MINISTRY OF EDUCATION AND SCIENCE OF THE RF
FEDERAL AGENCY FOR EDUCATION
ELETS STATE UNIVERSITY NAMED AFTER I.A.BUNINA
METHODOLOGY FOR STUDYING ALGEBRAIC, GEOMETRIC MATERIAL, QUANTITIES AND FRACTIONS
IN PRIMARY CLASSES
Tutorial
Yelets – 2006
BBK 65
Compiled by Faustova N.P., Dolgosheeva E.V. Methods for studying algebraic, geometric material, quantities and fractions in primary grades. - Yelets, 2006. - 46 p.
This manual reveals the methodology for studying algebraic, geometric material, quantities and fractions in the primary grades.
The manual is intended for students of the Faculty of Pedagogy and Methods of Primary Education, full-time and part-time, and can be used by primary school teachers, teachers of the Faculty of Pedagogical Education of universities and teacher training colleges.
The manual is compiled in accordance with the State Standards and the work program for this course.
Reviewers:
Candidate of Pedagogical Sciences, Associate Professor of the Department of Mathematical Analysis and Elementary Mathematics T.A. Poznyak
Leading specialist of the department of public education of the administration of the Yeletsk district of the Lipetsk region Avdeeva M.V.
© Faustova N.P., Dolgosheeva E.V., 2006
METHODOLOGY FOR STUDYING ALGEBRAIC MATERIAL IN PRIMARY SCHOOL CLASSES
1.1. General questions of methods for studying algebraic material.
1.2. Methods for studying numerical expressions.
1.3. Learning letter expressions.
1.4. Study of numerical equalities and inequalities.
1.5. Methods for studying equations.
1.6. Solving simple arithmetic problems by writing equations.
1.1. General questions of methods for studying algebraic material
The introduction of algebraic material into the initial course of mathematics makes it possible to prepare students for studying the basic concepts of modern mathematics (variables, equations, 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 equations provided by the curriculum and simple arithmetic problems by constructing 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.
Methodology for studying numerical expressions
In mathematics, an expression is understood as a sequence of mathematical symbols constructed 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:c; 5+(3+c) - literal expressions (expressions with a variable).
Objectives of studying the topic
2) Familiarize students with the rules for the order of performing arithmetic operations.
3) Learn to find numerical values of expressions.
4) Introduce identical transformations of expressions based on the properties of arithmetic operations.
The solution to the set tasks is carried out throughout all years of education in primary school, starting from the first days of the child’s stay at school.
The methodology for working on numerical expressions involves 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 level; at the third stage - about expressions containing two or more arithmetic operations of different levels.
Students are introduced to the simplest expressions - sum and difference - in the first grade (according to program 1-4) with the product and quotient in the second grade (with the term “product” in 2nd grade, with the term “quotient” in the third grade).
Let's consider the methodology for studying numerical expressions.
When performing operations on sets, children, first of all, learn the specific meaning of addition and subtraction, therefore, in entries of the form 3 + 2, 7-1, the signs of actions are recognized 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) several units, we increase (decrease) the number by the same number of units (reading: 3 increase by 2), then children learn the name of the action signs “plus” (reading: 3 plus 2), "minus".
In the topic “Addition and subtraction within 20,” children are introduced to the concepts of “sum” and “difference” as the names of mathematical expressions and as the name of the result of the arithmetic operations of addition and subtraction.
Let's look at a fragment of the lesson (2nd grade).
Attach 4 red and 3 yellow circles to the board using water:
How many red circles? (Write down the number 4.)
How many yellow circles? (Write down the number 3.)
What action must be performed on the written numbers 3 and 4 to find out how many red and how many yellow circles there are together? (the entry 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 a sum (Let’s say together: sum) and is 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 the full answer).
Likewise about 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. By calculating the values of these expressions, children practically master the rule about the order of arithmetic operations in expressions without parentheses, although they do not formulate it: 10-3+2=7+2=9. Such entries are the first step in performing identity transformations.
The method of familiarizing yourself with expressions with brackets can be different (Describe a fragment of the lesson in your notebook, prepare for practical lessons).
The ability to compose and find the meaning of an expression is used by children when solving arithmetic problems; at the same time, further mastery of the concept of “expression” occurs here, and the specific meaning of expressions in the recordings of problem solving is acquired.
Of interest is the type of work proposed by the Latvian methodologist J.Ya. Mencis.
A text is given, for example, like this: “The boy had 24 rubles, the cake costs 6 rubles, the candy costs 2 rubles,” it is suggested:
a) compose all types of expressions based on this text and explain what they show;
b) explain what the expressions show:
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 those consisting of a number and the product or quotient of two numbers. For example: 75-50:25+2. Where the order in which actions are performed does not coincide with the order in which they were written, brackets are used: 16-6:(8-5). Children must learn to read and write these expressions correctly and find their meanings.
The terms “expression” and “value of expression” 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 diagram that is compiled collectively and used when reading expressions:
1) I will determine which action is performed last.
2) I’ll think about what the numbers are called when performing this action.
3) I will read how these numbers are expressed.
The rules for the order of performing 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 to consider is the rule about the order of operations in expressions without parentheses, when numbers are either only addition and subtraction, or multiplication and division (3rd grade). The goal of the work at this stage is to rely on the practical skills of students acquired earlier, to pay attention to the order of performing actions in such expressions and to formulate a rule.
Leading children to the formulation of the rule and their awareness of it can be different. The main reliance is on existing experience, the greatest possible independence, creating a situation of search and discovery, evidence.
You can use the methodological 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 that he is confident are correct (answers are closed).
36:2 6=6, etc.
Invites children to find the meanings of 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 studying particular facts, formulate a rule (see mathematics textbook, 3rd grade).
Similarly, you can introduce the remaining rules for the order of actions: when expressions without brackets contain actions of the 1st and 2nd stages, 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, and therefore mathematicians decided to agree and formulated rules that must be strictly followed.
Transforming an expression is replacing a given expression with another with the same numerical value. Students perform such transformations of expressions, relying on the properties of arithmetic operations and consequences from them (p. 249-250).
When studying each property, students become convinced that in expressions of a certain type, actions can be performed in different ways, but the meaning of the expression does not change. In the future, students use knowledge of the properties of actions to transform given expressions into identical expressions. For example, tasks like: 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, students reason like this: on the left, from 76, subtract the sum of the numbers 20 and 4 , on the right, subtract 20 from 76; in order to get the same amount on the right as on the left, you must also subtract 4 from the right. Other expressions are transformed similarly, i.e., after reading the expression, the student remembers the corresponding rule. And, performing actions according to the rule, it receives a transformed expression. To ensure that the transformation is correct, children calculate the values of the given and transformed expressions and compare them.
Using knowledge of the properties of actions to justify calculation techniques, students in 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
Here it is also necessary that students not only explain on what basis they derive each subsequent expression, but also understand that all these expressions are connected by the “=” sign because they have the same meanings. To do this, children should occasionally be asked to calculate the meanings of expressions and compare them. This prevents errors of the form: 75 - 30 = 70 - 30 = 40+5 = 45, 24 12= (10 + 2) = 24 10+24 2 = 288.
Students in grades II-IV transform 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). Also based on the meaning of the multiplication action, more complex expressions are transformed: 8 4 + 8 = 8 5, 7 6-7 = 7 5.
Based on calculations and analysis of specially selected expressions, fourth grade students are led to the conclusion that if in expressions with brackets the brackets do not affect the order of actions, then they can be omitted. Subsequently, using the studied properties of actions and rules for the order of actions, students practice transforming expressions with brackets into identical expressions without brackets. For example, it is proposed to write these expressions without parentheses so that their values do not change:
(65 + 30)-20 (20 + 4) 3
96 - (16 + 30) (40 + 24): 4
Thus, 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 are convinced that the meaning of an expression does not change when changing the order of actions only if the properties of the actions are applied.
Lecture 7. The concept of the perimeter of a polygon
1. Methodology for considering the elements of algebra.
2. Numerical equalities and inequalities.
3. Preparing to become familiar with the variable. Elements of letter symbols.
4. Inequalities with a variable.
5. Equation
1. The introduction of algebra elements into the initial mathematics course allows, from the very beginning of training, to carry out systematic work aimed at developing in children such important mathematical concepts as: expression, equality, inequality, equation. Familiarization with the use of a letter as a symbol denoting any number from the field of numbers known to children creates conditions for generalizing many issues of arithmetic theory in the initial course, and is a good preparation for introducing children in the future to concepts in the variable of functions. Earlier familiarization with the use of the algebraic method of solving problems makes it possible to make serious improvements in the entire system of teaching children to solve a variety of word problems.
Tasks: 1. Develop students’ ability to read, write and compare numerical expressions.2. Introduce students to the rules for performing the order of actions in numerical expressions and develop the ability to calculate the values of expressions in accordance with these rules.3. To develop in students the ability to read, write letter expressions and calculate their meanings given the meanings of the letters.4. To acquaint students with equations of the 1st degree, containing the actions of the first and second stages, to develop the ability to solve them using the selection method, as well as on the basis of knowledge of the relationship between m / y components and the result of arithmetic operations.
The primary school program provides for introducing students to the use of letter symbols, solving elementary equations of the first degree with one unknown and applying them to problems in one step. These questions are studied in close connection with arithmetic material, which contributes to the formation of numbers and arithmetic operations.
From the first days of training, work begins to develop the concepts of equality among students. Initially, children learn to compare many objects, equalize unequal groups, and transform equal groups into unequal ones. Already when studying a dozen numbers, comparison exercises are introduced. First, they are performed with support on objects.
The concept of expression is formed in younger schoolchildren in close connection with the concepts of arithmetic operations. The methodology for working on expressions involves two stages. At 1, the concept of the simplest expressions (sum, difference, product, quotient of two numbers) is formed, and at 2, about complex expressions (the sum of a product and a number, the difference of two quotients, etc.). The terms “mathematical expression” and “value of a mathematical expression” are introduced (without definitions). After recording several examples in one activity, the teacher informs that these examples are otherwise called metamathematical expressions. When studying arithmetic operations, exercises on comparing expressions are included; they are divided into 3 groups. Studying the rules of procedure. The goal at this stage is, based on the practical skills of students, to draw their attention to the order of performing actions in such expressions and to formulate an appropriate rule. Students independently solve examples selected by the teacher and explain in what order they performed the actions in each example. Then they formulate the conclusion themselves or read it from a textbook. Identical transformation of an expression is the replacement of a given expression with another whose value is equal to the value of the given expression. Students perform such transformations of expressions, relying on the properties of arithmetic operations and the consequences arising from them (how to add a sum to a number, how to subtract a number from a sum, how to multiply a number by a product, etc.). When studying each property, students become convinced that in expressions of a certain type, actions can be performed in different ways, but the meaning of the expression does not change.
2. Numerical expressions are considered from the very beginning in inextricable connection with numerical equals and unequals. Numerical equalities and inequalities are divided into “true” and “false”. Tasks: compare numbers, compare arithmetic expressions, solve simple inequalities with one unknown, move from inequality to equality and from equality to inequality
1. An exercise aimed at clarifying students’ knowledge of arithmetic operations and their application. When introducing students to arithmetic operations, expressions of the form 5+3 and 5-3 are compared; 8*2 and 8/2. The expressions are first compared by finding the values of each and comparing the resulting numbers. In the future, the task is performed based on the fact that the sum of two numbers is greater than their difference, and the product is greater than their quotient; the calculation is used only to check the result. A comparison of expressions of the form 7+7+7 and 7*3 is carried out to consolidate students’ knowledge of the connection between addition and multiplication.
During the comparison process, students become familiar with the order of performing arithmetic operations. First, we consider expressions containing brackets of the form 16 - (1+6).
2. After this, the order of actions in expressions without brackets containing actions of one and two degrees is considered. Students learn these meanings as they complete the examples. First, the order of actions in expressions containing actions of one level is considered, for example: 23 + 7 - 4, 70: 7 * 3. At the same time, children must learn that if expressions contain only addition and subtraction or only multiplication and division, then they are performed in in the order in which they were written down. Then expressions containing the actions of both stages are introduced. Students are informed that in such expressions they must first perform the multiplication and division operations in order, and then addition and subtraction, for example: 21/3+4*2=7+8=15; 16+5*4=16+20=36. To convince students of the need to follow the order of actions, it is useful to perform them in the same expression in a different sequence and compare the results.
