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Development of a mathematics lesson in the 1st grade on the topic
"Adding a sum to a sum"
EMC "Perspective Primary School"
Sidorenko Irina Viktorovna -
primary school teacher MBOU secondary school №25
Lesson type: a lesson in discovering new knowledge
The goals of the teacher's activity: create conditions for familiarization with the methods of adding the amount to the amount; learn to apply the rule of adding the sum to the sum; continue the formation of skills to solve problems; develop speech skills, logical thinking.
Planned results(meta-subject universal learning activities) :
Regulatory: be aware of the need to control the result (retrospective), control the result at the request of the teacher; to distinguish between the correct task and the incorrect one.
Cognitive: use (build) tables, check against the table; compare, seriate, classify, choosing the most effective solution or the right solution (correct answer); build an oral explanation according to the proposed plan; to search for the necessary information to complete educational tasks, using the reference materials of the textbook; apply logical methods of thinking at an accessible level (analysis, comparison, classification, generalization).
Communicative: engage in dialogue (answer questions, ask questions, clarify incomprehensible); negotiate and come to a common decision, working in pairs; participate in a collective discussion of an educational problem; build productive interaction and cooperation with peers and adults for the implementation of project activities (under the guidance of a teacher).
Personal: to establish connections between the purpose of educational activity and its motive, in other words, between the result of learning and what induces to activity, for the sake of which it is carried out; The student should ask himself the question, “what meaning and what meaning does the teaching have for me?” and be able to answer it.
Equipment:
Chekin A.L. Mathematics. Grade 1: Textbook. At 2 o'clock - M.: Akademkniga / Textbook, 2014
Zakharova O.A., Yudina E.P. Mathematics in questions and tasks: Notebook for
independent work grade 1 (in 2 parts) - M .: Akademkniga / Textbook, 2014.
Cards with assignments for pair work (Appendix 2)
Task cards for groups (Appendix 3)
Presentation (Annex 1)
TSO (wall screen, laptop. multimedia projector, speakers)
Lesson script.
Motivation for learning activities.
Check readiness for the lesson. The presence of a general setting for the lesson. Greeting students.
Let's check the readiness for the lesson. (Slide 2. Presentation -Appendix 1 )
Emotional mood.Slides 3-4.
Smile at me, smile at each other.
Actualization and trial educational action.
Verbal counting.slide 5
Work in pairs. slide 6 .
1) The game "Cryptor"Envelopes with tasks on the tables(appendix 2).
- You will work in pairs. Envelope task. You must solve the expression together and write the answer next to it. When all the expressions are solved, it is necessary to enter the answers in the table in ascending order and write the letter under the answer. You will have a word.
Before you begin to complete the task, remember the rules for working in pairs.
What rules do you know. Let's read those rules that you did not name. Slide 7.
Get to work.
10 + 7 = ____ t
Which of the following expressions is redundant? Why? (9-4, since this is the difference, and all other sums)
In what order did you list your answers? (ascending)
What does ascending order mean? (From smallest number to largest)
Let's check your answers. slide 8.
What word came out? Slide 9
Zero comes after one
Number 10 on the page.
What can you say about this number?
( A person has TEN fingers on both hands. This is what led to the creation of the decimal number system. TEN is the smallest multi-digit number.)
The number 10 is the sum of the first four natural numbers. slide 10.
There are ten commandments in the bible.
In international (hundred-cell) checkers, the size of the board is 10×10 cells.
Chervonets is a monetary unit in the Russian Empire and the USSR. Chervonets, starting from the beginning of the 20th century, are traditionally called banknotes with a denomination of TEN units.
Diving is one of the water sports. The highest height from which these jumps are made is 10 meters.
2) The composition of the number 10.
- Let's remember the composition of the number 10? (table) slide 11
Where can you use this knowledge? Why do we need to know the composition of a number?
(Student answers)
- Let's see how you can solve problems.
