But the latter just usually does not happen, since students are kept at work with rational numbers for a long time, and thus their transition to "algebra" is delayed.

In other words, school algebra begins precisely when conditions are created for the transition from integers to real numbers, to expressing the measurement result as a fraction (simple and decimal - finite, and then infinite).

Moreover, the initial one can be acquaintance with the measurement operation, obtaining the final decimal fractions and learning how to act on them. If students already know this form of recording the measurement result, then this serves as a prerequisite for "throwing" the idea that a number can also be expressed as an infinite fraction. And it is advisable to create this prerequisite already within the elementary school.

If the concept of a fractional (rational) number is removed from the competence of school arithmetic, then the boundary between it and "algebra" will pass along the line of difference between integer and real numbers. It is it that "cuts" the course of mathematics into two parts. This is not a simple difference, but a fundamental "dualism" of sources - accounts and measurements.

Following Lebesgue's ideas regarding " general concept number", it is possible to ensure complete unity of teaching mathematics, but only from the moment and after the children are familiarized with the account and the whole (natural) number. Of course, the timing of this preliminary acquaintance may be different (in traditional elementary school programs they are clearly delayed), in the course elementary arithmetic, you can even introduce elements of practical measurements (which takes place in the program), - however, all this does not remove the difference between the bases of arithmetic and "algebra" as academic subjects. sections related to the measurement of quantities and the transition to real fractions took root. The authors of programs and methodologists strive to preserve the stability and "purity" of arithmetic as a school subject. This difference in sources is the main reason for teaching mathematics according to the scheme - first arithmetic (integer), then "algebra" (real number).

This scheme seems quite natural and unshakable, moreover, it is justified by many years of practice in teaching mathematics. But there are circumstances that, from a logical-psychological point of view, require a more thorough analysis of the legitimacy of this rigid teaching scheme.

The fact is that despite all the differences between these types of numbers, they refer specifically to numbers, i.e. to a special form of displaying quantitative relations. The belonging of the integer and real numbers to "numbers" serves as the basis for the assumption of genetic derivativeness and the very differences in counting and measurement: they have a special and single source corresponding to the very form of the number. Knowledge of the features of this unified basis for counting and measuring will make it possible to more clearly present the conditions of their origin, on the one hand, and the relationship, on the other.

What to turn to in order to find the common root of a branching tree of numbers? It seems that, first of all, it is necessary to analyze the content of the concept of magnitude. True, another term is immediately associated with this term - measurement. However, the legitimacy of such a connection does not exclude a certain independence of the meaning of "value". Consideration of this aspect allows us to draw conclusions that, on the one hand, bring together measurement with counting, and, on the other hand, operate with numbers with some general mathematical relations and patterns.

So, what is a "value" and what is its interest in constructing the initial sections of school mathematics?

In common use, the term "value" is associated with the concepts "equal", "more", "less", which describe a variety of qualities (length and density, temperature and whiteness). V.F. Kagan raises the question of what common properties these concepts have. He shows that they refer to collections - sets of homogeneous objects, the comparison of the elements of which allows us to apply the terms "greater than", "equal to", "less than" (for example, to the collections of all straight line segments, weights, speeds, etc.).

A set of objects is only transformed into a value when criteria are established that allow one to establish, with respect to any of its elements A and B, whether A will be equal to B, greater than B or less than B. At the same time, for any two elements A and B, one and only one of ratios: A=B, A>B, A<В.

These sentences constitute a complete disjunction (at least one occurs, but each excludes all the others).

V.F. Kagan identifies the following eight basic properties of the concepts "equal", "greater", "less": (, pp. 17-31).

1) At least one of the following relations holds: A=B, A>B, A<В.

2) If the relation A = B holds, then the relation A does not hold<В.

3) If the relation A=B holds, then the relation A>B does not hold.

4) If A=B and B=C, then A=C.

5) If A>B and B>C, then A>C.

6) If A<В и В<С, то А<С.

7) Equality is a reversible relation: the relation A=B always implies the relation B=A.

8) Equality is a reciprocal relation: whatever the element A of the set under consideration, A=A.

The first three sentences characterize the disjunction of the basic relations "=", ">", "<". Предложения 4 - 6 - их транзитивность при любых трех элементах А, В и С. Следующие предложения 7 - 8 характеризуют только равенство - его обратимость и возвратность (или рефлексивность). Эти восемь основных положений В.Ф.Каган называет поcтулатами сравнения, на базе которых можно вывести ряд других свойств величины.

These output properties of V.F. Kagan describes in the form of eight theorems:

I. The relation A>B excludes the relation B>A (A<В исключает В<А).

II. If A>B, then B<А (если А<В, то В>BUT).

III. If A>B holds, then A does not hold.

IV. If A1=A2, A2=A3,.., An-1=A1, then A1=An.

V. If A1>A2, A2>A3,.., An-1>An, then A1>An.

VI. If A1<А2, А2<А3,.., Аn-1<Аn, то А1<Аn.

VII. If A=C and B=C, then A=B.

VIII. If there is an equality or inequality A \u003d B, or A\u003e B, or A<В, то оно не нарушится, когда мы один из его элементов заменим равным ему элементом (здесь имеет место соотношение типа:

if A=B and A=C, then C=B;

if A>B and A=C, then C>B, etc.).

Comparison postulates and theorems, points out V.F. Kagan, "all those properties of the concepts "equal", "more" and "less" are exhausted, which in mathematics are associated with them and find application for themselves regardless of the individual properties of the set, to whose elements we apply them in various special cases" (, page 31).

The properties indicated in postulates and theorems can characterize not only those direct features of objects that we are used to associate with "equal", "greater", "less", but also with many other features (for example, they can characterize the relationship "ancestor - descendant"). This allows us to take a general point of view when describing them and consider, for example, from the point of view of these postulates and theorems, any three types of relations "alpha", "beta", "gamma" (in this case, it can be established whether these relations satisfy the postulates and theorems and under what conditions).