3. Exercises in which students learn and consolidate knowledge of the relationship between the components and results of arithmetic operations. They turn on already when learning the numbers ten.
In this group of exercises, students are introduced to cases where the results of actions change depending on a change in one of the components. Expressions in which one of the terms is changed (6+3 and 6+4) or reduced by 8-2 and 9-2, etc. are compared. Similar tasks are also included when studying table multiplication and division and are performed using calculations (5*3 and 6*3, 16:2 and 18:2), etc. In the future, you can compare these expressions without relying on calculations.
The exercises considered are closely related to the program material and contribute to its assimilation. Along with this, in the process of comparing numbers and expressions, students receive the first ideas about equality and inequality.
So, in grade 1, where the terms “equality” and “inequality” are not yet used, the teacher can, when checking the correctness of the calculations performed by the children, ask questions in the following form: “Kolya added eight to six and got 15. Is this decision correct or incorrect?” , or offer children exercises in which they need to check the solution to given examples, find the correct entries, etc. Similarly, when considering numerical inequalities of the form 5<6,8>4 and more complex ones, the teacher can ask a question in the following form: “Are these entries correct?”, and after introducing an inequality, “Are these inequalities correct?”
Starting from the 1st grade, children become familiar with transformations of numerical expressions, which are performed on the basis of the application of the studied elements of arithmetic theory (numbering, the meaning of actions, etc.). For example, based on knowledge of numeration and the place value of numbers, students can represent any number as the sum of its place parts. This skill is used when considering expression transformations in relation to the expression of many computational techniques.
In connection with such transformations, already in the first grade, children encounter a “chain” of equalities.
“Studying algebraic material in elementary school”
Performed by teacher of the highest category Averyakova N.N.
Introduction.
Chapter 1. General theoretical aspects of studying algebraic material in elementary school.
1.1. Experience in introducing algebra elements in elementary school.
1.2. Psychological basis for the introduction of algebraic concepts in elementary school.
1.3. The problem of the origin of algebraic concepts and its significance for the construction of an educational subject.
2.1. Teaching in primary school from the point of view of the needs of secondary school.
2.2. Comparing (contrasting) concepts in mathematics lessons.
2.3. Joint study of addition and subtraction, multiplication and division.
Chapter 3. Research work on the study of algebraic material in mathematics lessons in the primary grades of school No. 72.
3.1. Justification for the use of innovative technologies (UDE technology).
3.2. On the experience of familiarization with algebraic concepts.
3.3.Diagnostics of mathematics learning results.
Conclusion.
Bibliographic list.
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.
What is modern mathematics? Why is it needed? These and similar questions are often asked by children to teachers. 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, it appears that this statement has a significant defect. The language of mathematics is so widespread and so often effective precisely because mathematics cannot be reduced to it.
The outstanding Russian mathematician A.N. Kolmogorov wrote: “Mathematics is not just one of the languages. Mathematics is language plus reasoning, it’s like language and logic together. Mathematics is a tool for thinking. It contains the results of the precise thinking of many people. With the help of mathematics one can relate one reasoning to another...The apparent complexities of nature with its strange laws and rules, each of which admits of a very detailed separate 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 – (12))
Thus, mathematics allows us to form certain forms of thinking necessary to study the world around us.
Our education system is designed in such a way that for many, school provides the only opportunity to join a mathematical culture and master the values contained in mathematics.
What is the influence of mathematics in general and school mathematics in particular on the education of a creative personality? Teaching the art of solving problems in mathematics lessons provides us with an extremely favorable opportunity for developing a certain mindset in students. The need for research activities develops interest in patterns and teaches us to see the beauty and harmony of human thought. All this is an essential element of general culture. The mathematics course has an important influence 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 you to compare different hypotheses, find the best option, pose new problems, and look for ways to solve them. By maximizing the possibilities of human thinking, mathematics is the highest achievement.
The mathematics course (without geometry) is actually divided into 3 main parts: arithmetic (grades 1-5), algebra (grades 6), elements of analysis (grades 9-11). Each part has its own special “technology”. Thus, in arithmetic it is associated, for example, with calculations performed on multi-digit numbers, in algebra - with identical transformations, logarithmization, in analysis - with differentiation. But what are the deeper reasons associated with the conceptual content of each part? The next question concerns the basis for distinguishing between school arithmetic and algebra. Arithmetic includes the study of natural numbers (positive integers) and fractions (prime and decimal). However, a special analysis shows that combining these types of numbers in one school subject is unlawful. The fact is that these numbers have different functions: the first are associated with counting objects, the second with measuring quantities. From the point of view of measuring quantities, as A.N. Kolmogorov noted, “there is no such deep difference between rational and irrational real numbers. For pedagogical reasons, it is necessary to dwell on rational numbers, since they are easy to write in the form of fractions, but the use that is given to them from the very beginning should immediately lead to real numbers in all their generality” (12-p.9). Thus, there is a real opportunity, on the basis of natural (integer) numbers, to immediately form “the most general concept of number” (in the terminology of A. Lebesgue), the concept of a real number. But from the point of view of program construction, this means nothing more or less than the elimination of fraction arithmetic 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 30 years ago, are still relevant today. Is it possible to change the structure of teaching mathematics in primary school in this direction? What are the advantages and disadvantages of algebraization in elementary mathematics education? The purpose of this work is to try to answer the questions posed.
Realization of this goal requires solving the following tasks:
Consideration of general theoretical aspects of introducing algebraic concepts of magnitude and number in elementary school;
Study of specific methods for teaching these concepts in elementary school;
To show the practical applicability of the provisions under consideration in elementary school during mathematics lessons at Secondary School No. 72 by teacher N.N. Averyakova.
CHAPTER 1. GENERAL THEORETICAL ASPECTS OF STUDYING ALGEBRAIC MATERIAL IN PRIMARY SCHOOL.
- EXPERIENCE IN INTRODUCING ALGEBRA ELEMENTS IN PRIMARY SCHOOL.
The content of an academic subject depends on many factors - on life’s demands on students’ knowledge, on the level of relevant sciences, on the mental and physical age-related capabilities of children. Correct consideration of these factors is an essential condition for the most effective education of schoolchildren and expansion of their cognitive capabilities. But sometimes this condition is not met for a number of reasons. It seems that currently the teaching programs for some academic subjects, incl. mathematics, do not correspond to the new requirements of life, the level of modern sciences and new data from developmental psychology and logic. This circumstance dictates the need for theoretical and experimental testing of possible projects for new content of educational subjects. The foundation of math skills is laid 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 mathematics program in primary school (1-4) in its main features was formed 50-60 years ago and naturally reflects the system of mathematical, methodological and psychological ideas of that time.
Let's consider the characteristic features of the state standard in mathematics. Its main content is integers and operations on them, studied in a certain sequence. Along with this, the program involves the study of metric measures and time measures, mastering the ability to use them for measurement, knowledge of some elements of visual geometry - drawing a rectangle, square, measuring segments, areas, calculating volumes. Students must 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 operations - half the appropriate time is allocated for this. Solving problems helps students understand the specific meaning of an action, understand various cases of their application, establish relationships between quantities, and acquire basic skills of analysis and synthesis. From grades 1 to 4, children solve the following main types of problems (simple and composite): finding the sum and remainder, product and quotient, increasing and decreasing given numbers, difference and multiple comparison, simple triple rule, proportional division, finding an unknown from two differences and other types of problems. Children encounter different types of quantity dependencies when solving problems. But quite typically, students begin problems after and as they study numbers; The main thing that is required when solving is to find a numerical answer. Children have great difficulty identifying the properties of quantitative relations in specific, particular situations, which are usually considered arithmetic problems. Practice shows that manipulation of numbers often replaces the actual analysis of the conditions of the problem from the point of view of the dependencies of real quantities. Moreover, the problems introduced in textbooks do not represent systems in which more “complex” situations would be associated 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 in terms of the complexity of the plot (the number of actions increases), the rank of numbers (from ten to a billion), the complexity of physical dependencies (from distribution problems to movement problems) and other parameters. Only one parameter—deepening into the system of mathematical laws itself—is manifested weakly and indistinctly in them. Therefore, it is very difficult to establish a criterion for the mathematical difficulty of a particular problem. Why are problems on finding an unknown from two differences and finding out the arithmetic mean more difficult than problems on difference and multiple comparison? The technique does not answer 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 in the school course they are associated primarily with calculation techniques, or when solving problems, because the latter do not have the appropriate form and do not have the required system. The attempts of methodologists to improve teaching methods, although they lead to partial successes, 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 arithmetic program should be based on the following provisions:
The concept of number is not identical to the concept of the quantitative characteristics of objects;
Number is not the original form of expressing quantitative relations.
Let us provide the rationale for these provisions. It is well known that modern mathematics (in particular, algebra) studies aspects 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. (the relation “more”, “less”, “equal”). The presentation of initial mathematical concepts in modern manuals is carried out in such symbolism that does not necessarily imply the expression of objects by numbers. Thus, in E.G. Gonin’s book “Theoretical Arithmetic”, the main mathematical objects are denoted by letters and special signs from the very beginning. 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 only existing form of expression. It is noteworthy that many illustrations of individual mathematical definitions are given in graphical form, through the ratio of segments and areas. All basic properties of sets and quantities can be deduced and justified without involving numerical systems; Moreover, the latter themselves receive justification on the basis of general mathematical concepts.
In turn, numerous observations by psychologists and teachers show that quantitative ideas arise in children long before they acquire knowledge about numbers and how to operate them. True, there is a tendency to classify these ideas as “pre-mathematical formations” (which is quite natural for traditional methods that identify the quantitative characteristics of an object with a number), but this does not change the 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 significant for the development of a child’s own mathematical thinking than the intricacies of computer technology and the ability to find purely numerical dependencies. It is noteworthy that Academician A.N. Kolmogorov, characterizing the features of mathematical creativity, specially notes the following circumstance: “The basis of most mathematical discoveries is some simple idea: a visual geometric construction, a new elementary inequality, etc. You just need to properly apply this simple idea to solve a problem that at first glance seems inaccessible (12-p.17).
At present, a variety of ideas regarding the structure and ways of constructing a new program are appropriate. It is necessary to involve mathematicians, psychologists, logicians, and methodologists in the work on its construction. But in all specific options, it seems to have to satisfy the following requirements:
Overcome the existing gap between the content of mathematics in primary and secondary schools;
To provide a system of knowledge about the basic laws of quantitative relations of the objective world; in this case, the properties of numbers as a special form of expressing quantity should become a special, but not the main section of the program;
Instill in children methods of mathematical thinking, and not just calculation skills: this involves constructing a system of problems based on delving into the sphere of dependencies of real quantities (the connection of mathematics with physics, chemistry, biology and other sciences that study specific quantities);
Decisively simplify all calculation techniques, minimizing the work that cannot be done without appropriate tables, reference books, and other auxiliary tools.
The meaning of these requirements is clear: in elementary school it is possible to teach mathematics as a science about the laws of quantitative relationships, about the dependencies of quantities; computing 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 testing, carried out since the end of 1960, now allows us to talk about the possibility of introducing into school, starting from the 1st grade, a systematic mathematics course that provides knowledge about quantitative relationships and dependencies of quantities in algebraic form.
1.2. PSYCHOLOGICAL BASIS FOR INTRODUCING ALGEBRAIC CONCEPTS IN PRIMARY SCHOOL.
Recently, when modernizing programs, special importance has been attached to laying a set-theoretic foundation for the school course (this trend is manifested both here and abroad). The implementation of this trend in teaching (especially in the elementary grades, as is observed, for example, in an American school) will inevitably raise a number of difficult questions for child and educational psychology and for didactics, because now there are almost no studies revealing the features of a child’s assimilation of the meaning of set (unlike mastering counting and numbers, which has been studied very comprehensively).
Logical and psychological research in recent years (especially the work of J. Piaget) has revealed the connection between some mechanisms of children's thinking and general mathematical concepts. Below we specifically discuss the features of this connection and their significance for the construction of mathematics as an educational subject (we are talking about the theoretical side of the matter, and not about any particular version of the program).