I read texts of tasks. Children work in pairs and name the answer.
Here are eight rabbits walking along the path.
Two people run after them.
So how much is there in total along the forest path
Rushing to bunny school in winter? (ten)
slide 12.
The chicken went for a walk, gathered her chickens.
Seven ran ahead, three were left behind.
Count - guys, how many chickens were there. (ten)
About whom did I read the task to you? Name the answer. Let's check it out on the slide. slide 12 (click)
We had fun on the Christmas tree and danced and frolicked.
After the good Santa Claus brought us gifts.
He gave huge packages, they also have tasty items.
2 candies in blue papers, 5 nuts next to them,
Pear with apple, 1 golden tangerine.
Everything is in this bag, count all the items. Answer: 2+5+1+1+1=10.
About whom did I read the task to you? Name the answer. Let's check it out on the slide. slide 12 (click)
Group work.slide 13.
- I gave you worksheets with a task to complete, working in groups.
(appendix 3).
Consider expressions. Find their meaning. Write your answer on a piece of paper and stick it on the board.
(6 + 2) + (4 + 3) =
III. Identification of the location and cause of the difficulty. The topic of the lesson.
Checking (sheets on the board)
Consider the results of your work.
Why did not all groups find the meaning of expressions? (Answers of children).
Which expressions are easy to solve? Why were you able to solve them? (Such expressions were solved).
What knowledge helped you to cope with the task? (Adding a number to a sum, adding a sum to a number).
What was the difficulty? (We do not know how to add two sums). Slide 14.
What is the topic of the lesson? (Adding the sum to the sum). Slide 15.
What is the purpose of the lesson? What should be learned in class? Slide 16 ( I am correcting the answers of the children).
IV. Building a project to get out of trouble. Slide 17.
(There are fruit plates on the board).
Yellow apples - 6 Yellow pears - 3
Green apples -4 Green pears - 2
What do you see on the board? (plates with apples, pears) How to name the depicted objects in one word? (Fruits).
On what basis were the fruits laid out on plates? (by color and shape).
Make up different questions for this picture. Lead to an answer. (How many fruits are on 4 plates).
Misha answered this question in the following way. Appears slide 18.
Read the expression correctly.
On what basis did Misha add up the numbers? (by color). How did he find the amount of all the fruits? Explanation. Misha found the number of green fruits (6+3) and then found the number of yellow fruits (4+2). Then he added up the results.
Masha thought so. Slide 18 (click)
Read the math expression.
On what basis did Masha count? (by type of fruit) . How did Masha find the amount of all the fruits? Explanation. Masha found the number of apples (6+4), then found the number of pears (3+2). Then she added up the results.
Why are the amounts equal? Whose way do you like more? Why?
How is it more convenient to add the amount to the amount? (first add to 10, then the remaining numbers)
Remember, on what basis did Misha and Masha stack fruits? Do you think the sign is important in answering the question? Should I look for signs? Good.
Let's get back to the expression. An expression appears. slide 19.
(6+2)+(4+3)
How are we going to solve this expression? How can we solve this expression? Is the sign important in the decision? (Not important).
Why are these amounts equal? Explain.
Whose way do you like more? Why do you think so?
Let's make a conclusion? (To add the sums, we must add the number to 10., First add the first terms, and then the second)
Now could you solve the expression? How?
Fizkultminutka.slide 20.
V. Implementation of the constructed project.
Textbook work (pp. 56–57).Slide 21.
Open the textbook page 56, no. 2slide 22.
Read the entry on the left. Choose the entry on the right that shows a convenient way to solve this expression.
Why choose this method? How do we add two sums?
Task number 1.
- Consider the illustration for the problem.
- Name the condition of this task. (There were 3 green apples and 7 yellow apples, 4 green pears and 6 yellow pears on four plates.)
- Formulate the requirement of this task. (How many fruits are on four plates?)
– Explain how Misha solved the problem.