From this point of view, one can, for example, consider such a property of things as hardness (harder, softer, equal hardness), the sequence of events in time (following, precedence, simultaneity), etc. In all these cases, the ratios "alpha", "beta", "gamma" receive their specific interpretation. The task associated with the selection of such a set of bodies that would have these relationships, as well as the identification of signs by which "alpha", "beta", "gamma" could be characterized - this is the task of determining the comparison criteria in this set of bodies (practically, in some cases it is not easy to solve it). "By establishing the criteria for comparison, we turn the set into a value," wrote V.F. Kagan (, p. 41).

Real objects can be considered from the point of view of different criteria. Thus, a group of people can be considered according to such a criterion as the sequence of moments of birth of each of its members. Another criterion is the relative position that the heads of these people will take if they are placed side by side on the same horizontal plane. In each case, the group will be translated into a value that has the appropriate name - age, height. In practice, the value is usually denoted, as it were, not by the set of elements itself, but by a new concept introduced to distinguish between comparison criteria (the name of the value). This is how the concepts of "volume", "weight", "electrical voltage", etc. arise. “At the same time, for a mathematician, the value is quite definite when the set of elements and comparison criteria are indicated,” noted V.F. Kagan (, p. 47).

As the most important example of a mathematical quantity, this author considers the natural series of numbers. From the point of view of such a comparison criterion as the position occupied by numbers in a series (occupy one place, follows ..., precedes), this series satisfies the postulates and therefore represents a magnitude. According to the relevant comparison criteria, the set of fractions is also converted into a value.

Such, according to V.F. Kagan, the content of the theory of quantity, which plays a crucial role in the justification of all mathematics.

Working with quantities (it is advisable to fix their individual values with letters), you can produce a complex system of transformations, establishing the dependencies of their properties, moving from equality to inequality, performing addition (and subtraction), and when adding, you can be guided by commutative and associative properties. So, if the ratio A=B is given, then when "solving" problems, one can be guided by the ratio B=A. In another case, in the presence of the ratios A>B, B=C, we can conclude that A>C. Since for a>b there is a c such that a=b+c, we can find the difference between a and b (a-b=c), and so on. All these transformations can be performed on physical bodies and other objects by setting the criteria for comparison and the correspondence of the selected relations to the postulates of comparison.

The above materials allow us to conclude that both natural and real numbers are equally strongly associated with quantities and some of their essential features. Is it possible to make these and other properties the subject of a special study of the child even before the numerical form of describing the ratio of magnitudes is introduced? They can serve as prerequisites for the subsequent detailed introduction of the number and its different types, in particular for the propaedeutics of fractions, the concepts of coordinates, functions and other concepts already in the lower grades.

What could be the content of this initial section? This is an acquaintance with physical objects, criteria for their comparison, highlighting the value as a subject of mathematical consideration, familiarity with the methods of comparison and sign means of fixing its results, with methods for analyzing the general properties of quantities. This content must be expanded into a relatively detailed program of teaching and, most importantly, linked to the actions of the child through which he can master this content (of course, in the appropriate form). At the same time, it is necessary to establish experimentally, experimentally, whether children of 7 years old can master this program, and what is the expediency of its introduction for the subsequent teaching of mathematics in primary grades in the direction of convergence of arithmetic and elementary algebra.

Until now, our discussions have been theoretical in nature and were aimed at clarifying the mathematical prerequisites for constructing such an initial section of the course that would introduce children to the basic algebraic concepts (before the special introduction of a number).

The main properties that characterize quantities have been described above. Naturally, it is pointless for children of 7 years old to read "lectures" regarding these properties. It was necessary to find such a form of work for children with didactic material, by means of which they could, on the one hand, reveal these properties in the things around them, on the other hand, they would learn to fix them with certain symbolism and carry out elementary mathematical analysis highlighted relationships.

In this regard, the program should contain, firstly, an indication of those properties of the subject that are to be mastered, secondly, a description of didactic materials, and thirdly, and this is the main thing from a psychological point of view, the characteristics of those actions through which the child identifies certain properties of the object and master them. These "constituents" form the teaching program in the proper sense of the word.

It makes sense to describe the specific features of this hypothetical program and its "components" when describing the learning process itself and its results. Here is a diagram of this program and its main themes.

Theme I. Equalization and acquisition of objects (by length, volume, weight, composition of parts and other parameters).

Practical tasks for leveling and picking. Isolation of signs (criteria) by which the same objects can be equalized or completed. Verbal designation of these signs ("by length", by weight", etc.).

These tasks are solved in the process of working with didactic material (slats, weights, etc.) by:

The choice of the "same" subject,

Reproduction (construction) of the "same" object according to the selected (specified) parameter.

Theme II. Comparison of objects and fixation of its results by the equality-inequality formula.

1. Tasks for comparing objects and symbolic designation of the results of this action.

2. Verbal fixation of the comparison results (the terms "greater than", "less than", "equal to"). Letters ">", "<", "=".

3. Designation of the comparison result with a drawing ("copying", and then "abstract" - lines).

4. Designation of compared objects by letters. Recording the comparison result with formulas: A=B; BUT<Б, А>b.

A letter as a sign that fixes the directly given, private value of an object according to a selected parameter (by weight, by volume, etc.).

5. The impossibility of fixing the comparison result with different formulas. The choice of a specific formula for a given result (full disjunction of relations greater than - less than - equal to).

Topic III. Properties of equality and inequality.

1. Reversibility and reflexivity of equality (if A=B, then B=A; A=A).

2. The connection of the relations "greater than" and "less than" in inequalities with "permutations" of the compared sides (if A>B, then B<А и т.п.).

3. Transitivity as a property of equality and inequality:

if A=B, if A>B, if A<Б,

a B=C, a B>C, a B<В,

then A=B; then A>B; then A<В.