The natural number has been a fundamental concept in 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. In this case, it is assumed that while becoming familiar with numbers, the child simultaneously discovers for himself the initial features of quantitative relationships. Counting and number are the basis for 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 number, at the same time inadequately express its connection with other mathematical concepts, and inaccurately assess the place and role of 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 arise. It is necessary to specifically consider the actual connection of the concept of number with other concepts.
Many general mathematical concepts, and in particular the concepts of equivalence relations and order, are systematically considered in mathematics regardless of the numerical form. These concepts do not lose their independent character; on their basis, it is possible to describe and study a particular subject - various numerical systems, concepts that in themselves do not cover the meaning and meaning of the original definitions. Moreover, in the history of mathematical science, general concepts developed precisely to the extent that “algebraic operations,” a well-known example of which are the four operations of arithmetic, began to be applied to elements of a completely non-numerical nature.
Recently, attempts have been made to expand the stage of introducing a child to mathematics in teaching. This tendency finds its expression in methodological manuals, as well as in some experimental textbooks. Thus, in one American textbook intended for teaching children 6-7 years old, on the first pages tasks and exercises are introduced that specifically train children in establishing the identity of subject groups. Children are shown the technique of connecting sets, and the corresponding mathematical symbolism is introduced. Working with numbers is based on basic knowledge about sets. The content of specific attempts to implement this trend can be assessed differently, but it itself is quite legitimate and promising.
At first glance, the concepts of “relation,” “structure,” “laws of composition,” and other existing complex mathematical definitions cannot be associated with the formation of mathematical concepts in young children. Of course, the entire true and abstract meaning of these concepts and their place in the axiomatic structure of mathematics as a science is an object of assimilation for a head that is already well developed and “trained” in mathematics. However, some properties of things fixed by these concepts, one way or another, appear to the child relatively early: there is specific 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 develops complex systems of general ideas about the world around him and lays the foundation for meaningful and objective thinking. Moreover, based on relatively narrow empirical material, children identify general patterns of orientation in the spatio-temporal and cause-and-effect dependencies of things. These diagrams serve as a kind of framework for that “coordinate system”, within which the child begins to increasingly 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. They, figuratively speaking, are an intuitive form of organizing the child’s behavior (although, of course, they are increasingly reflected in judgments).
In recent decades, the issues of the formation of children's intelligence 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 colleagues. Some of his works are directly related to the problems of developing a child’s mathematical thinking, and therefore it is important for us to consider them in relation to issues of curriculum design.
In one of his latest books (17), J. Piaget provides 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 performing an inclusion operation (for example, A+A1=B) and its inverse operation (B- A1=A). seriation is the ordering of objects into systematic rows (for example, sticks of different lengths can be arranged in a row, each member of which is larger than all previous ones and smaller than all subsequent ones).
Analyzing the formation of classification, J. Piaget shows how from the initial form, from the creation of a “figurative aggregate” based only on the spatial proximity of objects, children move on to a classification based on the relationship of similarity (“non-figurative aggregates”), and then to the most complex form - to the inclusion of classes, determined by the connection 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 characteristics, and about developing in children the ability to change the basis of classification when adding new elements.
These studies pursued a very specific goal - to identify the patterns of formation of operator structures of the mind and, first of all, such a 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 understanding one of these directions causes ipso facto (by virtue of the fact itself) understanding of the other (17-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. Reversal occurs, for example, in the case when the spatial movement 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 and its inverse is an identical operation, or a zero transformation).
Reciprocity (or compensation) involves 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 was not annulled here, but it was compensated by the corresponding movement of its own body - and this is already a different form of transformation than circulation (17-p. 16). J. Piaget believes that the 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) allows us to accurately correlate the operator structures of thinking with algebraic structures, order structures and topological ones (17-p. 17) . 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 elements of a group also gives an element of the group; a direct operation corresponds to one and only one inverse operation; there is an identity operation; successive compositions are associative. In the language of intellectual actions this means:
The coordination of two systems of action constitutes a 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 is considered unchanged.
Let us consider the main provisions formulated by J. Piaget in relation to the issues of constructing a curriculum. First of all, the research of J. Piaget shows that during the period of 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 positions. Moreover, already at the stage of specific operations (from the age of 7), 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 Piaget allows us to draw a number of significant conclusions in relation to the design of a mathematics curriculum. First of all, factual data on the formation of a child’s intellect from 2 to 11 years old suggests that at this time not only are the properties of objects described through the mathematical concepts of “structure-relationship” not “alien” to him, but they themselves organically enter into the child’s thinking.
Traditional programs do not take this into account. Therefore, they do not realize many of the opportunities hidden in the process of a child’s intellectual development. By the age of 7, children have already sufficiently developed a plan for mental actions, and by teaching an appropriate program in which the properties of mathematical structures are given “explicitly” and children are given the means of analyzing them, it is possible to quickly bring children to the level of “formal” operations than in the time frame in which this is carried out during the “independent” discovery of these properties. 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 the ages of 7-11, are themselves inextricably linked with the forms of organization of learning characteristic of traditional elementary school.
Thus, at present there is factual data showing the close connection between the structures of children's thinking and general algebraic structures. The presence of this connection opens up fundamental possibilities for constructing an educational subject that develops according to the scheme “from simple structures to complex combinations.” This method can be a powerful lever for developing in children such thinking that is based on a fairly strong conceptual foundation.
1.3. THE PROBLEM OF THE ORIGIN OF ALGEBRAIC CONCEPTS AND ITS IMPORTANCE FOR THE CONSTRUCTION OF AN EDUCATIONAL SUBJECT.
The division of the school mathematics course into algebra and arithmetic is conditional. The transition occurs gradually. One of the central concepts of the initial course is the concept of a natural number. It is interpreted as a quantitative characteristic of the class of equivalent sets. The concept is revealed on a specific basis as a result of operating the set and measuring quantities. It is necessary to analyze the content of the concept “quantity”. True, another term is associated with this term - “dimension”. In general use, the term quantity is associated with the concepts “equal”, “more”, “less”, which describe a wide variety of qualities. A set of objects is only transformed into a quantity when criteria are established that make it possible 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. Moreover, for any two elements A and B, one and only one of the relationships holds : A=B, A B, A B.
V.F. Kogan identifies the following eight basic properties of the concepts “equal”, “more”, “less”.
1) at least one of the relationships holds: A=B, A B, A B;
2) if the relation A=B holds, then the relation A B does not hold;
3) if A=B holds, then the relation A B does not hold;
4) if A=B and B=C, then A=C;
5) if A is B and B is C, then A is C;
6) if A C and B C, then A C;
7) equality is a reversible relationship: A=B B=A;
8) equality is a reciprocal relation: whatever the element A of the set under consideration, A = A.
“By establishing comparison criteria, we transform multitude into magnitude,” wrote V.F. Kogan. In practice, a quantity usually denotes not the very set of elements, but a new concept introduced to distinguish comparison criteria (the name of the quantity. This is how the concepts of “volume”, “weight”, “length”, etc. arise. “At the same time, for a mathematician the value is completely defined when many elements and comparison criteria are indicated,” noted V.F. Kogan.
This author considers the natural series of numbers as the most important example of a mathematical quantity. From the point of view of such a comparison criterion as the position occupied by numbers in a series (occupies one place, follows ..., precedes ...), this series satisfies the postulates and therefore represents a quantity. Working with quantities (it is advisable to record individual values with letters), you can perform a complex system of transformations, establishing the dependence of their properties, moving from equality to inequality, performing addition and subtraction. 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 special study for the child even before the numerical form of describing the ratio of quantities is introduced? They can serve as prerequisites for the subsequent detailed introduction of 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 a quantity as a subject of mathematical consideration, familiarity with methods of comparison and symbolic means of recording its results, with techniques for analyzing the general properties of quantities. An initial section of the course is needed that would introduce children to basic algebraic concepts (before introducing numbers). What are the main key themes of such a program?
Topic 1. Leveling and completing objects (by length, volume, weight, composition of parts and other parameters).
Topic 2. Comparing objects and recording its results using the equality-inequality formula.
Tasks on comparing objects and symbolically designating the results of this action;
Verbal recording of comparison results (terms “more”, “less”, “equal”).
Written signs
Illustration of comparison results with a picture;
Designation of compared objects by letters.
Topic 3. Properties of equality and inequality.
Topic 4. Operation of addition (subtraction).
Topic 5. Transition from inequality of type A B to equality through the operation of addition (subtraction).
Topic 6. Addition and subtraction of equalities – inequalities.
With proper planning of lessons, improvement of teaching methods and a successful choice of teaching aids, this material can be fully mastered in three months.
Next, children become familiar with ways of obtaining a number that expresses the relationship of an object as a whole and its part. There is a line that is already implemented in grade 1 - transferring the basic properties of quantity and the operation of addition to numbers (integers). In particular, by working on the number line, children can quickly transform a sequence of numbers into a value. Thus, treating a number series as a quantity allows you to develop the very skills of addition and subtraction, and then multiplication and division, in a new way.
2.1. TEACHING IN PRIMARY SCHOOL FROM THE POINT OF VIEW 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 textbooks takes one and a half academic quarters. Secondary school mathematics teachers are dissatisfied with the preparation of primary school graduates. What is the reason for this situation? For this purpose, the most well-known elementary school mathematics textbooks today were analyzed: these are textbooks by the authors M.I. Moro, I.I. Arginskaya, N.B. Istomina, L.G. Peterson, V.V. Davydov, B.P. Geidman.
An analysis of these textbooks revealed several negative aspects, present to a greater or lesser extent in each of them and negatively affecting further learning. First of all, the assimilation of material in them is largely based on memorization. A clear example of this is memorizing the multiplication table. In elementary school, a lot of effort and time is devoted to memorizing it. But during the summer holidays the children forget it. The reason for such rapid forgetting is rote learning. Research by L.S. Vygotsky showed that meaningful memorization is much more effective than mechanical memorization, and the experiments conducted convincingly prove that material enters long-term memory only if it is remembered as a result of work corresponding to this material. When studying material in elementary school, reliance is placed on objective actions and illustrative clarity, which leads to the formation of empirical thinking. Of course, it is hardly possible to completely do without such visualization in primary school, but it should serve only as an illustration of this or that fact, and not as the basis for the formation of a concept. The use of illustrative clarity and substantive actions in textbooks often leads to the concept itself being “blurred.” For example, in the mathematics method of M.I. Moreau it is said that children have to perform division by arranging objects into piles or making a drawing for 30 lessons. With such actions, the essence of the division operation is lost as the inverse action of multiplication as a result of division is learned with the greatest difficulty and much worse than other arithmetic operations.
When teaching mathematics in elementary school, nowhere is there any talk about proving any statements. Meanwhile, remembering how difficult it will be to teach proof in high school, you need to start preparing for this already in the elementary grades. Moreover, this can be done on material that is quite accessible to primary schoolchildren. Such material, for example, can be the rule of 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 propaedeutics of algebra—work with letters and letter expressions. Most textbooks avoid using letters. As a result, children work almost exclusively with numbers for four years, after which, of course, it is very difficult to get used to working with letters. however, it is possible to provide propaedeutics for such work, to teach children to substitute a number instead of a letter in a letter expression already in elementary school. This is wonderfully done, for example, in the textbook by L.G. Peterson. From grade 1, alphabetic symbols are introduced along with numbers, and in some cases, ahead of them. All rules and conclusions are accompanied by a letter expression. For example, lesson 16 (grade 1, part 2) on the topic “Zero” introduces children to subtracting zero from a number and a number from itself and concludes with the following notation: a -0 = a a-a = 0
Lesson 30 on the topic “Comparison Problems” 1st grade includes work with comparison exercises of the form: a*a-3 c+4*c+5 c+0* c-0 d-1*d-2
These exercises force the child to think and look for evidence of the chosen solution.
2.2. COMPARING (CONTRASTING) CONCEPTS IN MATHEMATICS LESSONS.
The current program provides for the study in 1st grade of only two operations of the first stage: addition and subtraction. Limiting the first year of study to only two actions is, in essence, a departure from what was already 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 capabilities of first-graders. It is also worthy of attention that in schools in other countries, where education begins at the age of 6, the first school year includes initial acquaintance with all four operations of mathematics. Mathematics relies primarily on four actions, and the sooner they are included in the student’s thinking practice, the more stable and reliable the subsequent development of the mathematics course will be.
