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ALGORITHMIC BASIS OF LEARNING THE ACT OF SUBTRACTION
IBRAHiMOV FiRADUN NADiR OGLU,
Sheki branch of ADPU, doctor of pedagogical sciences, professor ORCID: https://orcid.org/0000-0002-0775-1048
GARAYEVA GULNARA BAHRAM GiZi,
Sheki branch of ADPU, doctoral student, head teacher ORCHID: https://orcid.org/my-orcid?orcid=0000-0002-8347-5145
ABDURRAHiMOVA ULDUZ ALi GiZi,
Sheki branch of ADPU, head teacher ORCHID:https://orcid.org/0000-0003-4298-7088
Key words:decimal composition of numbers; decreasing, subtracting and difference; exit with or without crossing the floor; written subtraction algorithm; verbal description of the algorithm; verbal subtraction algorithm; algorithmic culture; types of mental activity.
Summary. In the article the relevance of studying the methodical aspects of teaching subtraction based on algorithmic bases is substantiated, information about the purpose, meaning, and techniques of subtraction are brought to the attention of students, the description of the algorithm of subtraction in words (opening form) is presented, and the teaching units of the skills of practical application of that algorithm in students are presented. The scientific interpretation of the technology of adequately forming the range of natural numbers is reflected.
The actuality of the subject. Improving the pedagogical process and its subsystems is one of the perennial problems of didactics. The development of algorithmic culture is the central link of the mentioned processes, because in these processes the mental activity of students is controlled. In both analytical and heuristic types of mental activity, the place of algorithms and algorithm building experience is irreplaceable [4;166]. Focusing on the algorithmic basis of teaching the act of exiting is an element related to the formation of the algorithmic culture of students, in other words, it is included in the eternal problem of didactics (the problem of controlling the teaching of cognition). Therefore, we claim the relevance of the research topic.
The methodological basis of the research work: L.Bertalanfs idea of system analysis, "system-structure" approach formed as a branch of dialectics.
Interpretation of research materials. Subtraction and addition of multi-digit numbers are difficult for students to master. Because its algorithm contains a system of steps that is relatively difficult to execute.
It is easy to see when we consider the algorithm of withdrawal. The verbal description of that algorithm is as follows:
Xi, X2, X3,,. . ., Xft - decreasing, - subtracting, y1, у2,у3,,...,ут n>m — dir.
1. Each floor unit of the decrease is written under the condition that the corresponding floor unit of the subtraction is located; a minus sign is placed on the left; underlined; it goes to the second paragraph.
2. If it is xn > ym, the difference xn — ym is found and it is written down in the column of singularities floor, it is moved to the fourth item; otherwise, i.e. when xn < ym, it is passed to the 3rd paragraph.
3. The total is found and subtracted using the method of "shredding the first valuable floor unit (decrease) and giving shares to the floor units below it", the sum of xn + 10ym is found and ym is
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subtracted from it, the difference is written down in the column of singularities; The 5th item is passed.
4. If it is xn-1 > ym-1 the difference xn-1 — ym-1 is found, it is written under the decimal floor; Go to point 8, otherwise go to point 6.
5. If it is.xn-1 — 1 > ym-1 , the difference (xn-1 — 1) — ym-1 is found, the decimals are written down in the floor column, it is passed to the 8th point, otherwise it is passed to the 6th point.
6. The xn_i + 10 sum is found by using the method of "shredding the first valuable upper floor unit of the decrease and dividing it by the floor units below it", the difference xn-1 + 10 — ym-1 is written down in the decimal floor column; It goes to paragraph 8.
7. The total (xn-1 — 1) + 10 is found by using the method of "shredding the first valuable upper floor unit of the decrease and dividing it into lower floor units", ym-1 subtracting from the received number and writing down the decimals in the column of the floor; It goes to paragraph 8.
8. The 4 — 6 operations are carried out in an analogous x1 manner to the points, and the process continues until the appropriate floor unit is determined [5;57-62].