4. Transition from work with subject didactic material to assessments of the properties of equality-inequality in the presence of only literal formulas. Solving various problems that require knowledge of these properties (for example, solving problems related to the connection of relations of the type: it is given that A>B, and B=C; find out the relationship between A and C).

Topic IV. Addition (subtraction) operation.

1. Observations of changes in objects by one or another parameter (by volume, by weight, by duration, etc.). Image of increase and decrease by signs "+" and "-" (plus and minus).

2. Violation of the previously established equality with a corresponding change in one or another of its sides. The transition from equality to inequality. Writing formulas like:

if A=B, if A=B,

then A+K>B; then A-K<Б.

3. Ways of transition to a new equality (its "restoration" according to the principle: adding "equal" to "equal" gives "equal").

Working with formulas like:

then A+K>B,

but A+K=B+K.

4. Solving various problems that require the use of the operation of addition (subtraction) in the transition from equality to inequality and vice versa.

Topic V. Transition from inequality of type A<Б к равенству через операцию сложения (вычитания).

1. Tasks requiring such a transition. The need to determine the value of the value by which the compared objects differ. Possibility of recording equality with an unknown specific value of this quantity. How to use x (x).

Writing formulas like:

if A<Б, если А>B,

then A+x=B; then A-x=B.

2. Determining the value of x. Substitution of this value in the formula (familiarity with parentheses). Type formulas

3. Solving problems (including "plot-text") that require the performance of these operations.

Subject Vl. Addition-subtraction of equalities-inequalities. Substitution.

1. Addition-subtraction of equalities-inequalities:

if A=B if A>C if A>C

and M=D, and K>E, and B=G,

then A+M=B+D; then A+K>B+E; then A+-B>C+-D.

2. The possibility of representing the value of a quantity as the sum of several values. Type substitution:

3. Solving a variety of tasks that require taking into account the properties of relations that the children met in the process of work (many tasks require the simultaneous consideration of several properties, quick wits when evaluating the meaning of formulas; a description of the tasks and solutions are given below).

This is a program designed for 3.5 - 4 months. first half year. As the experience of experimental teaching shows, with proper lesson planning, with the improvement of teaching methods and the successful choice of didactic aids, all the material presented in the program can be fully assimilated by children in a shorter period (in 3 months).

How is our program going forward? First of all, children get acquainted with the method of obtaining a number, expressing the ratio of an object as a whole (the same value, represented by a continuous or discrete object) to its part. This ratio itself and its specific meaning is represented by the formula A / K \u003d n, where n is any integer, most often expressing the ratio with an accuracy of "one" (only with a special selection of material or when counting only "qualitatively" individual things, you can get absolutely exact integer). From the very beginning, children are "forced" to bear in mind that when measuring or counting, a residue may be obtained, the presence of which must be specially stipulated. This is the first step to further work with a fractional number.

With this form of obtaining a number, it is not difficult to lead children to describe an object with a formula like A = 5k (if the ratio was equal to "5"). Together with the first formula, it opens up opportunities for a special study of the relationships between the object, the base (measure) and the result of counting (measurement), which also serves as propaedeutics for the transition to fractional numbers (in particular, for understanding the basic property of a fraction).

Another line of program deployment, already implemented in class I, is the transfer to numbers (integers) of the basic properties of a quantity (disjunctions of equality-inequality, transitivity, reversibility) and the operation of addition (commutativity, associativity, monotonicity, the possibility of subtraction). In particular, when working on the number line, children can quickly turn a sequence of numbers into a value (for example, clearly evaluate their transitivity by making entries like 3<5<8, одновременно связывая отношения "меньше-больше": 5<8, но 5<3, и т.д.).

Acquaintance with some of the so-called "structural" features of equality allows children to approach the relationship of addition and subtraction in a different way. Thus, when passing from inequality to equality, the following transformations are performed: 7<11; 7+х=11; x=11-7; х=4. В другом случае дети складывают и вычитают элементы равенств и неравенств, выполняя при этом работу, связанную с устными вычислениями. Например, дано 8+1=6+3 и 4>2; find the relation between the left and right parts of the formula at 8+1-4...6+3-2; in case of inequality, bring this expression to equality (first you need to put the "less" sign, and then add "two" to the left side).

Thus, handling a number series as a quantity allows you to form the skills of addition-subtraction (and then multiplication-division) in a new way.

Chapter II. Guidelines for the study of algebraic material in elementary school 2.1 Teaching in elementary school in terms of the needs of secondary school

As you know, when studying mathematics in the 5th grade, a significant part of the time is devoted to repeating what children should have learned in elementary school. This repetition in almost all existing textbooks takes 1.5 academic quarters. This situation did not happen by chance. Its reason is the dissatisfaction of secondary school mathematics teachers with the preparation of primary school graduates. What is the reason for this situation? For this, five of the most well-known elementary school mathematics textbooks today were analyzed. These are the textbooks of M.I. Moro, I.I. Arginskaya, N.B. Istomina, L.G. Peterson and V.V. Davydov (, , , , ).

The analysis of these textbooks revealed several negative aspects, to a greater or lesser extent present in each of them and negatively affecting further learning. First of all, it is that the assimilation of the material in them is largely based on memorization. A striking example of this is the memorization of the multiplication table. In elementary school, a lot of time and effort is devoted to memorizing it. But during the summer holidays, the children forget it. The reason for such rapid forgetting is rote learning. Research L.S. Vygotsky showed that meaningful memorization is much more effective than rote memorization, and subsequent experiments convincingly prove that material gets into long-term memory only if it is memorized as a result of work corresponding to this material.

A way to effectively assimilate the multiplication table was found back in the 50s. It consists in organizing a certain system of exercises, performing which, the children themselves construct the multiplication table. However, this method is not implemented in any of the reviewed textbooks.