(7 + 6) + (3 + 4).
Explanation. Misha found the number of yellow fruits (7 + 6), then found the number of green fruits (3 + 4). Then he added up the results.
- Explain how Masha solved the problem.
(7 + 3) + (6 + 4).
Explanation. Masha found the number of apples (7 + 3), then found the number of pears (6 + 4). Then she added up the results.
Why do you think these amounts are equal?
-Which way of adding do you like more? Why? (Machine way is more convenient.)
Task number 2.
– Analyze these amounts.
– What unites them? (In these sums, each term is represented as the sum of two numbers.)
– Without doing the calculations for the sum on the left, find the sum on the right with the same value and underline it.
Will you pay attention to the order of the terms? (Not.)
Write: (8 + 5) + (2 + 5) = (8 + 2) + (5 + 5).
- Underline the part of the equation that makes it easier to calculate the value of the sum.
– Find the value of this sum using the rule of adding the sum to the sum.
VI.Primary consolidation with pronunciation in inner speech.
Task number 3. Work in TVET with. 76, No. 1slide 23.
open notebook page 76, no. 1(commenting)
Read the expression. How are we going to do it? Why?
Let's execute 2 expressions using a new technique. Find the value of the sums using Masha's experience.
Parents of modern children envy watching geeks - participants in the television shows "Best of All" and "Amazing People" - and worry that their children do not have an outstanding mind and super-smartness: they do not learn the primary school curriculum well, do not like to strain the brain and are afraid of lessons mathematics.
From the first grade, they count on fingers and sticks, they do not know the methods of oral counting, therefore they experience big problems in all subjects of the school course.
The methods of quick mental counting are simple and easy to learn, but it must be remembered that their successful mastery presupposes not mechanical, but quite conscious use of the methods and, in addition, more or less lengthy training.
Having mastered the elementary methods of mental counting, those who use them will be able to correctly and quickly perform instantaneous calculations in their minds with the same accuracy as in written calculations.
Peculiarities
There are a lot of techniques that contribute to learning fast counting in the mind. With all the visible differences, they have an important similarity - they are based on three "pillars":
- Training and experience. Regular practice, solving tasks from simple to complex qualitatively and quantitatively change the skill of oral calculations.
- Algorithm. Knowledge and application of "secret" techniques and laws greatly simplifies the process of counting.
- Abilities and natural gifts. A developed short-term memory and its considerable volume, as well as a high concentration of attention, are of great help in doing quick mental counting. A definite plus is the presence of a mathematical mindset and a predisposition to logical thinking.
Benefits of mental counting
People are not iron robots, but the fact that they create smart machines speaks of their intellectual superiority. A person needs to constantly keep his brain in good shape, which is actively promoted by training the counting skill in the mind.
For everyday life:
- successful mental counting is an indicator of an analytical mindset;
- regular mental counting will save you from early dementia and senile insanity;
- your ability to add and subtract well will not allow you to deceive in the store.
For successful study:
- mental activity is activated;
- develop memory, speech, attention, the ability to perceive what is said by ear, speed of reaction, ingenuity, the ability to find the most rational ways to solve the problem;
- confidence in their abilities is strengthened.
When should training start?
According to scientific minds (psychologists and teachers), by the age of 4, a child is already able to add and subtract. And by the age of 5, the baby can freely solve examples and simple tasks. But these are statistics, and children do not always adapt to it. So everything here is purely individual.
rules
The queen of sciences - mathematics - took care of schoolchildren and compiled a code of laws, algorithms and rules, having learned which and skillfully using them, children will love mathematics and mental work:
- The commutative property of addition: by swapping the components of an action, we get the same result.
- Associative property of addition: when adding three or more numbers, any two (or more) numerical values can be replaced by their sum.
- Addition and subtraction with the transition through a dozen: complement the larger component
- Up to round tens, and then add the remainder of the other component.
- We first subtract individual units from the number up to the sign of the action, and then subtract the remainder of the subtrahend from round tens.