In the first versions of the textbook by M.I. Moro for grade 1, multiplication and division were provided. However, the authors persistently held on to one “novelty” - coverage in the 1st grade of all cases of addition and subtraction within 100. But, since there was not enough time to study such an expanded volume of information, it was decided to shift multiplication and division completely to the next year of study. So, the fascination with the linearity of the program, i.e. purely quantitative expansion of knowledge (the same actions, but with larger numbers), took the time that was previously allocated for qualitative deepening of knowledge (studying all four actions within two dozen). Studying multiplication and division already in 1st grade means a qualitative leap in thinking, since it allows you to master condensed 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 the logical analysis of the question: the fact is that the complete table of addition of single-digit numbers is expanded within two tens (0+1= 1… 9+9=18). Thus, numbers within 20 form a complete system of relations in their internal connections; Hence, the expediency of preserving “20” in the form of a second holistic theme is clear (the first such theme is actions within the first ten). The case under discussion is precisely one where concentricity (preserving the second ten as a special theme) turns out to be more beneficial than linearity (dissolving the second ten in the “Hundred” theme).
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 examines actions within ten. There are experimental textbooks where the joint study of the numbering of the composition of numbers and actions is carried out within 10 at once in one section (Erdniev P.M.).
In the first lessons, the teacher should set the goal of teaching the student to use pairs of concepts, the content of which is revealed in the process of composing corresponding sentences with these words: more - less, longer - shorter, higher - lower, heavier - lighter, thicker - thinner, right - left , further - closer, etc. When working on pairs of concepts, it is important to use children's observations. Teaching the comparison process can be made more interesting by introducing so-called table exercises. Here the meaning of the concepts “column” and “row” is explained. The concept of left column and right column, top row and bottom row is introduced. Together with the children we show the semantic interpretation of these concepts. Such exercises gradually accustom children to spatial orientation and are important when subsequently studying the coordinate method of mathematics. Working on the number series is of great importance for the first lessons. It is convenient to illustrate the growth of a number series by adding one by one by moving to the right along the number line. If the (+) sign is associated with moving along the number line to the right by one, then the (-) sign is associated with the reverse movement to the left by one. (That’s why we show both signs at the same time in one lesson). Working on the number series, we introduce the following concepts: the beginning of the number series (the number zero) represents the left end of the ray; The number 1 corresponds to a unit segment, which must be depicted separately from the number series. Children work within three with the number beam. We select two adjacent numbers 2 and 3. Moving from number 2 to number 3, children reason like this: “The number 2 is followed by the number 3.” Moving from the number 3 to the number 2, they say: “Before the number 3 comes the number 2” or “The number 2 comes before the number 3.” This method allows you to determine the place of a given number in relation to both the previous and subsequent numbers; 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). The indicated transitions along the number series must be associated with the corresponding arithmetic operations. For example, the phrase “The number 2 is followed by the number 3” is symbolically depicted as follows: 2+1=3; however, it is psychologically beneficial to create the opposite connection: “Before the number 3 there is a number 2” and the entry: 3-1=2. To gain an understanding of the place of a number in a number series, paired questions should be asked:
1)What number is followed by the number 3? What number does the number 2 come before?
2) what number comes after the number 2? What number comes before the number 3? Etc.
It is convenient to combine working with a number series with comparing numbers by magnitude, as well as comparing the position of numbers on the 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; means 4 is greater than 3. And vice versa: the number 3 is to the left of the number 4, which means the number 3 is less than the number 4. This establishes a connection between pairs of concepts: to the right is more, to the left is less.
From the above, we see a feature of the integrated assimilation of knowledge: the entire set of concepts associated with addition and subtraction are offered together, in continuous transitions into each other. Learning experience shows the benefits of introducing pairs of mutually opposing concepts simultaneously, starting from the very first lessons. So, for example, the simultaneous use of three verbs: “add (add 1 to 2), “add” (add the number 2 with the number 1), which are depicted symbolically identically (2 + 1 = 3), helps children learn the similarity and proximity of these words by meaning (similar reasoning can be made regarding the words “subtract”, “subtract”, “reduce”.
Long-term tests have shown the advantages of monographic study of the first ten numbers. Each successive number is subjected to multilateral analysis, with all possible options for its formation being enumerated; within this number, all possible actions are performed, “all mathematics” is repeated, all acceptable grammatical forms of expressing the relationship between numbers are used. Of course, with this system of study, in connection with the coverage of subsequent numbers, previously studied examples are repeated, i.e. expansion of the number series is carried out with constant repetition of previously discussed 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 on these two operations are usually considered separately. But the simultaneous study of the dual operation “addition-decomposition into terms” is more preferable. Such work can be constructed as follows. Let the children solve the addition problem: “Add 1 stick to 3 sticks and you get 4 sticks.” Following this, we immediately pose the question: “What numbers does the number 4 consist of?” 4 sticks consist of 3 sticks (the child counts 3 sticks) and 1 stick (separates another 1 stick). The initial exercise can be decomposition of a number. The teacher asks the question: “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 do you get if you add 2 to 3?” (add 2 to 3 and you get 5). For the same purpose, it is useful to practice reading examples in two directions: 5+2=7. Add two to five and you get seven. (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 classroom abacus, which allow you to see the specific content of the corresponding operations. Calculation on the abacus is indispensable as a means of visualizing actions on numbers, and the value of a number within 10 is associated here with the length of the set of bones on one wire (this length is perceived visually by the student. So, when solving the example of addition (5+2=7), the student first counted by there are 5 stones in the abacus, then he added 2 to them and after that announced the sum: “Add 2 to 5 - you get 7” (the name of the resulting number 7 is determined by the student by recalculating the new set: 1-2-3-4-5-6- 7).
Student: Add 2 to 5 and you get 7.
Teacher: Show me what terms the number 7 consists of?
The student separates 2 bones to the right. The number 7 is 2 and 5. When performing these exercises, it is advisable to use from the very beginning the concept of “first term” (5), “second term” (2), “sum” (7). The following types of tasks are offered:
a) the sum of two terms is 7, find them;
c) what terms does the number 7 consist of?
c) decompose the sum 7 into 2 terms, 3, etc.
Mastering 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 are there in total?
Student: There are 5 sticks in total.
Teacher: How can I say more about this?
Student: Add 2 to 2 sticks - there will be 5 sticks.
Teacher: Make up this example using cut numbers. (the student makes an example from numbers).
Teacher: Now swap the chopsticks: from the left to the right, and from the right to the left. How many sticks are there in both hands now?
Student: There were only 5 in two hands, and now it’s 5 again.
Teacher: Why did this happen?
Student: Because we didn’t put aside or add sticks anywhere. As much as there was, so much remains.
The commutative law is also learned in exercises on decomposing a number into terms. When to introduce the displacement law? The main goal of teaching addition, already within the first ten, is to constantly emphasize the role of the commutative law in exercises. Let the children count 6 sticks, then add 3 sticks to them and by recalculation (seven-eight-nine) establish the sum: 6 and 3 will be 9. We immediately offer a new example: 3+6: a new sum can be established by recalculation, but gradually and purposefully a solution method should be formed in higher code, i.e. logically, without recalculation. If 6 yes 3 is 9 (answer recalculated), then 3 yes 6 (without recalculation) is 9.
L.G. Peterson introduces this method already in lesson 13, where children solve four expressions in letter symbols (T+K=F K+T=F F-T=K F-K=T), and then in numerical form: 2+1=3 1+2=3 3-2=1 3-1+2.
Compiling four examples is a means of expanding knowledge accessible to children. We see that the characterization of the addition operation should not happen sporadically, but should become the main logical means of strengthening correct numerical associations. The main property of addition - the mobility of terms - must be considered constantly in connection with the accumulation of new tabular results in memory. We see: the interconnection of more complex computational or logical operations by which a pair of “complex operations” are performed. The explicit opposition of complex concepts is based on the implicit 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 problems (3 tasks in each cycle):
1 a), b) multiplication with a constant multiplicand and division by content (together); c) division into equal parts.
2 a), b) decreasing and increasing the number several times (together), c) multiple comparison;
3 a), b) finding one part of a number and a number by the size of one of its parts (together) c) solving the problem “What part is one number of another?” Simultaneous study of multiplication and division in content. In lessons 2-3 devoted to multiplication, the meaning of the concept of multiplication as a condensed addition of equal terms is clarified. Typically, students are shown an entry on replacing addition with multiplication: 2+2+2+2=8 2*4=8 Here is the connection between addition and multiplication. It would be appropriate to immediately suggest an exercise designed to trigger the “multiplication-addition” feedback. Looking at this entry, the student should understand that the number 2 must be repeated as a addend as many times as the multiplier in the example 2*4=8 shows. The combination of both types of exercise is one of the important conditions that ensures the conscious assimilation of the concept of “multiplication.” It is very important to show for each of the corresponding cases of multiplication the corresponding case of division. In the future, it is beneficial to consider multiplication and division together.
When introducing the concept of division, it is necessary to recall the corresponding cases of multiplication in order to build on them to create the concept of a new action inverse to multiplication. Therefore, 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 “collapsed subtraction”, replacing the sequential “subtraction by 2 " The meaning of multiplication is comprehended not so much through the multiplication itself, but through constant transitions between multiplication and division, since division is a veiled, “altered” multiplication. All logical operations supported by practical activities must be well thought out. The result of the work will be multiplication and division tables:
By 2*2=4 4:by 2=2
2*3=6 6: 2=3 each
2*4=8 8: 2=4 each, etc.
The multiplication table is built using a constant factor of 1, and the division table is built using a constant divisor. The study of division into equal parts is introduced after the study of multiplication and division by 2. The task is given: “Four students brought 2 notebooks. How many notebooks did you bring?" When performing a practical activity, we collect notebooks (take 2 notebooks 4 times). Let’s create an inverse problem: “8 notebooks were distributed, 2 notebooks were distributed to each student.” The result is 4. The entry appears for 2t.*4=8t., 8t.: for 2t.=4t. At first, it is useful to write down the names in detail. Now we make up the 3rd task: “8 notebooks must be distributed equally to 4 students. How many notebooks will each person get? At first, division into equal parts should also be demonstrated on objects. Therefore, 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 condensed subtraction, replacing the sequential “subtraction by 2”. In this case, the explanation in the mathematics textbooks by L.G. Peterson and N.B. Istomina was very successfully constructed. a new concept is introduced into teaching using the activity method, i.e. children themselves “discover” its content, and the teacher guides their research activities and introduces them to generally accepted terminology and symbols. First, children repeat the meaning of multiplication and compose the product 2*4=8 from the picture. The learning of division actions is motivated by children's everyday practical activities. The teacher asks if in your life you have had to divide something equally, and offers a task: “We need to divide 36 candies equally among four people. How much should I give each? the difficulty that arises in connection with answering the question of the problem motivates research using subject models. Each person has 36 items (buttons, figures, tokens, etc.) prepared on their desks. They are laid out into 4 equal-sized piles, etc. The teacher shows the entry _ - divide into equal parts - this means finding the number of objects in each part. By completing a series of exercises, children come to the conclusion that the division operation is the inverse of the multiplication operation. When dividing nuts by 4, we get the number 2, which when multiplied by 4 gives us 8. 8:4=2 2*4=8. About the sign, children can be told that it is used in mathematics to designate sentences that express the same thing (equivalent sentence). While performing consolidation exercises, children make drawings and draw support diagrams.
At the end of the lesson, a conclusion is drawn and spoken out loud and extended to the general case of division - in order to divide the number a by the number b, you need to select a number c that, when multiplied by b, gives a:
A:B=C C*B=A and a supporting outline is drawn up. It is important to convey to children that mathematical expressions and formulas make it possible to identify general patterns and establish an analogy for phenomena that are completely different at first glance. Awareness of this fact will help students further understand the appropriateness of mathematical generalizations, the role and place of mathematics in the system of sciences.
CHAPTER 3. RESEARCH WORK ON STUDYING ALGEBRAIC MATERIAL IN MATHEMATICS LESSONS IN PRIMARY CLASSES of Secondary Educational Institution No. 72 WITH IN-DEPTH STUDY OF INDIVIDUAL SUBJECTS.