Another negative point affecting further education is that in many cases the presentation of the material in elementary school mathematics textbooks is structured in such a way that in the future children will have to be re-taught, and this, as you know, is much more difficult than teaching. In relation to the study of algebraic material, an example is the solution of equations in elementary school. In all textbooks, the solution of equations is based on the rules for finding unknown components of actions.

This is done somewhat differently only in the textbook by L.G. Peterson, where, for example, the solution of equations for multiplication and division is based on the correlation of the components of the equation with the sides and area of the rectangle and, as a result, also comes down to rules, but these are the rules for finding the side or area of the rectangle. Meanwhile, starting from the 6th grade, children are taught a completely different principle for solving equations, based on the application of identical transformations. This need for relearning leads to the fact that solving equations is a rather difficult moment for most children.

Analyzing textbooks, we also encountered the fact that when presenting the material in them, there is often a distortion of concepts. For example, the formulation of many definitions is given as implications, while it is known from mathematical logic that any definition is an equivalence. As an illustration, we can cite the definition of multiplication from the textbook by I.I. Arginskaya: "If all the terms in the sum are equal to each other, then addition can be replaced by another action - multiplication." (All terms in the sum are equal to each other. Therefore, addition can be replaced by multiplication.) As you can see, this is an implication in its purest form. Such a formulation is not only illiterate from the point of view of mathematics, not only incorrectly forms in children an idea of what a definition is, but it is also very harmful in that in the future, for example, when building a multiplication table, the authors of textbooks use the replacement of the product by the sum of identical terms , which the present formulation does not allow. Such incorrect work with statements written in the form of an implication forms an incorrect stereotype in children, which will be overcome with great difficulty in geometry lessons, when children will not feel the difference between a direct and inverse statement, between a sign of a figure and its property. The error when the inverse theorem is used in solving problems, while only the direct one is proved, is a very common one.

Another example of the incorrect formation of concepts is the work with the relation of literal equality. For example, the rules for multiplying a number by one and a number by zero in all textbooks are given in literal form: a x 1 \u003d a, and x 0 \u003d 0. The equality relation, as you know, is symmetrical, and therefore, such a notation provides not only that when multiplied by 1, the same number is obtained, but also that any number can be represented as the product of this number and one. However, the verbal formulation proposed in the textbooks after the letter notation speaks only of the first possibility. The exercises on this topic are also aimed only at working out the replacement of the product of a number and one by this number. All this leads not only to the fact that a very important point does not become the subject of children's consciousness: any number can be written as a product, which in algebra, when working with polynomials, will cause appropriate difficulties, but also to the fact that children, in principle, do not know how to correctly work with equality. For example, when working with the difference of squares formula, children, as a rule, cope with the task of decomposing the difference of squares into factors. However, those tasks where a reverse action is required in many cases cause difficulties. Another vivid illustration of this idea is the work with the distributive law of multiplication with respect to addition. Here, too, despite the literal notation of the law, both its verbal formulation and the system of exercises work out only the ability to open brackets. As a result, taking the common factor out of brackets in the future will cause significant difficulties.

Quite often in elementary school, even when a definition or rule is formulated correctly, teaching encourages reliance not on them, but on something completely different. For example, when studying the multiplication table by 2, all the textbooks reviewed show how to construct it. In the textbook M.I. Moro did it like this:

2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2

With this method of work, children will very quickly notice the pattern of the resulting number series.

Already after 3–4 equalities, they will stop adding twos and begin to write down the result, based on the observed pattern. Thus, the method of constructing the multiplication table will not become the subject of their consciousness, which will result in its fragile assimilation.

When studying the material in elementary school, reliance is placed on objective actions and illustrative visualization, which leads to the formation of empirical thinking. Of course, it is hardly possible to do without such visibility in elementary school. But it should serve only as an illustration of this or that fact, and not as a basis for the formation of a concept. The use of illustrative visualization and objective actions in textbooks often leads to the fact that the concept itself is "blurred". For example, in the methodology of mathematics for grades 1–3, M.I. Moreau says that children have to perform division, putting objects into piles or making a drawing for 30 lessons. Behind such actions, the essence of the division operation as an action, the inverse of multiplication, is lost. As a result, division is assimilated with the greatest difficulty and much worse than other arithmetic operations.

When teaching mathematics in elementary school, nowhere is it a question of proving any statements. Meanwhile, bearing in mind the difficulty of teaching proof in secondary school, it is necessary to start preparing for this already in the primary grades. Moreover, this can be done on material that is quite accessible to younger students. Such material, for example, can be the rules for dividing a number by 1, zero by a number, and a number by itself. Children are quite capable of proving them using the definition of division and the corresponding multiplication rules.

The elementary school material also allows for the propaedeutics of algebra - work with letters and literal expressions. Most textbooks avoid using letters. As a result, for four years, children work almost exclusively with numbers, after which, of course, it is very difficult to teach them to work with letters. However, it is possible to ensure the propaedeutics of such work, to teach children how to substitute a number instead of a letter into a letter expression, already in elementary school. This is done, for example, in the textbook by L.G. Peterson.

Speaking about the shortcomings of teaching mathematics in primary school, which hinder further learning, it is necessary to emphasize the fact that often the material in textbooks is presented without looking at how it will work in the future. A very striking example of this is the organization of the assimilation of multiplication by 10, 100, 1000, etc. In all the textbooks reviewed, the presentation of this material is structured in such a way that it inevitably leads to the formation of the rule in the minds of children: "To multiply a number by 10, 100, 1000, etc., you need to assign as many zeros to it on the right as there are in 10, 100, 1000 etc." This rule is one of those that are very well learned in elementary school. And this leads to a large number of errors when multiplying decimal fractions by integer bit units. Even having memorized the new rule, children often automatically add zero to the decimal fraction on the right when multiplying by 10. In addition, it should be noted that when multiplying a natural number, and when multiplying a decimal fraction by integer bit units, in fact, the same thing happens: each digit of the number is shifted to the right by the corresponding number of digits. Therefore, it makes no sense to teach children two separate and completely formal rules. It is much more useful to teach them the general method of action in solving such tasks.