- Representing the minuend as the sum of tens and ones, we remove the smaller from the tens of the larger and add the units of the minuend to the answer.
- When adding and subtracting round tens (they are also called “round” numbers), tens can be counted in the same way as units.
- Addition and subtraction of tens and ones. It is more convenient to add tens to tens, and units to ones.
Adding a number to a sum
The methods are as follows:
- We calculate its value, and then add this value to it.
- We add it to the first term, and then we add the second term to the result.
- We add the number to the second term, and then we add the first term to the answer.
Adding a sum to a number
The methods are as follows:
- Calculate its reading, and then add to the number.
- Add the first term to the number, and then add the second term to the result.
- Add the second term to the number, and then add the first term to the result.
Addition of two sums. Adding two sums, we choose the most convenient method of calculation.
Using the main properties of multiplication
The methods are:
- Commutative property of multiplication. If you swap the factors in places, their product does not change.
- Associative property of multiplication. When multiplying three or more numbers, any two (or more) numbers can be replaced by their product.
- Distributive property of multiplication. To multiply a sum by a number, you must multiply each of its components by this number and add the resulting products.
Multiplication and division of numbers by 10 and 100
- To multiply any number by 10, you must add one zero to the right of it.
- To do the same 100 times, you need to add two zeros to it on the right.
- To reduce the number by 10, you need to discard one zero on the right, and to divide by 100 - two zeros.
Multiplying a sum by a number
- 1st way. Calculate the amount and multiply it by this value.
- 2nd way. We multiply the number with each of the terms, and add the answers obtained.
Multiplying a number by a sum
- 1st way. Find the sum and multiply the number by what we get.
- 2nd way. We multiply the number by each of the terms, and add the resulting products.
Dividing a sum by a number
- 1st way. Calculate the sum and divide it by the number.
- 2nd way. We divide each of the terms by a number and add the resulting partials.
Dividing a number by a product
Options:
- 1st way. Divide the number by the first factor, and then divide the result by the second factor.
- 2nd way. Divide the number by the second factor, and then divide the result by the first factor.
Kinds
In the lessons, a meager time is allotted for oral counting, but this does not detract from its importance for the development of the mental activity of the children. Oral computing skills are formed in mathematics lessons in elementary school when performing various types of tasks and exercises.
Find the value of a mathematical expression
Compare math expressions
These tasks are different:
- determine the equality or inequality of two given expressions (having previously found and compared their values);
- to the relation given by the sign and one of the expressions, compose a second expression or supplement an unfinished sentence;
- in such exercises, single-digit, two-digit, three-digit numbers and quantities and all four arithmetic operations can be used in expressions. The main purpose of such tasks is a solid assimilation of theoretical material and the development of computational skills.
- Solve equations. They help to learn the connections between the components and results of arithmetic operations.
- Solve a problem. These can be both simple and complex tasks. With their help, theoretical knowledge is strengthened, computational skills and abilities are developed, and the mental activity of children is activated.
Oral counting techniques
Signs of divisibility of numbers:
- by 2: everything that exceeds it, and in the number series go through one;
- by 3 and 9: if the sum of the digits is a multiple of these indicators without a remainder;
- by 4: if the last two digits in the entry sequentially form a number that is divided by 4;
- on 5: round tens and those where 5 is at the end;
- by 6: numbers that are multiples of two and three are divided;
- by 10: numeric values that end with 0;
- by 12: numbers are divided that can be divided into three and four at the same time;
- by 15: numbers that are divided simultaneously by integer single-digit components are the number of factors.
Forms of counting in elementary school
It is well known that the main activity of preschoolers and younger students is the game, which is useful to include in all stages of the lesson. Some forms of oral counting are given below.
Silent game
Promotes attention and discipline. Silence can consist of examples in one action, two or more. It is played in all elementary school classes with both abstract integers and named numbers.