The process of mastering the subtraction algorithm for primary school students is carried out with a special methodical system, on the basis of concentrations.
It should be noted that, in addition to the written subtraction algorithm, we can also mention the visual (material)-operational and verbal subtraction algorithm, the latter of which has a special, important role in mastering the first one. It is required that they know well the written subtraction table and the numbering of multi-digit numbers (they know the ratio of oddities of adjacent floors and be able to quickly reduce each floor unit to another). Also, they should be able to determine and execute the sequence of steps of the subtraction algorithm [6;193-204].
The work system aimed at the perfect mastery of the exit algorithm by the students, while continuing on the concentrations, contains the following steps: a) exit without crossing the floor; b) subtract numbers with zero (or zeros) when decreasing; c) exit by crossing floors.
Mastering the checkout algorithm is a long-term process. In this process, we can distinguish the following teaching stages that are of particular importance:
1) mastering the algorithm of subtracting numbers in the circle of 10;
2) mastering the algorithm of subtracting numbers in the circle of 20 without exceeding ten and exceeding ten;
3) mastering the algorithm of subtracting numbers in the circle of 100 without exceeding ten and exceeding ten;
4) mastering the algorithm of verbal subtraction of numbers in the circle of 1000;
5) mastering the written subtraction algorithm of numbers in the circle of 1000;
6) Mastering the algorithm for subtracting multi-digit (millions) numbers [9; 38-43].
It should be noted that there are specific characteristics of the work at each stage and they should be taken into account in the real pedagogical process.
As it is known, the exercises of subtracting singularities are done with the same tools as adding singularities, that is, with the help of the tools applied in the latter, and consistency is expected there. The next stage of work is subtraction of two (duality). In connection with the subtraction of two, the student has the opportunity to master the algorithm of action (the most elementary). It is visual (material)-practical in nature. Threes and fours are performed on the basis of subtraction, division into their components and sequential subtraction. For example, subtracting 3: 1) determining the composition of 3: 1+1+1; 1+2; 2+1; 2) consecutively subtract the singularities: 9-1=8; 8-1=7; 7-1=6/ 9-3=6; 9-1=8; 8-2=6/ 9-3=6; 9-2=7; 7-1=6 /9-3=6 [10; 123].
This process should be carried out according to the methodical approach as follows: first, with the help of visual aids and didactic materials, then by solving problems, and finally on anonymous numbers. Students should learn by heart the subtraction table of oddities, twos, threes, and fours in 10 circles according to such exercises [7;194]. Teachers with advanced experience expect a certain
sequence, order when subtracting 3, 4 (together with addition). For example, we can differentiate the following sequence of subtraction by 3:
1) 3+3=6; 6+3=9.
9-3=6; 6-3=3.
2) 1+3=4; 4+3=7; 7+3=10.
10-3=7; 7-3=4; 4-3=1.
3) 2+3=5; 5+3=8;
8-5=3; 5-3=2.
Subtraction with groups is considered as the main calculation method in subtraction of five, six, seven, eight and nine. When the subtraction and subtraction are close numbers in a subtraction operation, it works well to teach students to use the "rounding method" to simplify the calculation algorithm. Let's show an example to illustrate our point. Suppose you need to subtract 7 from 9. The calculation in "Subtraction with singles and groups" is long and difficult. But when students use rounding down the subtraction, the calculation process becomes much simpler. Undoubtedly, here too, it is necessary to rely on students' knowledge of the composition of numbers.
It is necessary to pay special attention to two cases of subtraction of numbers in the circle of twenty: 1) cases of subtraction related to numbering; 2) withdrawals according to the collection schedule; 3) subtraction with a zero number.
