Let's consider the traditional approach to learning the multiplication table.
In the traditional method, there are 3 stages:
Stage 1 - preparatory.
At this stage, students study the basic theoretical issues on which table multiplication is based (theoretical basis):
a) the meaning of multiplication,
b) the name of the components and the result of multiplication,
V) special cases multiplying one and zero by a number,
d) commutative property of multiplication,
e) the relationship between the components and the result of multiplication,
g) special cases of multiplication with the number 10,
h) study of multiplication cases corresponding to the multiplication table of two,
Stage 2 - compiling tables.
At this stage, students create multiplication tables and columns of corresponding multiplication cases. It is possible to highlight the features of compiling these tables:
Drawing up a table relies on actions with objects and the use of numerical figures;
The compilation of each table begins with the case of multiplication of identical factors;
By studying each column of the multiplication table, 3 more columns are added to it. Column 2 data includes:
1st column - multiplying a number according to the first constant sign;
Column 2 - multiplication by the second constant sign (based on commutability);
Stage 3 - memorizing tables.
Since in modern elementary school We are talking about the formation of conscious computational skills, then the compilation of multiplication tables is preceded by the study of theoretical issues, which are the basis of those computational techniques that students can use when compiling these tables. But the sequence of compiling tables and organizing student activities aimed at mastering them may be different.
The set-theoretic interpretation of the meaning of the action of multiplication is easily translated into the language of objective actions and allows for the active use of previously studied material to master a new concept.
Let's take a closer look at the methodology of the traditional program edited by Moro M.I.
Mastering the meaning of multiplication allows students to independently cope with the preparation of multiplication tables. The commutative property of multiplication allows you to reduce the number of table cases that need to be memorized. So memorizing cases 2 · 3 guarantees knowledge of case 3 · 2, etc. This allows each subsequent table to begin with the case of multiplying identical factors. As a result, the number of cases in each of the following tables is reduced:
6 6 6 7 6 8 6 9 6 10
To study subsequent cases of multiplication from the table, it is necessary to create a second column. As we already said, based on the commutative property of multiplication: 7 6 8 6 9 6
When memorizing tables, students experience great difficulties due to the large volume of multiplication cases that are immediately offered to students for memorization.
In the first lesson, students make up all four columns of the table that they must remember. And in subsequent lessons, children perform a variety of exercises aimed at memorizing table cases of multiplication. At this stage, it is important for the teacher to skillfully select tasks that successfully solve a given problem.
Let's consider the methodology for studying the table using the example of multiplication of four and the corresponding cases of division. 4 4 4 5 4 6 4 7 4 8 4 9 4 10
IN preparatory work You can include exercises on finding an unknown factor (· 2 = 8, 3 · = 15), you can repeat the multiplication table for two and three and the corresponding cases of division, you must also repeat all the examples known to children for multiplication with the number 4.
Then they move on to compiling a multiplication table for four by the constant first factor.
The last entries to be made are for case 4 · 4. Next, students are asked to consider all the expressions in the first table and say what interesting things they noticed. Children must answer that the first factors are the same, the second factors increase by one, and the product increases by 4 units. The records of other columns are also compared. In this way, children establish patterns when compiling tables, which will help them memorize them meaningfully, as well as use them in calculations in appropriate cases of multiplication (based on the commutative property of multiplication).
Having memorized all the table cases of multiplication, perform exercises to consolidate them.
1) The first stage - compiling and mastering multiplication tables is included in the content of the course. Students learn multiplication table cases as they learn the meaning of multiplication. This makes it possible to offer students interesting, meaningful exercises and tasks, the completion of which helps them involuntarily memorize the multiplication tables.” The results of the work on the formation of table multiplication skills are summarized in general lessons on the topic “Multiplication”, where students are given a task, during which they can check how each of them has mastered the multiplication table. From the above, we can conclude that multiplication table skills are first developed. At the same time, the work associated with compiling and mastering the multiplication tables is distributed over time and organically included in the content of the course.
The following features of this approach to skill formation table multiplication:
2) compiling and mastering the multiplication table begins with cases of multiplying the number 9 (from more difficult to easier), which allows students not only to practice addition and subtraction of two-digit and single digit numbers with the transition through ten, replacing the product with a sum, but also focus on cases of the multiplication table that are difficult to remember: 9 · 8, 9 · 7, 9 · 6, in relation to which a memorization setting is given.