2.1 Comparison (opposition) of concepts in mathematics lessons

The current program provides for the study in grade I of only two actions of the first stage - addition and subtraction. The limitation of the first year of study to only two actions is, in essence, a departure from what had already been achieved in the textbooks that preceded the current ones: not a single teacher ever complained then that multiplication and division, say, within 20, was beyond the power of first graders. . It is also noteworthy that in schools in other countries, where education begins at the age of 6, the first academic year includes the initial acquaintance with all four operations of arithmetic. Mathematics relies primarily on four actions, and the sooner they are included in the schoolchild's thinking practice, the more stable and reliable the subsequent development of the mathematics course will be.

In fairness, it should be noted that in the first versions of M. I. Moro's textbooks for grade I, multiplication and division were provided. However, chance prevented the matter: the authors of the new programs persistently held on to one "novelty" - coverage in the first grade of all cases of addition and subtraction within 100 (37 + 58 and 95-58, etc.). But, since there was not enough time to study such an expanded amount of information, it was decided to shift multiplication and division completely to the next year of study.

So, the passion for the linearity of the program, i.e., the purely quantitative expansion of knowledge (the same actions, but with large numbers), took the time that was previously allotted for the qualitative deepening of knowledge (the study of all four actions within two dozen). The study of multiplication and division already in the first grade means a qualitative leap in thinking, since it allows you to master folded thought processes.

According to tradition, the study of addition and subtraction within 20 used to be a special topic. The need for this approach in systematizing knowledge is visible even from a logical analysis of the issue: the fact is that the complete addition table of single-digit numbers expands within two tens (0 + 1 = 1, ...,9+9=18). Thus, the numbers within 20 form in their internal connections a complete system of relations; this explains the expediency of preserving the "Twenty" in the form of a second integral theme (the first such theme is actions within the first ten).

The case under discussion is precisely one where concentricity (keeping the second ten as a special topic) proves to be more advantageous than linearity (the "dissolution" of the second ten into the "Hundred" topic).

In the textbook by M.I. Moro, the study of the first ten is divided into two isolated sections: first, the composition of the numbers of the first ten is studied, and the next topic deals with actions within 10. In the experimental textbook by P.M. Erdniev, in contrast to this, a joint study of numbering, the composition of numbers and operations (addition and subtraction) within 10 at once in one section was carried out. With this approach, a monographic study of numbers is used, namely: within the number under consideration (for example, 3), all “available mathematics” is immediately comprehended: 1 + 2 = 3; 2 + 1 = 3; 3 - 1 = 2; 3 - 2 = 1.

If, according to the current programs, 70 hours were allotted for studying the top ten, then in the case of experimental training, all this material was studied in 50 hours (moreover, some additional concepts that were not in the stable textbook, but structurally related to the main material, were considered beyond the program).

Particular attention in the methodology of elementary education requires the question of the classification of tasks, the names of their types. Generations of methodologists worked to streamline the system of school problems, to create their effective types and varieties, right up to the selection of successful terms for the names of problems intended for study at school. It is known that at least half of the study time in mathematics lessons is devoted to their solution. School tasks, of course, need to be systematized and classified. What kind (type) of tasks to study, when to study, what type to study in connection with the passage of a particular section - this is a legitimate object of study of the methodology and the central content of the programs. The significance of this circumstance is evident from the history of the methodology of mathematics.

In the author's experimental teaching aids, special attention is paid to the classification of tasks and the distribution of their necessary types and varieties for teaching in a particular class. At present, the classical names of the types of problems (to find the sum, the unknown term, etc.) have disappeared even from the table of contents of the stable first grade textbook. In the trial textbook P.M. Erdniev, these names "work": they are useful as didactic milestones not only for the student, but also for the teacher. Let us present the content of the first topic of a trial mathematics textbook, which is characterized by the logical completeness of concepts.

first ten

Comparison of the concept of above - below, to the left - to the right, between, shorter - longer, wider - narrower, thicker - thinner, older - younger, further - closer, slower - faster, lighter - heavier, little - a lot.

Monographic study of the numbers of the first ten: the name, designation, comparison, postponing numbers on the accounts and the designation of numbers on the numerical beam; signs: equal (=), not equal (¹), greater than (>), less than (<).

Straight and curved lines; circle and oval.

Point, line, segment, their designation with letters; measuring the length of a segment and laying off segments of a given length; designation, naming, construction, cutting out equal triangles, equal polygons. Polygon elements: vertices, sides, diagonals (their designation with letters).

Monographic study of numbers within the number in question:

composition of numbers, addition and subtraction.

The name of the components of addition and subtraction.

Four examples of addition and subtraction:

3 + 2 = 5, 5 - 2 = 3, 2 + 3 = 5, 5 - 3 = 2.

Deformed examples (with missing numbers and signs):

X + 5 = 7; 6 - X = 4; 6 = 3A2.

Solving problems on finding the sum and term, difference, reduced and subtracted. Compilation and solution of mutually inverse problems.

Three tasks: to increase and decrease the number by several units and to make a difference comparison. Comparison of segments by length.

Commutative law of addition. The change in the amount depending on the change in one term. Condition when the amount does not change. The simplest literal expressions: a + b = b + a, a + 0 = a, a - a = 0.

Drawing up and solving problems by expression.

In the following presentation, we will consider the main issues of the method of presenting this initial section of school mathematics, keeping in mind that the method of presenting the subsequent sections should be largely similar to the process of mastering the material of the first topic.

At the very first lessons, the teacher should set himself the goal of teaching the student to apply pairs of concepts, the content of which is revealed in the process of compiling appropriate sentences with these words. (First, we master the comparison at a qualitative level, without using numbers.)