M.I. Moro, MABantova, GVBelyutkova consider it expedient to focus on cases 12-2, 12-10 of the deduction related to numbering. They show that subtraction principles based on knowledge of the decimal composition of second decimal numbers should be disclosed [7;194]. For this purpose, it is useful to use didactic materials. Students perform practical operations on sets using sticks or strips with circles and are encouraged to explain: 10+5 (1 tens and 5 ones make 15); 16-6 (the number 16 consists of 1 tens and 6 singles; if we subtract 6 singles, we get 1 tens or 10); 14-10 (number 14 consists of 1 decimal and 4 odd ones, if 1 decimal is subtracted from 4 odd ones by 1 decimal, 4 odd ones or 4 is obtained) Here it is useful to repeat the rule of subtracting one of the added ones from the sum of the two added ones.
We think that it is necessary to give an important place to the case of subtracting all its singles or tens from a two-digit number, and for this case the methodical approach mentioned above is also acceptable. However, at the stage under discussion, it is not appropriate to be satisfied with this situation alone. It would not be bad to pay attention to the case of subtracting a few odd numbers from a two-digit number. The implementation of this work with didactic means does not cause any difficulties and greatly facilitates the mastering of the subtraction act at a later stage.
Subtracting one-digit and two-digit numbers from twenty allows students to learn how to use the decimal part of a number. According to our understanding, at this stage, students should be taught to subtract a two-digit number from any two-digit number (not exceeding twenty) (on visual basis). Here, the application of the completion method in cases where the decrement and subtraction are close to each other strengthens the students' mental activity. The most important thing is that students are prepared to understand the written subtraction algorithm.
We believe that the two ways of subtracting numbers up to twenty, which are considered a special stage in the circle of 100, should be used comprehensively - the act of subtracting by exceeding ten and not exceeding ten. In both cases, the teaching work should be aimed at mastering the subtraction algorithm. We call it subtraction when the subtraction is greater than the unity of the subtraction by exceeding ten, and when it is small by not exceeding ten, or rather, we are inclined to classify the work aimed at mastering subtraction in this way. In general, it is appropriate to include methods of subtraction by exceeding ten with visual (material) and didactic means, on practical grounds, in the teaching process in the following order:
1. The case of subtracting all its components or decimals from a two-digit number (for example: 17-7; 13-10);
2. The case of subtracting several ones from a two-digit number; with the condition that the
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number of singularities of a two-digit number is greater than that of a one-digit number (for example: 15-3).
The algorithm for subtracting several ones from a two-digit number is as follows, which is practically performed with straws:
1) 15=10+5;
2) (10+5)-3=10+(5-3);
3) 10+2=12.
3. The case of subtracting one-digit and two-digit numbers (less than 20) from twenty (for example: 20-6; 20-14).
The algorithm for subtracting one-digit and two-digit numbers from twenty is as follows:
20-6 20-14
1) 20=10+10 1)14=10+4
2) (10+10)-6=10+(10-6) 2) 20-10=10
3) 10+4=14 3) 10-4=6
20-6=14. 20-14=6.
4. The case of subtracting a two-digit number from a two-digit number not exceeding twenty (for example: 17-14; 18-15). The fourth case is relatively difficult. But the difficulty is overcome when this case is clearly explained by means of a class calculator, rods, bars and cubes.
The process can be set up like this. First, 10 bars out of 17 bars (a tied set) are taken, then four more bars are taken and the difference is determined (3 bars). The operation has the following algorithmic basis:
First step, dividing 14 into components consisting of 10 and 4 is an elementary step for the student. This is done based on the student's knowledge of numbering the second decimal.
Second stepand it is elementary because it refers to the case of subtracting all its singles or tens from a two-digit number, and this case is known to students.
The third step, 10 is the tabular case of subtraction in circle.
The following sequence of work on teaching subtraction in the circle of 100 is considered acceptable: 22-20; 60-20; 100-30; 36-20; 36-2; 30-4; 60-24; 35-7; verbal subtraction in the circle of 100 by exceeding ten; 40-8; 50-24; 52-24; written subtraction of numbers in the hundred circle (see: [2-3; 8]).