3) Considering that not all children can involuntarily memorize the multiplication table in the process of completing educational tasks, in the textbook, in a certain system, instructions are given for memorizing three or four table cases. At the same time, the setting for memorizing the table is focused on memorizing certain table cases.
4) For organization independent work Students are encouraged to record all cases of table multiplication on a card. For example, on one side is an expression, and on the other is its meaning. This will help students act when memorizing table cases of multiplication, as well as exercise self-control.”
We will also consider the features of the approach according to I.I.’s textbook. Arginskaya. When studying table multiplication, the author singled out only two stages in the students’ work:
Stage 1 - familiarization with theoretical information, including the order of action in expressions.
Stage 2 - studying the multiplication table using the Pythagorean table.
I.I. Arginskaya distinguishes two approaches - direct and indirect, giving them a detailed description, pointing out the advantages of the indirect one.
“The direct approach is characterized by the presence finished sample performing the operation under study and a large number of ready-made training exercises, during which students master a skill based on reproductive activity, where mastery of the skill acts as an end in itself according to the principle “solve in order to learn to solve.” Reproductive activity is characterized by the fact that the student receives ready-made information, perceives it, understands, realizes, remembers, and then reproduces it himself. The main goal of this type of activity is the formation of students’ knowledge of learning, the development of attention and memory.”
The main advantage here is the very rapid achievement of the required result, which is why it is so widespread and occupies a strong position in school practice. However, there is also negative aspects. I.I. Arginskaya considers the direct approach “unnatural, because a person masters the technical side of any business not as an end in itself, but for the sake of solving problems that are relevant to him. The predominance of reproductive activity in the formation of computational skills significantly provides the opportunity to promote children in development, and currently the development of schoolchildren is a priority task of education in any system.”
Why does the system prefer an indirect approach to the formation of computing skills?
The fact is that almost any task should contribute to the advancement of children in development, and the direct approach completely excludes this component. To shape the development of children's cognitive interests, it is necessary to interest them, which requires active forms and methods of teaching to awaken in children an active perception of the material. Students contribute to the best assimilation and memorization of material various means clarity, as well as tables, drawings, diagrams used in each lesson.
Teacher's tasks: 1) Form a concept about the specific meaning of multiplication and division;
2) Study multiplication and division tables;
3) Bring your knowledge of tables to automaticity.
To prepare for the introduction specific meaning of multiplication include counting pairs and triplets of objects. Students are given tasks to find the sum of equal and unequal terms. It is useful to illustrate such tasks with objects or drawings. Should include reverse exercises: Using these pictures, create problems (examples) for addition.
When solving such problems and examples, students notice that there are sums with identical terms and count how many such terms there are. Next the sum of identical terms is replaced by the product(6 + 6+6 + 6 = 24; 6·4 = 24).
The following task is also given: Present the numbers (6,8,10, 32) as a sum of identical terms.
12= 2+2+2+2+2+2+2
Revealing the specific meaning of multiplication, several exercises to replace a sum with a product. 2+2+2+2=8 2·4=8 Students learn different readings of the expression: 2 multiplied by 4
take 2 4 times
take 2 4 times
When calculating some sums of identical terms, it is advisable to familiarize children with method of grouping terms. For example, when calculating the sum 2 + 2 + 2 + 2 + 2 + 2 + 2, you need to draw children’s attention to the fact that the sum of five terms is equal to 10, and to 10 it is easy to add the sum of the remaining terms: 10 + 4 = 14. This technique is used later when compiling multiplication tables.
The specific meaning of division is revealed in the process of solving simple problems of division by content and into equal parts. Students must learn to perform, according to the conditions of the problem, the operation of dividing a given set into a number of equal subsets and associate this operation with the action of division, learn to write down the solution to problems using this action.
The first computational method of division is based on knowledge of the specific meaning of the action of division: students find the quotient by performing actions with objects. For example, to find the quotient of 8:4, they take 8 circles, lay them out in 4s and count how many times they get 4 circles, or lay out 8 circles into 4 equal parts and count how many circles they get in each part.