Here are examples of the most common pairs of concepts that should be used in lessons not only in mathematics, but also in the development of speech:

More - less, longer - shorter, higher - lower, heavier - lighter, wider - narrower, thicker - thinner, right - left, further - closer, older - younger, faster - slower, etc.

When working on such pairs of concepts, it is important to use not only the illustrations in the textbook, but also the observations of children; so, for example, from the classroom window they see that there is a house behind the river, and they make up phrases: “The river is closer to the school than the house, and the house is further from the school than the river.”

Let the student hold a book and a notebook alternately in his hand. The teacher asks: what is heavier - a book or a notebook? What is easier? "A book is heavier than a notebook, and a notebook is lighter than a book."

Having lined up in front of the class next to the highest and lowest student of the class, we immediately compose two phrases: "Misha is higher than Kolya, and Kolya is lower than Misha."

In these exercises, it is important to achieve a grammatically correct replacement of one judgment with a dual one: "A stone house is higher than a wooden one, which means that a wooden house is lower than a stone one."

When familiarizing yourself with the concept of “longer - shorter”, you can show a comparison of objects in length by superimposing one on the other (which is longer: a pen or a pencil case?).

In the lessons of arithmetic and speech development, it is useful to solve logical problems aimed at teaching the use of opposite concepts: “Who is older: father or son? Who is younger: father or son? Which one was born first? Who is later?

“Compare the book and briefcase in width. What is wider: a book or a briefcase? What is already - a book or a portfolio? Which is heavier: a book or a briefcase?

Learning the comparison process can be made more interesting by introducing the so-called matrix (tabular) exercises. A table of four cells is built on the board and the meaning of the concepts "column" and "row" is explained. We introduce the concepts of "left column" and "right column", "top row" and "bottom row".

Together with students, we show (imitate) the semantic interpretation of these concepts.

Show the column (children move their hand from top to bottom).

Show the left column, the right column (children hold two hand swings from top to bottom).

Show the line (hand wave from left to right).

Show top line, bottom line (two hand waves showing top line, bottom line).

It is necessary to ensure that students accurately indicate the position of the cell: “upper left cell”, “lower right cell”, etc. The inverse problem is immediately solved, namely: the teacher points to some cell of the table (matrix), the student gives the appropriate name for that cell. So, if a cell is indicated that lies at the intersection of the top row and left column, then the student should name: "Upper left cell." Such exercises gradually accustom children to spatial orientation and are of great importance when subsequently studying the coordinate method of mathematics.

Of great importance for the first lessons of elementary mathematics is the work on the number series.

The growth of the number series by adding one by one is conveniently illustrated by moving to the right along the number line.

If the sign (+) is associated with moving along the number series to the right by one, then the sign (-) is associated with the reverse movement to the left by one, etc. (Therefore, we show both signs simultaneously in the same lesson.)

Working with a number series, we introduce concepts: the beginning of a number series (the number zero) represents the left end of the ray; number 1 corresponds to a single segment, which must be depicted separately from the number series.

Let the students work with the number series within three.

We select any two neighboring numbers, for example, 2 and 3. Moving from the number 2 to the number 3, the children reason like this: "The number 2 is followed by the number Z." Moving from the number 3 to the number 2, they say:

"Before the number 3 comes the number 2" or: "The number 2 precedes the number Z."

This method allows you to determine the place of a given number in relation to both the previous and the subsequent number; it is appropriate to immediately pay attention to the relativity of the position of the number, for example: the number 3 is simultaneously both subsequent (behind the number 2) and previous (before the number 4).

These transitions along the numerical series must be associated with the corresponding arithmetic operations.

For example, the phrase "The number 2 is followed by the number Z" is depicted symbolically as follows: 2 + 1 = 3; however, it is psychologically beneficial to create an opposite connection of thoughts immediately after it, namely: the expression “Before the number 3 comes the number 2” is supported by the entry: 3 - 1 = 2.

To achieve an understanding of the place of any number in the number series, paired questions should be asked:

1. What number is followed by the number 3? (The number 3 follows the number 2.) Which number is preceded by the number 2? (The number 2 comes before the number 3.)

2. What number follows the number 2? (The number 2 is followed by the number 3.) What number comes before the number 3? (The number 3 comes before the number 2.)

3. Between which numbers is the number 2? (The number 2 is between the number 1 and the number 3.) What number is between the numbers 1 and 3? (Between the numbers 1 and 3 is the number 2.)

In these exercises, mathematical information is contained in functional words: before, behind, between.

It is convenient to combine work with a number series with a comparison of numbers in magnitude, as well as with a comparison of the position of numbers on a number line. Connections of judgments of a geometric nature are gradually developed: the number 4 is on the number line to the right of the number 3; so 4 is greater than 3. And vice versa: the number 3 is on the number line to the left of the number 4; it means that the number 3 is less than the number 4. This establishes a connection between pairs of concepts: to the right - more, to the left - less.

From the foregoing, we see a characteristic feature of the enlarged assimilation of knowledge: the entire set of concepts related to addition and subtraction is offered together, in their continuous transitions (recodings) into each other.

The main means of mastering the numerical ratios in our textbook are colored bars; it is convenient to compare them in length, establishing how many cells are more or less than them in the upper or lower bar. In other words, we do not introduce the concept of "difference comparison of segments" as a special topic, but students get acquainted with it at the very beginning of studying the numbers of the first ten. In the lessons devoted to the study of the first ten, it is convenient to use colored bars, which allow you to perform propaedeutics of the main types of tasks for the actions of the first stage.

Consider an example.

Let two colored bars, divided into cells, be superimposed on each other:

in the lower - 3 cells, in the upper - 2 cells (see Fig.).