3.1. RATIONALE FOR THE USE OF INNOVATIVE TECHNOLOGIES (UDE TECHNOLOGY).
In my work, I successfully use the technology of enlargement of didactic units (UDE), developed by P.T. Erdniev. The author put forward the scientific concept of a “didactic unit” more than 30 years ago. His system of consolidating didactic units in primary school equips schoolchildren with an algorithm for creative development of educational information. This technology is relevant and promising, as it has the power of long-range action, instills in the child the traits of intelligence, and contributes to the formation of an active personality.
P.M. Erdniev identifies four main ways to enlarge didactic units:
1) joint and simultaneous study of interrelated actions and operations;
2) the use of deformed exercises;
3) widespread use of the inverse problem method;
4)increasing the proportion of creative tasks.
Each of the methods contributes to the actualization of thinking reserves. The first way is to jointly study interrelated actions, operations - addition - subtraction, multiplication - division. In the first grade, studying the first ten, children get acquainted with examples of the form: 3+4=7 using the technology of enlargement of didactic units, I introduce the commutative property of addition: 4+3=7 the answer is the same, the record takes the form: 3+4= 7
I offer children examples of subtraction, and the notation looks like this: 7 -3=4
4=3. Knowledge is summarized and combined and records are brought together. Similarly, you can construct work on multiplication and division. For example: 8+8+8+8+8=40 8*5=40 5*8=40 40:5=8 40:8=5
Children learn to distinguish between opposing concepts and operations while simultaneously studying related actions. “Nervous habits,” according to K.D. Ushinsky, are fixed in a person not separately, but in pairs, rows, rows, groups. This presentation of material creates conditions for the development of independence and initiative in children.
The second way to enlarge didactic units is the method of deformed exercises, in which the required element is not one, but several elements. For example, in the first grade you can offer a task where you need to determine the sign of the action and the unknown component: 8 = 2. In such examples, the student first selects the sign of the action based on the comparison and then finds the missing component. When solving such an example, the child reasons as follows: 8 2, which means the minus sign. 8 consists of 2 and 6, which means the example is 8-6 = 2. This way, attention is activated and students’ thinking develops based on solving logical chains.
The third way to enlarge didactic units is to solve a direct problem and transform it into inverse and similar ones. Solving problems in elementary school is of central importance for the development of students' thinking: when solving, children become familiar with the dependence of quantities, with various aspects of life, learn to think, reason, and compare. When teaching problem solving, it is necessary to teach children how to create inverse problems. Each method is based on the great information law of living nature - the law of feedback. When working on tasks, it is advantageous to use when in a series of tasks the next one differs from the previous one in only one element. In this case, the transition from one problem to another is easier, and the information obtained from solving the previous problem helps in finding solutions to subsequent problems. This technique is especially useful for weak and slow children. For example, a problem to find a sum, let’s create its inverse problems. “Father gave Masha 11 apples, and mother added 5 more apples. How many apples did Masha’s parents give in total?”
- We carry out analysis on the questions: “What is known in the problem? What do you need to know? Write down the task briefly. How can you find out how many apples Masha’s parents gave her? (12+5=17)
- Drawing up an inverse problem, where the unknown is the number of apples given by the father. “Father gave several apples, and mother added 5 more apples. In total, Masha now has 17 apples. How many apples did Masha’s father give?”
- You can create another inverse problem, where the unknown will be the number of apples given to Masha by her mother. “Father gave Masha 12 apples, and mother added a few more apples. In total, Masha now has 17 apples. How many apples did Masha’s mother give?” (17-12=5). In notebooks we keep short notes on all 3 tasks. Interrelated tasks merge into a group of related tasks as a large unit of assimilation and form three tasks. So, the main technological novelty of the system for enlarging didactic units is the presence of tasks for which the student practices independently composing inverse problems based on an analysis of the conditions of the direct problem, identifying a logical chain.
The fourth method of consolidation is to increase the share of creative tasks. For example, a task is given with a “window”: +7-50=20. Children are looking for the answer using the selection method, but you can solve this task by reasoning along the arrow, using the inverse operation: 20+59-7=63. The required number is 63. Creative tasks must be present in every lesson. With the help of such exercises, the child becomes accustomed to the independent continuation of thought, to the restructuring of judgment, which is of decisive importance in the future for the formation of an active, creative mind of a person, so valuable in its manifestation in any field of work.
3.2. ABOUT THE EXPERIENCE OF ABOUT ALGEBRAIC CONCEPTS.
Already in the 1st grade, I teach children to independently establish signs by which they can compare certain objects. The teacher shows the children 2 weights of different colors. “By what criteria can they be compared?” The children give the answer: “They can be compared by weight, height, bottom.” What can we say? - they are unequal (in weight, height). How to express this more precisely? - the black weight is heavier, larger, thicker. What does heavier mean? - Heavier, more in weight. Similar work with leading questions is carried out in relation to other characteristics. Together with the teacher, we establish that “heavier” means more in weight, “longer” means more in length (height, height), etc. The conclusion of this work was to find out that if you can find a sign by which objects are compared, then they will be either equal or unequal. This can be written with special signs “=” and “=”. L.G. Peterson very successfully compares these concepts, and only then the signs are clarified - less or more. Children are very willing to solve these inequalities. We also carry out reverse tasks - different objects are selected using the “less than” or “greater than” signs. In this case, a unique task immediately arises - defining the concepts “from left to right” - 5 is less than 10. In addition, it is successfully possible to write not only with numbers, but also with different figures and lines. During this period, the letter form of recording is introduced on this basis. When working with various kinds of tasks, it is necessary to give children the understanding that the letters themselves do not write down the result of a comparison; they need a sign connecting them. And only the whole formula speaks about this result - a comparison of the weight, length of 2 objects or more.
Work on this topic is of paramount importance for the development of the entire initial section of mathematics, since it is essentially connected with the construction in the child’s activity of a system of relations that identifies quantities as the basis for further transformations. Literal formulas, replacing a number of preliminary recording methods, for the first time transform these relations into an abstraction, because the letters themselves denote any specific values of any specific quantities, and the entire formula is any possible relationship of equality or inequality of these values. Now, relying on formulas, you can study your own properties of the selected relations, turning them into a special subject of analysis.
- DIAGNOSTICS OF MATHEMATICS TRAINING RESULTS.
The importance of diagnostics is great, since with its help it is established that the child’s achievements meet the mandatory requirements for learning outcomes. By analyzing the results, we can draw conclusions about what changes occur with the child during the learning process, why it was not possible to teach, what was not taken into account, how to adjust the learning process, what kind of help the student needs. Tests can serve as a diagnostic tool. For each content line, in accordance with the mandatory minimum content of primary education, test tasks are compiled, and such tests are also widely presented in ready-made printed publications. They help identify learning gaps. In my class, the following problems were identified in the study of algebra elements:
Some students experience some difficulties when solving letter expressions (finding the numerical value of a letter expression given the given values of the letters included in it);
When solving equations, mistakes are made in using the rules for finding unknown components (dependence between the components of addition, subtraction, multiplication and division);
When checking the roots of an equation, some children do not calculate the left side of the equation, but automatically put an equal sign;
With a more complex structure of equations of the form X+10=30-7 or X+(45-17)=40, when transforming and simplifying the equation, some children lose the variable, getting carried away with arithmetic calculations.
Having received the test data and analyzed the results, I make a work plan for myself to correct gaps and shortcomings.
A sample test to test students' knowledge.
- Add to 10 9, 5, 8, 4, 7, 0.
- Write the number on the card: 8+5 17-9
8+2+ 17-7-
- Guess what number should be written on the card:
3, 6, 9, 12, * A(13), B(15), C(18), G(other number)
- Write a number on the card so that the equality is true:
9=17-* A(6), B(15), C(4), G (another number)
- . 8+7=19-* A(3), B(15), C(4), G(another number).
6 Indicate the correct equalities:
A) 12+1=11 B)14-5=9 C)17+3=20 D)20-1=9 E)18+2=20 F)8-5=13 H)6+9=15
7. Arrange the expressions in decreasing order of their values: A)7-5 B)7+6 C)3+7
8. What numbers can replace *?
1)12 1* A(0, 1, 2) B(3, 4, 5, 6, 7, 8, 9) C(0, 1)
9. Where is the correct order of actions? A) 12-3+7 B) 19-9-5+3
10.Write down numerical expressions and find the values: from the number 12, subtract the sum of the numbers 3 and 5
A) (3+5)-12 B) 12-3+5 C) 12-(3+5) D) other answer:
This test shows which of the children has not clearly mastered the numbering of the second ten numbers. These are children who received less than 18 points. Corrective work needs to be carried out with them, which includes all possible cases of using the acquired knowledge, where children navigate similar exercises quite well. A plan for working with the parents of these children is outlined and consultation is provided for those parents who need it. The final diagnostic tests the knowledge of the entire 1st grade course of study. I do another work with them to test their mastery of addition and subtraction of numbers within 20, and then 100. Children should be able to perform actions using the techniques they have learned: find the unknown component of addition and subtraction, compare numbers and numerical expressions, be able to find the inverse action . As for the programs of other authors, it can be observed that the early introduction of algebraic material is quite acceptable for all children. Having worked through different programs and studied the teaching methods of different mathematics authors, I use all the elements I need from any textbook to make the lesson more effective and productive. Interesting exercises that develop thinking, logic, teach you to think, invent, and combine are included in every mathematics lesson. My children's favorite subject is mathematics. The use of printed notebooks and screening tests helps identify gaps in knowledge.
When studying all content areas of mathematics, learning results are constantly monitored and teaching diagnostics are carried out. Children constantly perform intermediate tests and assessments, so it is easy to monitor student progress.
In elementary school, during grade-free education (1-2 grades), I use the following levels and criteria for the development of knowledge of algebraic material: high level (20-25 points) - at this level, the child consciously masters the studied material, concepts on the topic have been mastered, and can work independently on the topic , completes tasks without errors;
average level (14-9 points) - the topic has been mastered, can answer indirect questions, answers correctly on the topic with the help of leading questions, makes 1-2 mistakes, finds them and corrects them independently;
low level (less than 14 points) - makes mistakes in most tasks, does not always answer the teacher’s direct question correctly, corrective exercises and additional individual work are necessary.
Also, when processing diagnostic work, I conduct an element-by-element analysis of the test results: errors and the reasons for their occurrence. When solving equations (in the process of searching for a number, the substitution of which turns the equation into a correct numerical equality), the following errors are possible and occur:
In choosing an arithmetic operation when finding an unknown component (the reason for such an error is the inability to determine the relationship between components or ignorance of this material);
Computational errors (reasons in the use of addition, subtraction, multiplication and division algorithms; a detailed analysis was not carried out at some stage of the algorithm).
When solving literal expressions with given values of the letters included in it, the following errors are made:
When using algorithms (specific computational techniques);
With a specific choice of a given letter value (carelessness, no analysis of the correspondence of a given letter to a certain number was carried out).
When comparing numbers and numerical expressions they make mistakes:
In the formulation of more and less signs (the reason is ignorance of specific concepts, the bitwise and class composition of numbers has not been analyzed, ignorance of the numbering of natural numbers, the place meaning of numbers);
In arithmetic calculations.
When finding the value of a compound numeric expression, errors are made:
In order of action,
Incorrect recording of action components (cause of errors - failed to determine the structure of the original expression and accordingly apply the necessary rule, did not know the algorithm for performing actions). By carefully analyzing the results of monitoring knowledge, abilities, skills, the teacher identifies gaps and errors in performance, and further work can be correctly planned to eliminate deficiencies in training.
Below are examples of tests and diagnostics of the sections and checks performed.