Students count in their minds and silently, when called by the teacher, write on the blackboard the answers to the examples given to them. Correct answers are met with light clapping, and wrong answers are met with silence.
Game "Loto"
There may be several types corresponding to those sections of mathematics that are studied and need to be consolidated. For example, a lotto with examples of multiplication and division within "hundreds".
To add more interest to the game, tires with answers can be made from a cut picture. If all the examples are solved correctly, a picture is obtained from the tires.
Game "Arithmetic mazes"
They look like concentric circles with gates that have numbers. To get to the center, you need to dial the number in the center. Labyrinths for solution may require either one action (addition), or several. It should be noted that these problems have several solutions.
The game "Catch up with the pilot" (a kind of "Ladder")
Drawing on the board: an airplane with loops, in which examples. Two called students write down the answers to the left and right of the loops. Whoever decides correctly and quickly will catch up with the pilot.
Game "Circular Examples"
The didactic material is a set of cards arranged in envelopes; each of them has 8 cards, each of which contains one example.
Numerical examples in each envelope are different in their content and are selected according to the principle of self-control: when solving them, the result of one example will be the beginning of the next one.
Circular examples can be offered in the form of ladders.
Development Methods and Techniques
Considering ways to teach children 6 years old to count quickly in the mind, it is impossible not to note the uniqueness and simplicity of the Japanese method of counting "Soroban". The Soroban method allows you to teach children aged 4 to 11 years, developing their mental abilities and expanding the range of intellectual abilities of kids. It is easy to teach any schoolchild to count examples in mathematics in his mind, using the Japanese method of counting on the soroban. By practicing mental mental counting, we include the whole brain in the work., thereby unloading the left hemisphere, which is responsible for solving mathematical problems.
Mental arithmetic allows even the "figurative" hemisphere to be interested in computational operations, which increases the efficiency of the brain.
Large numbers require written methods of calculation, although there are individuals who hone their skills in working with them too.
Counting math examples in your mind is a vital necessity, since school exams are now taking place without the use of calculators, and the ability to count in the mind is included in the list of required skills for graduates of grades 9 and 11.
Rule of thumb for mental addition: 
Subtraction features: reduction to round numbers
Single-digit subtrahends are rounded up to 10, two-digit ones - up to 100. Subtract 10 or 100 and add the correction. Acceptance is relevant for small amendments.
Mind subtracting three-digit numbers
Based on a good knowledge of the composition of the numbers of the 1st ten, you can subtract in parts in this order: hundreds, tens, ones.
You can multiply and divide without problems, knowing the multiplication table - a "magic wand" to the rapid development of counting in the mind. It is noteworthy that the village children of pre-revolutionary Russia knew the continuation of the so-called Pythagorean table - from 11 to 19, and it would be nice for modern schoolchildren to know the table up to 19 * 9 by memory.
To captivate children with mathematics and make difficult moments in the school curriculum closer and more accessible, there are ways and methodological techniques turning difficulties into fun and interesting:
- To multiply any single-digit number by 9, we will show everyone our empty palms. We bend the finger corresponding in order (counting from the thumb of the left hand) to the number of the first factor. We look at how many fingers to the left of the bent one - these will be tens of the desired product, and to the right - its units.
- Multiplication by 11 of any two-digit number, the sum of the digits of which does not reach 10, is carried out amusingly and simply: let's mentally expand the digits of this number and put their sum between them - the answer is ready.
- If the sum of the digits of the number multiplied by 11 turns out to be equal to 10 or more than 10, then between the mentally spaced digits of this number, you should put their sum and add the first two digits on the left, leaving the other two unchanged - got the product.
Question 5. Oral methods of addition and subtraction within 100. The associative property of addition.
Oral computing techniques for adding and subtracting two-digit numbers.
At the preparatory stage, the methods of addition and subtraction within 10, the table of addition and subtraction within 10, computational methods of the form 40 + 5, 45-5, 45-40, based on the knowledge of numbering, are repeated.