Comparing the number of cells in the upper and lower bars, the teacher makes two examples of reciprocal actions (2 + 1 = 3, 3 - 1 = 2), and the solutions of these examples are read in pairs in all possible ways:

2 + 1 = 3 3 – 1 = 2

a) add 1 to 2 - you get 3; a) subtract 1 from 3 - you get 2;

b) 2 increased by 1 - you get 3; b) reduce 3 by 1 - you get 2;

c) 3 is more than 2 by 1; c) 2 is less than 3 by 1;

d) 2 yes 1 will be 3; d) 3 without 1 will be 2;

e) add the number 2 to the number 1 - e) subtract the number 1 from the number 3 -

it will turn out 3. it will turn out 2.

Teacher. If 2 is increased by 1, how much will it be?

Student. If you multiply 2 by 1, you get 3.

Teacher. Now tell me what you need to do with the number 3 to get 2?

Student. Reduce 3 by 1 to get 2.

Let us pay attention here to the need for a methodically competent implementation of the opposition operation in this dialogue. ,

Confident mastering by children of the meaning of paired concepts (add - subtract, increase - decrease, more - less, yes - without, add - subtract) is achieved through their use in one lesson, based on the same triple of numbers (for example, 2 + 1 = =3, 3-1=2), based on one demonstration - comparing the lengths of two bars.

This is the fundamental difference between the methodological system of enlarging units of assimilation and the system of separate study of these basic concepts, in which the contrasting concepts of mathematics are introduced, as a rule, separately into the speech practice of students.

Learning experience shows the advantages of simultaneously introducing pairs of mutually opposite concepts from the very first lessons of arithmetic.

So, for example, the simultaneous use of three verbs: “add” (add 1 to 2), “add” (add the number 2 to the number 1), “increase” (2 increase by 1), which are symbolically depicted in the same way (2 + 1 = 3), helps children learn the similarity, closeness of these words in meaning (similar reasoning can be carried out with respect to the words "subtract", "subtract", "reduce").

In the same way, the essence of difference comparison is acquired in the course of repeated use of comparing pairs of numbers from the very beginning of training, and in each part of the dialogue in the lesson, all possible verbal forms of interpretation of the solved example are used: “Which is greater: 2 or 3? How much more is 3 than 2? How much must be added to 2 to get 3? etc. Of great importance for mastering the meaning of these concepts is the change in grammatical forms, the frequent use of interrogative forms.

Years of testing have shown the benefits of a monographic study of the numbers of the first ten. At the same time, each successive number is subjected to a multilateral analysis, with an enumeration of all possible options for its formation; within this number, all possible actions are performed, “all available mathematics” is repeated, all permissible grammatical forms of expression of the dependence between numbers are used. Of course, with this system of study, in connection with the coverage of subsequent numbers, the previously studied examples are repeated, i.e., the expansion of the number series is carried out with the constant repetition of the previously considered combinations of numbers and varieties of simple problems.

2.3 Joint study of addition and subtraction, multiplication and division

In the methodology of elementary mathematics, exercises for these two operations are usually considered separately. Meanwhile, it seems that the simultaneous study of the two-unit operation "addition - expansion into terms" is more preferable.

Let the students solve the addition problem: "Add 1 stick to three sticks - you get 4 sticks." Following this task, the question should immediately be asked: “What numbers does the number 4 consist of?” 4 sticks consist of 3 sticks (child counts 3 sticks) and 1 stick (separates 1 more stick).

The initial exercise can also be the decomposition of a number. The teacher asks: "What numbers does the number 5 consist of?" (The number 5 consists of 3 and 2.) And immediately a question is asked about the same numbers: “How much will it be if 2 is added to 3?” (Add 2 to 3 to get 5.)

For the same purpose, it is useful to practice reading examples in two directions: 5+2=7. Add 2 to 5, you get 7 (read from left to right). 7 consists of terms 2 and 5 (read from right to left).

It is useful to accompany verbal opposition with such exercises on class accounts that allow you to see the specific content of the corresponding operations. Calculations on the accounts are indispensable as a means of visualizing actions on numbers, and the value of numbers within 10 is associated here with the length of the set of bones located on one wire (this length is perceived by the student visually). One cannot agree with such an “innovation” when existing textbooks and programs completely abandoned the use of Russian accounts in the lessons.

So, when solving an addition example (5 + 2 = 7), the student first counted 5 bones on the accounts, then added 2 to them, and then announced the sum: “Add 2 to 5 - you get 7” (the name of the resulting number is 7, while the student establishes by recalculating a new set: "One - two - three - four - five - six - seven").

Student. Add 2 to 5 to get 7.

Teacher. Now show what terms the number 7 consists of.

Student (first separates two bones to the right, then speaks). The number 7 is made up of 2 and 5.

When performing these exercises, it is advisable to use from the very beginning the concepts of “first term” (5), “second term” (2), “sum”.

The tasks of the following types are offered: a) the sum of two terms is equal to 7; find terms; b) what terms does the number 7 consist of?; c) decompose the sum of 7 into 2 terms (into 3 terms). Etc.

The assimilation of such an important algebraic concept as the commutative law of addition requires a variety of exercises, initially based on practical manipulations with objects.

Teacher. Take 3 sticks in your left hand, and 2 in your right hand. How many sticks were there in total?

Student. There were 5 sticks in total.

Teacher. How can I say more about this?

Student. Add 2 sticks to 3 sticks - there will be 5 sticks.

Teacher. Make this example with cut numbers. (The student makes an example: 3+2=5.)

Teacher. Now swap the sticks: the sticks lying in the left hand, shift to the right, and the sticks from the right hand shift to the left. How many sticks are now in two hands together?

Student. In total, there were 5 sticks in two hands, and now it turned out to be 5 sticks again.

Teacher. Why did it happen?

Student. Because we didn’t put off anywhere and didn’t add sticks. As much as it was, so much remains.

Teacher. Compose solved examples from split numbers.

Student (postpones: 3+2=5, 2+3=5). Here was the number 3, and now the number 2. And here was the number 2, and now the number 3.