Test number | Developed skills and abilities |
10-11 | The score is within 20, 100. Addition and subtraction table. Finding the value of a numerical expression in 2-4 steps. Read, write, compare within 100. The name and designation of the operations of addition and subtraction. Solving problems in 1-2 steps. Ability to compare and classify. Spatial representations. Knowledge of quantities. Level of formation of basic skills and mathematical development. |
Final diagnostic results for 1st grade
10-11 | level |
|||||||||||
Antonov A. Batraeva D. Bashlovkin D. Belova V. Bobyleva E. Gabrielyan G. Gasnikova M. Goroshko A. Guzaeva E. Dvugrosheva M. Kondratyev D. Konstantinov I. Kopylov V. Mikhailova V. Mikhailova I. Morozova A. Podgorny I. Razin N. Romanov D. Sinitsyna K. Suleymanov R. Sulyoznov A. Teplyakova Yu. Frolov D. Shirshaeva K. | Short Short Average Average High Average Average High High Short High High High High Average High Short Average Average High High Average Average Average average |
Checking the level of memory development
auditory | visual | motor | Visual-auditory |
||
Antonov A. Batraeva D. Bashlovkin D. Belova V. Bobyleva E. Gabrielyan G. Gasnikova M. Goroshko A. Guzaeva E. Dvugrosheva M. Kondratyev D. Konstantinov I. Kopylov V. Mikhailova V. Mikhailova I. Morozova A. Podgorny I. Razin N. Romanov D. Sinitsyna K. Suleymanov R. Sulyoznov A. Teplyakova Yu. Frolov D. Shirshaeva K. | 0.4 average 0.2 low 0.6 average 0.8average 1 high 0.7 average 0.7 average 1 high 1 high 0.5 low 1 high 1 high 1 high 1 high 0.9 average 1 high 0.4 low 0.7 average 0.7 average 1 high 1 high 0.7 average 1 high 0.7 average 0.6 average | 0.4 low 0.3 low 0.8 average 0.9 average 1 high 0.6 average 1 high 1 high 1 high 0.4low 1 high 1 high 1 high 1 high 1 high 1 high 0.4low 0.9average 1 high 1 high 1 high 0.8average 0.9average 0.9 average 0.8average | 0.8 average 0.4 low 1 high 1 high 1 high 0.9average 1 high 1 high 1 high 0.8average 1 high 1 high 1 high 1 high 1 high 1 high 0.5low 0.8average 0.7 average 1 high 0.9 average 0.8average 1 high 0.8average 0.5low | 0.7 average 0.4 low 0.9 average 0.9 average
0.8 average 0.9 average
0.5 low
0.4 low 0.9 average 0.9 average
0.8 average 0.9 average 0.8 average 0.5 average |
С=а:N С – memory coefficient, at С=1 – optimal option – high level
C=0.7 +/-0.2 - average level, C - less than 0.5 – low level of development
CONCLUSION
Currently, quite favorable conditions have arisen for a radical improvement in the organization of mathematics education in primary school:
- the primary school was converted from a three-year to a four-year school;
- hours are allocated for studying mathematics in the first four years, i.e. 40% of the total time devoted to this subject throughout high school?
- Every year, an increasing number of people with higher education work as primary school teachers;
- The possibilities for better providing teachers and schoolchildren with educational and visual aids have increased; most of them are produced in color.
There is no need to prove the decisive role of initial mathematics instruction for the development of a student’s intelligence in general. The wealth of various associations acquired by a student during the first four years of study, if done correctly, becomes the main condition for self-expansion of knowledge in subsequent years. If this stock of initial ideas and concepts, trains of thought, basic logical techniques is incomplete, inflexible, and impoverished, then when moving to high school, schoolchildren will constantly experience difficulties, regardless of who will teach them next or what textbooks they will study from.
As you know, primary schools have been functioning in our and other countries for many centuries, therefore the theory and practice of primary education are much richer in traditions than education in high schools.
Precious methodological discoveries and generalizations on primary mathematics teaching were made by L.N. Tolstoy, K.D. Ushinsky, V.A. Latyshev and other methodologists already in the last century. Significant results have been obtained in recent decades using the methods of elementary mathematics in the laboratories of L.V. Zankov, A.S. Pchelko, as well as in research on the consolidation of didactic units.
With reasonable consideration of the available scientific results obtained in the last 20 years using the methods of primary education by various creative teams, there is now every opportunity to achieve “learning with passion” in primary school. In particular, exposing students to basic algebraic concepts will undoubtedly have a positive impact on students' acquisition of relevant knowledge in high school.
BIBLIOGRAPHICAL LIST
- Current problems in teaching mathematics in primary school./Ed. M.I.Moro, A.M.Pyshkalo. -M.: Pedagogy, 1977.
- I.I. Arginskaya, E.A. Ivanovskaya. Mathematics: Textbook for grades 1,2,3,4 of a four-year primary school. - Samara: Publishing house. house "Fedorov", 2000.
- M.A. Bantova, G.V. Beltyukova. Methods of teaching mathematics in primary classes. - M.: Pedagogika, 1984.
- P.M. Erdniev. Integrated knowledge as a condition for joyful learning./ Elementary school. - 1999 No. 11, pp. 4-11.
- V.V. Davydov. Mental development in primary school age./ Ed. A.V. Petrovsky. - M.: Pedagogy, 1973.
- A.Z.Zak. Development of mental abilities of younger schoolchildren.
- I.M. Doronina. Using the UDE methodology in mathematics lessons. //Primary school.-2000, No. 11, p.29-30.
- N.B. Istomina. Methods of teaching mathematics in primary school. - M.: Publishing Center "Academy", 1998.
- M.I. Voloshkina. Activation of cognitive activity of junior schoolchildren in mathematics lessons.//Elementary school-1992 No. 10.
- V.F.Kogan. On the properties of mathematical concepts. -M. : Science, 1984.
- G.A.Pentegova. Development of logical thinking in mathematics lessons. //Elementary school.-2000.-No. 11.
- A.N. Kolmogorov. About the profession of mathematics. M.-Pedagogy. 1962.
- M.I.Moro, A.M.Pyshkalo. Methods of teaching mathematics in elementary school. - M. Pedagogy, 1980.
- L.G. Peterson. Mathematics grades 1-4. - Methodological recommendations for teachers - M.: “Ballas”, 2005.
- Diagnostics of the results of the educational process in a 4-year primary school: Educational and methodological manual / Ed. Kalinina N.V. / Ulyanovsk: UIPKPRO, 2002.
- Independent and test work for elementary school (-4). M. - “Ballas”, 2005.
- J. Piaget. Selected psychological works. SP-b.: Publishing house "Peter", 1999.
- A.V. Sergeenko. Teaching mathematics abroad. - M.: Academy, 1998.
- Stoilova L.P. Mathematics. M. - Academy, 2000.
- W.W. Sawyer Prelude to Mathematics, M.-Prosveshchenie.1982.
- Tests: Primary school. 1, 2, 3, 4 grades: Educational and methodological manual / L.M. Zelenina, T.E. Khokhlova, M.N. Bystrova and others - 2nd ed., stereotype. - M .: Bustard, 2004.
Send your good work in the knowledge base is simple. 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.
Posted on http://www.allbest.ru/
Methods for studying algebraic material
Lecture 1. Mathematical expressions
1.1 Learning the concept of "mathematical expression"
Algebraic material is studied starting from grade 1 in close connection with arithmetic and geometric material. The introduction of algebra elements promotes the communication of concepts about number, arithmetic operations, and mathematical relationships, and at the same time prepares children for studying algebra in subsequent grades.
The main algebraic concepts of the course are “equality”, “inequality”, “expression”, equation". There are no definitions of these concepts in the primary school mathematics course. Students understand these concepts at the level of ideas in the process of performing specially selected exercises.
The mathematics program in grades 1-4 provides for teaching children to read and write magmatic expressions: to familiarize them with the rules of the order of actions and teach them to use them in calculations, to familiarize students with identical transformations of expressions.
When forming the concept of a mathematical expression in children, it is necessary to take into account that the action sign placed between the numbers has a double meaning; on the one hand, it denotes an action that must be performed on numbers (for example, 6+4 - add 4); on the other hand, the action sign serves to indicate the expression (6+4 is the sum of the numbers 6 and 4).
The methodology for working on expressions involves two stages. At the first of them, the concept of simple expressions (sum, difference, product, quotient of two numbers) is formed, and at the second - about complex ones (sum about, products and numbers, difference of two quotients, etc.).
Introducing the first expression - the sum of two; numbers occurs in the 1st grade when studying addition and subtraction within 10. By performing operations on sets, children, first of all, learn the specific meaning of addition and subtraction, therefore, in entries of the form 5+1, 6-2, they understand the signs of actions as a short designation of words "add", "subtract". This is reflected in the reading (add 1 to 5 equals 6, subtract 2 from 6 equals 4). In the future, the concepts of these actions deepen. Students learn that adding a few units increases a number by the same number of units, and subtracting a number decreases it by the same number of units. This is also reflected in the new form of reading notes (4 increase by 2 equals 6, 7 decrease by 2 equals 5). Then children learn the names of the action signs: “plus”, “minus” and read examples, naming the action signs (4+2 =6, 7-3 =4),
After becoming familiar with the names of the components and the result of addition, students use the term "sum" to refer to the number that is the result of addition. Based on the children's knowledge of the names of numbers in addition, the teacher explains that in addition examples, a record consisting of two numbers connected by a plus sign is called the same as the number on the other side of the equal sign (9 the sum "6+3 is also a sum). It is clearly depicted like this:
In order for children to learn the new meaning of the term “sum” as the name of an expression, the following exercises are given: “Write down the sum of the numbers 7 and 2; calculate what the sum of the numbers 3 and 4 is equal to; read the entry (6 + 3), say what the sum is equal to; replace number by the sum of numbers (9= ?+?); compare the sums of numbers (6+3 and 6+2), say which one is greater, write it down with a “greater than” sign and read the entry.” In the process of such exercises, students gradually realize the double meaning of the term “sum”: in order to write down the sum of numbers, they must be connected with a “plus” sign; To find the value of the sum, you need to add the given numbers.
In approximately the same way, we are working on the following expressions: the difference, product and quotient of two numbers. However, now each of these terms is introduced immediately both as the name of the expression and as the name of the result of the action. The ability to read and write expressions and find their meaning using the appropriate action is developed through repeated exercises similar to exercises with sums.
When studying addition and subtraction within 10, expressions consisting of three or more numbers connected by the same or different action signs of the form are included: 3+1+1, 4-1-1, 2+2+2. By calculating the meanings of these expressions, children in expressions master the rule about the order of performing Actions in expressions without parentheses, although they do not formulate it. Somewhat later, children are taught to transform expressions in the process of calculations: for example: 7+5=3+5=8. Such entries are the first step in performing identity transformations.
Introducing first-graders to expressions of the form: 10 - (6+2), (7-4)+5, etc. prepares them to study the rules for adding a number to a sum, subtracting a number from a sum, etc., to write down solutions to compound problems, and also contribute to a deeper understanding of the concept of expression.
Methodology for introducing students to expressions of the form: 10+(6-2), (7+4)+5, etc. prepares them to study the rules for adding a number to a sum, subtracting a number from a sum, etc., to write down solutions to compound problems, and also contribute to a deeper understanding of the concept of expression.
The method of introducing students to expressions of the form: 10+(6-2), (5+3) -1 can be different. You can immediately teach how to read ready-made expressions by analogy with the example and calculate the meanings of expressions, explaining the sequence of actions. Another possible way to familiarize children with expressions of this type is to compose these expressions by students from a given number and the simplest expression.
The ability to compose and find the meaning of expressions is used by students when solving compound problems; at the same time, further mastery of the concept of expression occurs here, and the specific meaning of expressions in records of solutions to problems is acquired. An exercise is useful in this regard: the condition of the problem is given, for example, “The boy had 24 rubles. Ice cream costs 12 rubles, and candy costs 6 rubles.” Children should explain what the following expressions show in this case:
In the second grade, the terms “mathematical expression” and “meaning of expression” (without definition) are introduced. After recording several examples in one activity, the teacher informs that these examples are otherwise called mathematical expressions.
As instructed by the teacher, the children themselves make up various expressions. The teacher suggests calculating the results and explains that the results are otherwise called the values of mathematical expressions. Then more complex mathematical expressions are considered.
Later, when performing various exercises, first the teacher and then the children use new terms (write down expressions, find the meaning of an expression, compare expressions, etc.).
In complex expressions, action signs connecting the simplest expressions also have a double meaning, which is gradually revealed by students. For example, in the expression 20+(34-8), the “+” sign indicates the action that must be performed on the number 20 and the difference between the numbers 34 and 8 (add the difference between the numbers 34 and 8 to 20). In addition, the plus sign serves to indicate a sum - this expression is a sum in which the first term is 20, and the second term is expressed by the difference between the numbers 34 and 8.
After children become familiar in the second grade with the order of performing actions in complex expressions, they begin to form the concepts of sum, difference, product, quotient, in which individual elements are specified by expressions.
Subsequently, in the process of repeated exercises in reading, composing and writing expressions, students gradually master the ability to establish the type of complex expression (in 2-3 steps).