The presented structure of the process of teaching subtraction in the circle of 100 is acceptable from the point of view of its algorithm being mastered by students. These structural elements of the process of teaching subtraction in the circle of 100 can be combined into two groups: 1. Subtraction without exceeding ten; 2. Do not subtract by exceeding ten.
In cases of subtraction belonging to the first group, the singularities of the decrement are more than the singularities of the subtractor or there are no singularities (57-35; 80-33). In cases of subtraction included in the second group, the singularities of the decrement are less than the number of singularities of the subtracted.
The following cases of subtraction without exceeding ten can be distinguished:
1.Subtract the ones or tens from a two-digit number. This case requires students to know only numbering.
2.A single-digit number subtracted from a two-digit number is less than the units of the two-digit number (for example: 36-2). In this case, the method of subtraction consists of subtracting the units of the unit from the units of the unit without touching the decimals of the unit. This process contains the following steps: 1)36=30+6; 2) 6-2=4; 3) 30+4=34.
3.Subtract round decimals from a two-digit number and subtract two-digit numbers with the same number of odd ones (for example: 68-30; 68-38). Here, the method of subtraction in the first case is as follows: the decimals are subtracted from the decimals of the two-digit number. The second case consists of the subtraction of decimal components.
4.Subtract two-digit numbers (eg: 57-26).
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Two methods can be applied here:
The first method.In this method, processes are carried out sequentially, and there is no need to remember intermediate results: 1) 26=20+6; 2) 57-20=37; 3) 37-6=31.
This method is convenient during oral calculations.
The secondmethod:1) 57=50+7; 2) 26=20+6; 3) 50-20=30; 4) 7-6=1; 5) 30+1=31/57-26=31.
By the way, let's note that the second method has an advantage in terms of applying the written calculation algorithm. Here, the transition to columnar written calculation is not difficult. Columnar calculation is understood in its essence. At this stage, the teaching process should be structured in such a way that each student is able to explain his activity in the following order:
1) I write the tens under the tens, the odd ones under the odd ones;
2) I leave the singulars from the singulars: 7-6=1, I write 1 under the singulars;
3) I subtract the tens from the tens: 5-2=3, I write 3 under the tens;
4) I read the answer: the difference is equal to 31.
5.Subtract single-digit numbers from round decimals (eg: 30-4).
In this case, a decimal is separated from the rounded decimals, an incomplete difference is found based on knowledge of subtraction in the circle of 10, this difference is added to the unused decimals, that is, as follows: 1) 30=20+10; 2) 10-4=6; 3) 20+6=26 / 30-4=26.
6.Subtracting a two-digit number from round decimals (eg: 60-24). Here, subtracted decimals are divided into their constituents and successive subtraction operations are performed: 1) 24=20+4;
2) 60-20=40; 3) 40-4=36/ 60-24=36.
At this stage, the student is encouraged to apply column subtraction. He should be able to articulate the steps he takes in the following structure and content (approximately):
1) I write singulars under singulars, tens under tens;
2) I come out alone; 4 cannot be subtracted from 0, I take 1 decimal from 6 decimals (I put a dot on top of 6 so as not to forget); 1 tens=10 ones: 10-4=6; I write 6 under singularities;
3) I subtract tens: there were 6 tens; when the singulars came out, we took 1 decimal; 5 decimals left: 5-2=3; I write 3 under the tens;
4) I read the answer: the difference is equal to 36.
The following can be attributed to subtraction by exceeding ten:
1.Subtract from a two-digit number a single-digit number that is greater than its unity (eg: 524; 35-7).
The implementation scheme for example 52-4 is as follows: 1) 4=2+2; 2) 52-2=50; 3) 50-2=48/ 52-4=48.
2. Subtracting a two-digit number from a two-digit number with the singularity of the decrement less than the singularity of the subtractor (for example: 52-24).