Teacher. We swapped the numbers 2 and 3, but the result is the same:

5. (An example is formed from split numbers: 3 + 2 = 2 + 3.)

The commutative law is also assimilated in exercises on decomposing a number into terms.

When to introduce the commutative law of addition?

The main goal of teaching addition - already within the first ten - is to constantly emphasize the role of the displacement law in exercises.

Let the children first count 6 sticks; then we add three sticks to them and by counting (“seven - eight - nine”) we set the sum: 6 yes 3 - it will be 9. It is necessary to immediately offer a new example: 3 + 6; a new sum can first be established again by recalculation (i.e., in the most primitive way), but gradually and purposefully, a solution method should be formed on a higher code, i.e., logically, without recalculation.

If 6 yes 3 will be 9 (the answer is set by recalculation), then 3 yes 6 (without recalculation!) will also be 9!

In short, the commutative property of addition must be introduced from the very beginning of the exercises for adding different terms, so that it becomes a habit to compose (pronounce) the solution of four examples:

6 + 3 = 9, 9 - 3 = 6, 3 + 6 = 9, 9 – 6 = 3.

Drawing up four examples is a means of enlarging knowledge accessible to children.

We see that such an important characteristic of the addition operation as its portability should not pass sporadically, but should become the main logical means of strengthening correct numerical associations. The main property of addition - the transferability of terms - must be constantly considered in connection with the accumulation in memory of all new tabular results.

We see: the interconnection of more complex computational or logical operations is based on a similar pairwise relationship (proximity) of elementary operations, through which a pair of “complex” operations is performed. In other words, the explicit opposition of complex concepts is based on the implicit (subconscious) opposition of simpler concepts.

It is advisable to carry out the initial study of multiplication and division in the following sequence of three cycles of tasks (three tasks in each cycle):

I cycle: a, b) multiplication with a constant multiplicand and division by content (jointly); c) division into equal parts.

II cycle: a, b) decrease and increase in the number several times (together); c) multiple comparison.

III cycle: a, b) finding one part of a number and a number by the value of one of its parts (together); c) solution of the problem: "What part is one number from another?"

The methodological system for studying these problems is similar to that described above for simple problems of the first stage (for addition and subtraction).

Simultaneous study of multiplication and division by content. In two or three lessons (no more!), devoted to multiplication, the meaning of the concept of multiplication as a folded addition of equal terms is clarified (the action of division has not yet been discussed in these lessons). This time is enough to study the multiplication table of the number 2 by single digits.

Usually, students are shown a record for replacing addition with multiplication: 2+2+2+2=8; 2*4=8. Here the connection between addition and multiplication goes in the direction of "addition-multiplication". It is appropriate to immediately offer students an exercise designed for the appearance of feedback of the “multiplication-addition” type (equal terms): considering this entry, the student should understand that it is required to repeat the number 2 with the terms as many times as the multiplier in the example shows (2 * 4 \u003d eight).

The combination of both types of exercise is one of the important conditions that ensure the conscious assimilation of the concept of "multiplication", meaning folded addition.

In the third lesson (or fourth, depending on the class), each of the known cases of multiplication is given the corresponding case of division. In the future, it is advantageous to consider multiplication and division by content only together in the same lessons.

When introducing the concept of division, it is necessary to recall the corresponding cases of multiplication in order, starting from them, to create the concept of a new action, the inverse of multiplication.

Consequently, the concept of “multiplication” acquires a rich content: it is not only the result of the addition of equal terms (“generalization of addition”), but also the basis, the initial moment of division, which, in turn, represents a “folded subtraction”, replacing the sequential “subtraction by 2":

The meaning of multiplication is comprehended not so much in the multiplication itself, but in the constant transitions between multiplication and division, since division is a veiled, "altered" multiplication. This explains why it is advantageous subsequently to always study multiplication and division at the same time (both tabular and extra-table; both oral and written).

The first lessons on the simultaneous study of multiplication and division should be devoted to the pedantic processing of the logical operations themselves, supported in every possible way by extensive practical activities in collecting and distributing various objects (cubes, mushrooms, sticks, etc.), but the sequence of detailed actions should remain the same same.

The result of such work will be the multiplication and division tables, which are written side by side:

2*2=4, 4:2=2,

2*3=6, 6:2=3,

by 2*4=8, 8: by 2=4,

2*5= 10, 10: 2=5, etc.

Thus, the multiplication table is built on a constant multiplicand, and the division table - on a constant divisor.

It is also useful to offer students paired with this task a structurally opposite exercise on the transition from division to subtraction of equal subtrahends.

In repetition exercises, it is useful to offer tasks of this type: 14:2 ==.

The study of division into equal parts. After the multiplication of the number 2 and division by 2 have been studied or repeated together, one of the lessons introduces the concept of “dividing into equal parts” (the third type of problem of the first cycle).

Consider the problem: “Four students brought 2 notebooks each. How many notebooks did they bring in total?

The teacher explains: take 2 4 times - you get 8. (A record appears: 2 * 4 = 8 each.) Who will make the inverse problem?

And a generalization of the experience of teachers in conducting mathematics lessons on this topic. The course work consists of an introduction, two chapters, a conclusion, a list of references. CHAPTER I

Still doesn't highlight the issue. Since the question of the method of teaching the transformation of tasks is covered to the least extent, we will continue to study it. Chapter II. Technique for learning to transform tasks. 2.1. Transformation tasks in mathematics lessons in elementary school. Since there is very little specialized literature on the transformation of tasks, we decided to conduct a survey among teachers...

When studying new material, it is recommended that the lesson be structured in such a way that the work begins with a variety of demonstrations conducted by the teacher or student. The use of visualization in mathematics lessons in the study of geometric material allows children to firmly and consciously learn all program issues. The language of mathematics is the language of symbols, conventional signs, drawings, geometric ...