A diagram that is compiled collectively and used when reading expressions greatly facilitates children’s work:
determine which action is performed last;
remember what numbers are called when performing this action;
Exercises in reading and writing complex actions using simple expressions help children learn the rules of order.
1.2 Learning the rules of procedure
Rules for the order of performing actions in complex expressions are studied in 2nd grade, but children practically use some of them in 1st grade.
First, we consider the rule about the order of operations in expressions without parentheses, when numbers are performed either only addition and subtraction, or only multiplication and division. The need to introduce expressions containing two or more arithmetic operations of the same level arises when students become familiar with the computational techniques of addition and subtraction within 10, namely:
Similarly: 6 - 1 - 1, 6 - 2 - 1, 6 - 2 - 2.
Since to find the meanings of these expressions, schoolchildren turn to objective actions that are performed in a certain order, they easily learn the fact that arithmetic operations (addition and subtraction) that take place in expressions are performed sequentially from left to right.
Students first encounter number expressions containing addition and subtraction operations and parentheses in the topic "Addition and Subtraction within 10." When children encounter such expressions in 1st grade, for example: 7 - 2 + 4, 9 - 3 - 1, 4 +3 - 2; in 2nd grade, for example: 70 - 36 +10, 80 - 10 - 15, 32+18 - 17; 4*10:5, 60:10*3, 36:9*3, the teacher shows how to read and write such expressions and how to find their meaning (for example, 4*10:5 read: 4 multiply by 10 and divide the resulting result at 5). By the time they study the topic “Order of Actions” in 2nd grade, students are able to find the meanings of expressions of this type. The goal of the work at this stage is to draw their attention, based on the practical skills of students, to the order of performing actions in such expressions and formulate the corresponding rule. Students independently solve examples selected by the teacher and explain in what order they performed them; actions in each example. Then they formulate the conclusion themselves or read from a textbook: if in an expression without parentheses only the actions of addition and subtraction (or only the actions of multiplication and division) are indicated, then they are performed in the order in which they are written (i.e., from left to right).
Despite the fact that in expressions of the form a+b+c, a+(b+c) and (a+b)+c the presence of parentheses does not affect the order of actions due to the associative law of addition, at this stage it is more advisable to orient students to that the action in parentheses is performed first. This is due to the fact that for expressions of the form a - (b + c) and a - (b - c) such a generalization is unacceptable and it will be quite difficult for students at the initial stage to navigate the assignment of brackets for various numerical expressions. The use of parentheses in numerical expressions containing addition and subtraction operations is further developed, which is associated with the study of such rules as adding a sum to a number, a number to a sum, subtracting a sum from a number and a number from a sum. But when first introducing parentheses, it is important to direct students to do the action in the parentheses first.
The teacher draws the children's attention to how important it is to follow this rule when making calculations, otherwise you may get an incorrect equality. For example, students explain how the meanings of the expressions are obtained: 70 - 36 +10 = 24, 60:10 - 3 = 2, why they are incorrect, what meanings these expressions actually have. Similarly, they study the order of actions in expressions with brackets of the form: 65 - (26 - 14), 50: (30 - 20), 90: (2 * 5). Students are also familiar with such expressions and can read, write and calculate their meaning. Having explained the order of actions in several such expressions, children formulate a conclusion: in expressions with brackets, the first action is performed on the numbers written in brackets. Examining these expressions, it is not difficult to show that the actions in them are not performed in the order in which they are written; to show a different order of their execution, and parentheses are used.
The following introduces the rule for the order of execution of actions in expressions without parentheses, when they contain actions of the first and second stages. Since the rules of procedure are accepted by agreement, the teacher communicates them to the children or the students learn them from the textbook. In order for students to understand the introduced rules, along with training exercises, they include solving examples with an explanation of the order of their actions. Exercises in explaining errors in the order of actions are also effective. For example, from the given pairs of examples, it is proposed to write down only those where the calculations were performed according to the rules of the order of actions:
After explaining the errors, you can give a task: using parentheses, change the order of actions so that the expression has the specified value. For example, in order for the first of the given expressions to have a value equal to 10, you need to write it like this: (20+30):5=10.
Exercises on calculating the value of an expression are especially useful when the student has to apply all the rules he has learned. For example, the expression 36:6+3*2 is written on the board or in notebooks. Students calculate its value. Then, according to the teacher’s instructions, the children use parentheses to change the order of actions in the expression:
An interesting, but more difficult, exercise is the reverse exercise: placing parentheses so that the expression has the given value:
Also interesting are the following exercises:
1. Arrange the brackets so that the equalities are true:
25-17:4=2 3*6-4=6
2. Place “+” or “-” signs instead of asterisks so that you get the correct equalities:
3. Place arithmetic signs instead of asterisks so that the equalities are true:
By performing such exercises, students become convinced that the meaning of an expression can change if the order of actions is changed.
To master the rules of the order of actions, it is necessary in grades 3 and 4 to include increasingly complex expressions, when calculating the values of which the student would apply not one, but two or three rules of the order of actions each time, for example:
90*8- (240+170)+190,
469148-148*9+(30 100 - 26909).
In this case, the numbers should be selected so that they allow actions to be performed in any order, which creates conditions for the conscious application of the learned rules.
1.3 Introduction to Expression Conversion
Converting an expression is replacing a given expression with another whose value is equal to the value of the given expression. Students perform such formations of expressions, relying on the properties of arithmetic operations and the consequences arising from them.
As students study each rule, they become convinced that in expressions of a certain type, actions can be performed in different ways, but the meaning of the expression does not change. In the future, students use knowledge of the properties of actions to transform given expressions into expressions equal to them. For example, tasks like this are offered: continue recording so that the "=" sign is preserved:
56- (20+1)=56-20...
(10+5) * 4=10*4...
60:(2*10)=60:10...
When completing the first task, students reason like this: on the left, subtract the sum of the numbers 20 and 1 from 56; on the right, subtract 20 from 56; in order to get the same amount on the right as on the left, you must also subtract 1 from the right. Other expressions are transformed similarly, i.e., after reading the expression, the student remembers the corresponding rule and, performing actions according to the rule, receives the transformed expression. To ensure that the transformation is correct, children calculate the values of the given and transformed expressions and compare them. Using knowledge of the properties of actions to justify calculation techniques, students in grades 2-4 perform transformations of expressions of the form:
54+30=(50+4)+20=(50+20)+4=70+4=74
72:3=(60+12):3=60:3+12:3=24
16 * 40=16 * (3 * 10)=(16 * 3) * 10=540
Here it is also necessary that students not only explain on what basis they derive each subsequent expression, but also understand that all these expressions are connected by an “=” sign because they have the same meanings. To do this, children should sometimes be asked to calculate the meanings 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 in grades 2 and 3 transform expressions not only based on the properties of actions, but also on the basis of definitions of actions. 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. Also based on the meaning of the multiplication action, more complex expressions are transformed: 8 * 4+8 = 8 * 5, 7 * 6 - 7 = 7 * 5.
Based on calculations and analysis of specially selected expressions, 3rd grade students are led to the conclusion that if in expressions with brackets the brackets do not affect the order of actions, then they can be omitted: (30+20)+10=30+20+10, (10-6):4=10-6:4, etc. Subsequently, using the studied properties of actions and rules for the order of actions, students practice transforming expressions with brackets into identical expressions without brackets. For example, it is proposed to write these expressions without parentheses so that their values do not change: (65+30) - 20 (20+4) * 3
Explaining the solution to the first of the given expressions based on the rule for subtracting a number from a sum, children replace it with the expressions: 65+30 - 20, 65 - 20+30, 30 - 20+65, explaining the procedure for performing actions in them. By performing such exercises, students are convinced that the meaning of an expression does not change when the order of actions is changed only if the properties of the actions are applied.
Thus, the familiarization of primary schoolchildren with the concept of expression is closely related to the formation of computational skills. At the same time, the introduction of the concept of expression makes it possible to organize appropriate work on the development of students’ mathematical speech.
Lecture 2. Letter symbols, equalities, inequalities, equations
2.1 Methodology for familiarization with letter symbols
In accordance with the mathematics program, letter symbols are introduced in the 3rd grade.
Here, students become familiar with the letter a as a symbol to denote an unknown number or one of the components of an expression when solving expressions of the form: write the letter a instead of the “box”. Find the values of the sum a+6 if a=8, a=7. Then, in subsequent lessons, they become familiar with some letters of the Latin alphabet, denoting one of the components in the expression. The letter x, as a symbol for denoting an unknown number when solving equations of the form: a + x = b, x - c = b - is introduced in the 4th quarter of the 3rd grade.
The introduction of a letter as a symbol to denote a variable makes it possible to begin work on the formation of the concept of a variable already in the elementary grades, and to introduce children to the mathematical language of symbols earlier.
Preparatory work for revealing the meaning of a letter as a symbol to designate a variable is carried out at the beginning of the school year in 3rd grade. At this first stage, children are introduced to some letters of the Latin alphabet (a, b, c, d, k) to represent a variable, i.e. one of the components in the expression.
When introducing letter symbols to designate a numerical variable, a skillful combination of inductive and deductive methods plays an important role in the system of exercises. In accordance with this, exercises involve transitions from numerical expressions to alphabetic ones and, conversely, from alphabetic expressions to numerical ones. For example, a poster with three pockets is hung on the board, on which it is written: “1 term”, “2 term”, “sum”.
While talking with students, the teacher fills the pockets of the poster with cards with numbers and mathematical expressions written on them:
Next, it becomes clear whether it is still possible to compose expressions, how many such expressions can be composed. Children make up other expressions and find something in common in them: the same action is addition and different action is different terms. The teacher explains that, instead of writing down different numbers, you can designate any number that can be a term by some letter, for example a, any number that can be a second term, for example, b. Then the amount can be denoted as follows: a + b (the corresponding cards are placed in the pockets of the poster).
The teacher explains that a+b is also a mathematical expression, only in it the terms are designated by letters; each of the letters denotes any numbers. These numbers are called letter values.
The difference of numbers is introduced similarly as a generalized notation of numerical expressions. In order for students to realize that the letters included in an expression, for example, b + c, can take on many numerical values, and the letter expression itself is a generalized notation of numerical expressions, exercises are provided for the transition from letter expressions to numerical ones.
Students are convinced that by giving letters personal numerical values, they can get as many numerical expressions as they want. In the same way, work is being done to concretize the literal expression - the difference between numbers.
Further, in connection with the work on expressions, the concept of a constant value is revealed. For this purpose, expressions are considered in which a constant value is fixed using a number, for example: a±12, 8±c. Here, as in the first stage, exercises are provided to transition from numerical expressions to expressions written using letters and numbers, and vice versa.
For this purpose, at first, a poster with three pockets is used.
As students fill the pockets of the poster with cards with numbers and mathematical expressions written on them, they notice that the values of the first term change, but the second term does not change.
The teacher explains that the second term can be written using numbers, then the sum of the numbers can be written as follows: m + 8, and the cards are inserted into the corresponding pockets of the poster.
In a similar way, you can obtain mathematical expressions of the form: 17±a, in ±30, and later - expressions of the form: 7* in, c*4, a:8, 48:in.
In grade 4, exercises like: Find the meaning of the expression a:b if
a=3,400 and b=2;
a=2,800 and b=7.
Once students understand the meaning of letter symbols, letters can be used as a means of summarizing the knowledge they are developing.
The specific basis for the use of letter symbols as a generalization tool is knowledge about arithmetic operations and the knowledge that is formed on their basis.
These include concepts about arithmetic operations, their properties, connections between components and results of actions, changes in the results of arithmetic operations depending on a change in one of the components, etc.
Thus, the use of letter symbols helps to increase the level of generalization of knowledge acquired by primary school students and prepares them for studying a systematic algebra course in subsequent grades.
2.2 Numerical equalities, inequalities
The concept of equalities, inequalities and equations is revealed in interrelation. Work on them is carried out from the 1st grade, organically combined with the study of arithmetic material.
According to the new program, the task is to teach children to compare numbers, as well as compare expressions in order to establish relationships “more”, “less”, “equal”; teach how to write comparison results using the signs ">", "<", "=" и читать полученные равенства и неравенства.