Oral addition techniques are also based on knowledge of the associative (associative) law of addition (see table).
For addition, the associative law (a + b) + c \u003d a + (b + c) is valid, which is a consequence of the associativity of the union of specific sets whose pairwise intersection is an empty set.
In elementary school, the law is revealed with the help of the rules for adding a number to a sum and a sum to a number.
They can try to deduce the associative property on their own. The teacher must convince students that to calculate the expressions (a + b) + c and a + (b + c), actions can be performed in any order, that is, the values of the expressions do not depend on the order in which the actions are performed. The assimilation of these rules does not cause difficulties if their mathematical content is revealed based on the intuitive ideas of children.
To study the rule of adding a number to the sum (a + b) + c, a series of problems is proposed that have a different plot, but the same mathematical content.
“The boy found 2 white mushrooms, 3 boletus, 4 boletus. How many mushrooms did the boy find in total?
Work on these tasks is carried out according to the following plan:
the condition of the problem is concretized, on the typesetting canvas there is an illustration with the help of geometric figures, which is gradually supplemented and the recording (2 + 3) + 4 is performed.
then another version of the same problem is compiled, the canvas is filled in, and a mathematical notation (3 + 4) + 2 is compiled.
similar to (4+2)+3.
the conclusion is made: the problem can be solved in three different ways, the result does not change.
The result may not be calculated.
Thus, the meaning of the law is revealed:
on the image;
on numbers;
in literal form.
Then it is proposed to compose a problem according to a numerical expression of the form:
And rephrase its condition so that it is solved using expressions:
(a+c)+b and (b+c)+a
The rule for adding a number to the sum is formed:
You can add a number to a sum by adding the numbers in any order. Memorizing a more detailed formulation (“to add a number to the sum, you can first ...”) is inappropriate, as it contributes to the formal assimilation of the essence of the rule. It is more important to teach how to address problems if the rule is forgotten.
The rule for adding a sum to a number is introduced similarly.
Also, for proof, students can explore these expressions on graphical models. Consider 2 expressions. Changing the order of operations can change the result, so you need to match the expressions and find out if they are equal.
The teacher reports that the resulting property is called associative and offers to express its meaning in words. The associative property can be formulated in different ways:
To add a third number to the sum of two numbers, you can add the sum of the second and third to the first number.
to add the sum of two numbers to a number, you can first add the first term to it, then the second.
the value of the sum does not depend on the choice of actions.
II. Acquaintance stage.
View Reception: 20+30
The abacus is first filled with two strips of one dozen circles each, then three more strips. In total, there are 2 + 3 strips in the abacus, or 5 tens.
Thus, the method of adding round tens is reduced to adding single-digit numbers, that is, 2 tens + 3 tens = 5 tens.
Reception of subtraction of a kind: 60-40 is entered similarly.
The theoretical basis is the specific meaning of the operations of addition and subtraction.
Then addition techniques are introduced, based on the knowledge of the properties of adding a number to a sum and adding a sum to a number:
22+5 (20+2)+5 theoretical basis - adding a number to the sum.
45+30 (40+5)+30=40+(5+30)
20+13 theoretical basis - adding the sum to the number
20+35=20+(30+5)=(20+30)+5
22+35=22+(30+5)=(22+30)+5=52+5=57
25+36=25+(30+6)=(25+30)+6=55+6=61
Cases of the form 28+5 have two ways of finding the result.
28+5=(20+8)+5=20+(8+5)=33 theoretical basis - adding a number to the sum.
Reasoning algorithm: replace, get an example, it's more convenient here.
28+5=28+(2+3)=(28+2)+3=33 theoretical basis-
2 3 adding the sum to the isl.
Studying the methods of oral addition of two-digit numbers, students should come to the conclusion that it is easier to add two two-digit numbers if you add tens of the second to the tens of the first, add the units of both terms and add to the sum of the tens.