The implementation scheme for the example of 52-24 is as follows: 1) 24=20+4; 2) 52-20=32;
3) 4=2+2; 4) 32-2=30; 30-2=28/ 52-24=28.
It is necessary to try so that the student is able to perform the subtraction in the form of a column independently, provided that he explains it in the following structure and content:
1) I write singulars under singulars, tens under tens;
2) I subtract odd numbers: 4 cannot be subtracted from 2, I take a decimal from 5 tens (I put a dot above 5 so that I don't forget), 1 decimal and 2 odd ones, it makes 12 odd ones: 12-4=8, 3 I write under singularities;
3) I subtract the decimals (here I act as before): 4-2=2, I write 2 under the decimals;
4) I read the answer: the difference is equal to 28.
In our opinion, mastering the algorithm of verbal subtraction of numbers in the range of 1000 should be included among the important issues. Scientific sources distinguish different cases of teaching oral methods of speech [9;38-43]. We prefer the following system of the mentioned cases:
* Subtract round hundreds;
*Do not subtract hundreds, tens, and odd numbers from a three-digit number;
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*Do not subtract round tens without exceeding hundredths;
* Subtract round hundreds from a three-digit number (round tens);
* Subtracting a two-digit number from a three-digit number by exceeding a hundred;
* Subtraction of a three-digit number with rounded decimals not exceeding a hundred;
* Subtract singles, round tens, two-digit numbers from round hundreds;
* Subtract round hundreds from round hundreds;
*Do not round off decimals beyond a hundred.
Undoubtedly, there are cases other than those mentioned here.
Based on the generalizations we made on the research materials, it can be argued that it is useful to pay attention to the following directions for students to master the written subtraction algorithm in the circle of thousands:
■Each digit of the decrement is greater than the corresponding digit of the subtraction (567345; 356-232);
■ The odd number of the decrement is less than the odd number of the subtractor (642-227; 791-524) or the odd number of the decrement is equal to zero (650-234);
■ The number of decimals of the decrement is smaller than the number of decimals of the minus (428-265) or this number is equal to zero (608-241);
■ The odd and decimal digits of the declinator are less than the corresponding digits of the deductor (823-256) or both of these digits of the declinator are equal to zero (500-216);
■ There is a zero in the middle of the decrement, and its odd number is less than the odd number of the subtraction (403-217).
In order to create the habit of written subtraction of multi-digit numbers, students need to memorize the subtraction table and know the numbering of multi-digit numbers well (know the ratio of odd numbers of adjacent floors and be able to quickly break one floor unit into another) and have the ability to use the algorithm that includes all cases of subtraction of three-digit numbers [ 1; 125].
According to our understanding, the work of teaching subtraction on multi-digit numbers should be continued according to the following system:
■ The digits of the decrement are greater than the corresponding digits of the deductor (57894117);
■Subtraction contains zeros (26088-4004);
■ Some digits of the decrement are smaller than the corresponding digits of the deductor (72351876);
■ There is one zero in the decrement (3508-1776);
■Decrement contains two or more zeros next to each other (140004-8547).
Undoubtedly, each level has its own options, these options should be reflected in the exercises.
Scientific novelty of the work. The technology of adequately forming the skills of applying the subtraction algorithm to the expansion of the range of natural numbers by educational units has been developed.
The theoretical importance of the work. The "gaps in the theory" related to the inclusion of the subtraction algorithm in the process of teaching mathematics in primary classes have been largely eliminated.
Workpractical importance. The development of the technology of adequately forming the skills of applying the subtraction algorithm to the expansion of the range of natural numbers in educational units will have a positive effect on the formation of an environment that conditions the elimination of mistakes that may be made in the teaching activities of practical educators.
The result. Adequate application of the technology of forming the skills of the practical application of the subtraction algorithm on natural numbers to the expansion of the range of natural numbers in educational units is a more effective methodical approach in terms of forming algorithmic culture and improving the quality of education.
LiTERATURE
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