Students obtain numerical equalities and inequalities based on comparisons of given numbers or arithmetic expressions. Initially, younger schoolchildren develop concepts only about true equalities and inequalities (5>4, 6<7, 8=8).
Subsequently, when students gain experience working on expressions and inequalities with a variable, after considering the concepts of true and false (true and false) statements, they move on to such a definition of the concepts of equality and inequality, according to which any two numbers, two expressions connected by one of the signs “greater than ", "less" is called inequality. At the same time, true and false equalities and inequalities are distinguished. In grade 3 the following exercises are offered: check if the given equations are correct (4th quarter): 760 - 400=90*4; 630:7=640:8.
But these exercises are not enough. In grade 4, similar exercises and others are offered, like: check if the inequalities are true: 478 * 24<478* (3*9); 356*10*6>356*16.
Familiarization with equalities and inequalities in the primary grades is directly related to the study of numbering and arithmetic operations. mathematical algebra equation
Comparison of numbers is carried out first on the basis of comparison of sets, which is performed, as is known, by establishing a one-to-one correspondence. This method of comparing sets is taught to children in the preparatory period and at the beginning of learning the numbering of the first ten numbers. At the same time, the elements of sets are counted and the resulting numbers are compared. In the future, when comparing numbers, students rely on their place in the natural series: 9<10, потому что при счете число 9 называют перед числом 10, и т.д.
Established relationships are written using the signs ">", "<", "=", учащиеся упражняются в чтении и записи равенств и неравенств. Впоследствии при изучении нумерации чисел в пределах 100, 1000, а также нумерации многозначны: чисел сравнение чисел осуществляется либо на основе сопоставления их по месту в натуральном ряду, либо на основе разложения чисел по десятичному составу сравнения соответствующих разрядных чисел, начиная с высшего разряда.
The comparison of named numbers is first carried out based on the comparison of the values of the quantities themselves, and then carried out on the basis of a comparison of abstract numbers, for which the given named numbers are expressed in the same units of measurement.
Comparing named numbers causes great difficulties for students, therefore, in order to teach this operation, it is necessary to systematically offer a variety of exercises in grades 2-4:
1 dm * 1 cm, 2 dm * 2 cm
Replace with an equal number: 7 km 500 m = _____ m
3) Select the numbers so that the entry is correct: ____ h< ____ мин, ___ см=__ дм и т.д.
4) Check whether the equalities given are true or false, correct the sign if the equalities are incorrect:
4 t 8 c=480 kg, 100 min.=1 hour, 2 m 5 cm=250 cm.
The transition to comparing expressions is carried out gradually. First, in the process of learning addition and. subtraction within 10, children spend a long time practicing comparing expressions and numbers. The first inequalities of the form 3+1>3, 3 - 1<3 полезно получать из равенства (3=3), сопровождая преобразования соответствующими операциями над множествами. В дальнейшем выражение и число учащиеся сравнивают, не прибегая к операциям над множествами: находят значение выражения и сравнивают его с заданным числом, что отражается в записях:
After becoming familiar with the names of expressions, students read equalities and inequalities like this: the sum of the numbers 5 and 3 is greater than 5.
Based on operations on sets and comparison of sets, students practically learn the important properties of equalities and inequalities (if a = b, then b = a). Comparing two expressions means comparing their meanings. Comparing numbers and expressions is first included when studying numbers within 20, and then when studying actions in all concentrations, these exercises are systematically offered to children.
When studying actions in other concentrations, exercises for comparing expressions become more complicated: expressions become more complex, students are asked to insert a suitable number into one of the expressions so as to obtain correct equalities or inequalities, and compose correct equalities or correct inequalities from these expressions.
Thus, when studying all concentrations, exercises to compare numbers and expressions, on the one hand, contribute to the formation of concepts about equalities and inequalities, and on the other hand, to the acquisition of knowledge about numbering and arithmetic operations, as well as the development of computational skills.
2.3 Methodology for familiarizing yourself with inequalities with a variable
Inequalities with a variable of the form: x+3< 7, 10 - х >5 are introduced in 3rd grade. At first, the variable is denoted not by a letter, but by a “window”, then it is denoted by a letter.
The terms “solve an inequality” and “solve an inequality” are not introduced in primary grades, since in many cases they are limited to selecting only a few values of a variable, which results in a true inequality. Exercises are performed under the guidance of a teacher.
Exercises with inequalities reinforce computational skills and also help master arithmetic knowledge. Selecting letter values in inequalities and equalities of the form: 5 + x = 5, 5 - x =5 10 * x = 10, 10* x<10, учащиеся закрепляют знания особых случаев действий. Но самым важным является то, что работая с неравенствами, учащиеся закрепляют представление о переменной и подготавливаются к решению неравенств в 5 классе. В соответствии с программой в 1-4 классах рассматриваются упражнения первой степени с одним неизвестным вида: 7+х=10, х* (17 - 10)=70.
Exercises in elementary grades are considered as true equalities; solving the equation comes down to finding the value of the letter (an unknown number) for which the given expression has the specified value. Finding an unknown number in such equalities is performed based on knowledge of the relationship between the result and the components of arithmetic operations. These program requirements determine the methodology for working on equations,
2.4 Methodology for studying equations
At the preparatory stage for the introduction of the first equations when studying addition and subtraction within 10, students learn the connection between the sum and the terms. In addition, by this time children have mastered the ability to compare expressions and numbers and receive their first ideas about numerical equalities of the form: 8 = 5 + 3, 6 + 4 = 40. Of great importance in terms of preparing for the introduction of equations are exercises for finding the missing number in equalities of the form: 4 + * = 6, 5- * = 2. In the process of performing such exercises, children get used to the idea that not only the sum or difference can be unknown, but also one of the terms.
The concept of an equation is introduced in 3rd grade. Equations are solved orally, using the method of selection, i.e. children are offered simple equations of the form: x + 3 = 5. To solve such equations, children remember the composition of numbers within 10, in this case the composition of the number 5 (3 and 2), which means x = 2.
In grade 4, the teacher shows a record of solving an equation, based on the children’s knowledge of the connections between the components and the result of arithmetic operations. For example, 6+x=15. We don't know the second term. To get the second term we need to subtract the first term from the sum.
Recording the solution:
Examination:
It is necessary to explain to students that when we perform a check, it is necessary, after substituting the resulting number instead of x, to find the value of the resulting expression.
Later, in the next step, the equations are solved based on knowledge of the rules for finding the unknown component.
A separate lesson is given for each case.
Posted on Allbest.ru
...Similar documents
The concept of inequality, its essence and features, classification and varieties. Basic properties of numerical inequalities. Technique for graphically solving second degree inequalities. Systems of inequalities with two variables, with a variable under the modulus sign.
abstract, added 01/31/2009
Trigonometric equations and inequalities in the school mathematics course. Analysis of material on trigonometry in various textbooks. Types of trigonometric equations and methods for solving them. Formation of skills for solving trigonometric equations and inequalities.
thesis, added 05/06/2010
Theoretical information on the topic "Tests for the equality of triangles." Methodology for studying the topic "Signs of equality of triangles." The topic of the lesson is "Triangle. Types of triangles." "Properties of isosceles and equilateral triangles."
course work, added 01/11/2004
Types of equations that allow reduction of order. Linear differential equation of higher order. Theorems on the properties of partial solutions. Wronski's determinant and its application. Using Euler's formula. Finding the roots of an algebraic equation.
presentation, added 03/29/2016
The concept and mathematical description of the elements of a differential equation as an equation connecting the desired function of one or more variables. Composition of incomplete and linear differential equations of the first order, their application in economics.
abstract, added 08/06/2013
A method for analytically solving (in radicals) an algebraic equation of the nth degree with a return to the roots of the original equation. Eigenvalues for finding functions of matrices. Stability of solutions to linear differential and difference equations.
scientific work, added 05/05/2010
Type of the Riccati equation for an arbitrary fractional-linear transformation of the dependent variable. Properties of the reflecting function, its construction for nonlinear differential equations of the first order. Formulation and proof of the lemma for the OF Riccati equation.
course work, added 11/22/2014
The main directions of development of the line of equations and inequalities in the school mathematics course, its connection with the numerical and functional system. Features of the study, analytical and graphical methods for solving equations and inequalities containing parameters.
course work, added 02/01/2015
Systematization of information about linear and quadratic dependencies and related equations and inequalities. Isolation of a complete square as a method for solving some non-standard problems. Properties of the function |x|. Equations and inequalities containing moduli.
thesis, added 06/25/2010
Analysis of the features of developing a computer program. General characteristics of the simple iteration method. Introduction to the basic methods of solving a nonlinear algebraic equation. Consideration of the stages of solving an equation by the bisection method.
For quite a long time, the prevailing opinion in psychology was that the elements of algebra should be studied not in the elementary grades, but in the senior grades due to the peculiarities of the thinking of a junior schoolchild and his inability to form abstractions of a higher level. However, such prominent psychologists as P.Ya. Galperin, V.V. Davydov, D.B. Elkonin, etc., and teachers - A.I. Merkushevich, A.M. Pyshkalo, etc., found that children 6 -10 years old, with a certain organization of training, can fully master the content of some algebraic concepts. Based on this, algebraic material was included in the primary school mathematics curriculum in 1969.
When studying the elements of algebra, younger schoolchildren receive initial information about numerical expressions, numerical equalities and inequalities, inequalities with a variable, expressions with a variable, with two variables, and equations.
Algebraic material is studied from 1st grade. in close connection with arithmetic and geometric. The introduction of algebra elements contributes to the generalization of concepts about number, arithmetic operations, and mathematical relations, and at the same time prepares children for studying algebra in the following grades.
The main stages of studying and the content of algebraic material
1. METHODOLOGY FOR STUDYING NUMERICAL EXPRESSIONS
Numeric expression -
1. every number is a numerical expression.
2. if a and b are numerical expressions, then their sum a+b, difference a-b, product a∙b and quotient a:b are also numerical expressions.
Numeric expression value- this is the number obtained as a result of performing all actions. indicated in numerical terms.
The mathematics program provides:
Introduce the rules for the order of actions and teach them how to use them in calculations,
Introduce students to identical transformations of expressions.
The methodology for familiarizing yourself with CVs can be divided into 3 stages:
Stage 1. Familiarization with expressions containing one action (sum, difference, product, quotient of two numbers).
Acquaintance with the first expression - the sum - occurs in 1st grade. when studying the concentration “10”.
1. When performing operations on sets, children first of all learn the specific meaning of addition and subtraction, therefore, in notations of the form 5 + 1,6-2, they understand the signs of actions as a short designation of the words “add”, “subtract” (reading: add 1 to 5, you get 6, subtract 2 from 6, you get 4).
2. In the future, the concept of these actions deepens. Students learn that adding a few units increases a number by the same number of units, and subtracting a number decreases it by the same number of units.
(reading: 5 increase by 1, 6 decrease by 2).
3. Then children learn the name of the action signs: “plus”, “minus”
(reading: 5 plus 1,6 minus 1).
4. Children learn the names of the components of the CV.
(reading: 1 term. 5, 2 terms 1, sum equal to 6).
In approximately the same way, work is underway on the following expressions: difference (1st grade), product and quotient (2nd grade).
Stage 2. Familiarization with CVs containing actions of one stage .
Before studying expressions with brackets, students are offered expressions of the form 8+1-7 10-5+4
In these cases, the value of the expression enclosed in an oval is first found, then the number in the square is subtracted from the resulting result. In this case, students use the rule for the order of actions in an implicit form and perform the first identical transformations (8+1-7=9-7=2).
Later the brackets 6+4-1=(6+4)-1 are introduced.
The rule is formed: the action written in brackets is performed first.
To master the introduced rule, various training exercises are included. At the same time, children learn to correctly read and write these expressions:
Write and calculate: .
1. Subtract 10 from the sum of numbers 9 and 7.
2. To 10 add the difference between the numbers 9 and 7.
Subsequently, the concepts of a numerical expression (ostensive, by showing) and the meaning of a numerical expression are introduced. 2 classes With. 68
After this, children read or write down expressions, find their meanings, and make up expressions themselves.
Mastering new terms allows them to read expressions in new ways ( write down expressions, find the meaning of an expression, compare expressions etc.) 2nd grade p.58 No. 1,2, 6; p.69 No. 2.
In complex expressions, the action signs connecting the expressions have a double meaning, which is revealed to students.