Table cases of multiplication and division. The preparatory work includes: 1. Familiarization with the specific meaning of multiplication and division;
2. Establishing a connection between multiplication and division;
3. Techniques for finding a work.
These include:
Replacing the product with the sum 2·5=2+2+2+2+2
Using the answer from the previous and next example.12 6= (2 5)+2=12
Method of grouping terms 2·8=2·5+2+2+2=2·5+2·3
Using the techniques you've learned , compiling a multiplication table for two, which children will have to gradually memorize. When compiling a multiplication table of two, the result is found by addition using visual aids, for example, a square with a corner, or circle 9 rows of cells in a notebook, 2 cells per row.
2·3=2+2+2=6 3·2 6:2 6:3
2·4 4·2 8:2 8:4
………….. ……………………
2·8= 2+2+2+2+2 +2+2+2=2·5+2·3=16 …………………...
2·9= 9·2 18:2 18:9
Next we study the commutative property of multiplication. Knowing this property is necessary, first of all, for mastering the operation of multiplication, and in addition, knowledge of this property makes it possible to almost halve the number of cases that need to be memorized. Instead of two cases (8*3 and 3*8), students remember only one.
Based on the commutative property of multiplication a multiplication table by 2 is compiled. Students are asked to create this table themselves.
The connection between the components and the result of the multiplication action is revealed with the help of visual aids. Students are asked to create an example of multiplication based on the picture:
Students make up an example: 4·2=8. Using the same picture, the task is given to create two examples of division. (8:4 = 2, 8:2=4.)
After completing several similar exercises, students do conclusion: If the product of two numbers is divided by one of the factors, we get the other factor.
Introduced tables of division by 2 and division with quotient 2(answer 2)
Table multiplication and division are studied together, i.e., from each case of multiplication, corresponding cases of division are obtained.
First, all table cases of multiplication and division with the number 3 are considered, then 4, 5, etc.
Each multiplication table for the constant first factor is compiled starting with the case of equal factors (3 3, 4 4, etc.), since the cases preceding these have already been considered earlier in other tables.
To remember better multiplication and division tables you can use the following methods:
1) often repeat cases of multiplication and division;
2) repetition of tables at random;
3) use of tabular cases in mathematical dictations;
4) reproduction of tabular cases of multiplication by result (24=6·4, 24=3·8, etc.);
5) gaming methods;
6) compiling triplets of examples (1 for multiplication and 2 for division)
©2015-2017 site
All rights belong to their authors. This site does not claim authorship, but provides free use.
Agreement on the use of site materials
We ask you to use the works published on the site exclusively for personal purposes. Publishing materials on other sites is prohibited.
This work (and all others) is available for download completely free of charge. You can mentally thank its author and the site team.
Submitting your good work to the knowledge base is easy. Use the form below
Students, graduate students, young scientists who use the knowledge base in their studies and work will be very grateful to you.
Similar documents
Mathematical foundations for learning table multiplication and related cases of division. Age characteristics junior schoolchildren. Methodological approaches to studying the topic "Tabular multiplication and corresponding cases of division." Types of independent work.
course work, added 02/26/2010
Formation of computing skills among students primary classes when studying tabular cases of multiplication and division. An experimental study on the formation of strong skills in table multiplication and division in mathematics lessons at school.
thesis, added 01/09/2014
Mathematical foundations of studying tabular multiplication and division in elementary school, the formation of computational skills in the traditional education system. Features of the didactic system of L.V. Zankova: full-fledged computing skill, quality, tasks.
thesis, added 08/31/2011
Metaknowledge: the concept and content of metasubjectivity in modern education. Methods and techniques for studying multiplication tables in primary school; research and analysis diagnostic work“at the input” and “at the output”, aimed at meta-subject results.
thesis, added 09/17/2011
Formation of computational skills and abilities in junior schoolchildren in the initial mathematics course. Methodological and mathematical foundations for the formation of table multiplication skills. Characteristics of methodological techniques that promote memorization of the multiplication table.
course work, added 03/19/2016
Mental calculations, arithmetic tables, multiplication tables. Laws of arithmetic operations. An axiomatic approach to defining the concepts of work and private. Pedagogical foundations for the formation of computing skills. Analysis of the program and textbook.
course work, added 02/10/2015
Federal state educational standard for primary general education. The concept and essence of meta-subject matter in modern education. The essence and content of meta-subject results. Features of learning the multiplication table in elementary school.
course work, added 11/14/2011