Subtraction techniques use properties.
Subtraction of a number from the sum: 45-3, 40-5, 45-30
Subtracting the sum from a number: 45-9, 45-23, 45-28.
They are studied according to the same plan as the properties of addition. The various subtraction methods are based on relevant questions from a theory course in mathematics.
45-3=(40+5)-3=40+(5-3)=40+2=42 (the number 3 is subtracted from the number of units being reduced);
theoretical basis - subtracting a number from a sum
45-9=45-(5+4)=(45-5)-4=40-4=36
theoretical basis - subtracting a sum from a number
45-23=45-(20+3)=(45-20)-3=25-3=22
the theoretical basis is the subtraction of the sum from the number.
All these operations, if necessary, can be performed on a demonstration abacus, students on an individual abacus. Mathematical expression is written on the blackboard and in notebooks.
When studying the techniques of oral addition and subtraction of numbers, different approaches can be traced.
I An approach.
According to the traditional program, the main way to introduce a computational technique is to show a sample of an action, which in some cases is explained at the subject level, and then consolidated in the process of performing training exercises.
The process of forming computational skills is focused on mastering the mode of action for particular cases of addition and subtraction of numbers.
The study of any property is carried out according to one plan:
disclosure of the essence of the property (using visual aids);
applying the property when performing tasks;
selection of rational methods of computation (based on properties).
Thus, the first approach is related to the study of the properties of arithmetic operations.
II The approach is associated with the study of the associative law of addition with access to generalization: when adding numbers, it is convenient to add units to units, tens to tens. This conclusion carries over to subtraction techniques.
III An approach.
The process of forming computational skills is focused on mastering the general method of action, which is based on children's awareness of writing numbers in the decimal number system (bit composition of a number) and the meaning of addition and subtraction.
The main way to introduce a new computational technique is not to show a pattern of actions, but to perform actions with models of tens and ones and correlate these actions with mathematical notation.
In the process of such activity, students observe a change in the numbers indicating the number of tens (ones) in the record, with an increase (decrease) in the number by several tens (units).
Observation of a change in the notation of numbers is accompanied by an active interpretation of the methods of analysis and comparison, classification, and generalization.
The problem is how to organize the productive activity of students in mastering the technique.
N.Ya. Vilenkin, L.G. Peterson developed a training technology that is practically expedient and reflects the main theoretical results of psychological and pedagogical research. In their curriculum and textbooks on mathematics for elementary school, they offer the following approach to the introduction of computational techniques.
Techniques are introduced in a problematic way, when the teacher does not explain all the material himself, but leads the children to the “discovery” of new knowledge. It is of fundamental importance that children themselves derive new rules for actions with numbers by analyzing and generalizing their own objective actions with models of these numbers.
Green triangles with ten red circles are used as models: a red circle represents units, a green triangle represents tens, and ten red circles on a green triangle represent hundreds.
The structure of the introduction lesson:
Statement of the educational task.
Students perform independent work, in which, among the known cases of addition and subtraction, they encounter a case unknown to them. A problem situation arises that motivates the study of new material.
Construction of subject models.
To resolve the problem situation, the example that caused the difficulty is modeled and discussed frontally. As a result of this discussion, students "invent" a new way of action (using triangles, bunches of sticks).
Construction of graphic models.
Students use the new mode of action to build graphical models of a new type. In this case, the resulting conclusion is again spoken out.
Iconic modeling.
An example is written in a more compact form, using numbers and signs of arithmetic operations (notation as a numeric expression). Now students apply a new computational technique without relying on a visual model. If a written reception, then the teacher introduces the children to a more convenient form of writing examples of a new type in a column.
Self-control and self-esteem.
Students independently solve an example for a new computational technique and make sure that they have mastered the new method of action. The problem situation is resolved. Then a new computational technique is used to solve word problems. The solution is carried out with commenting, without graphic models, without an abacus.
