Methods for studying algebraic material in elementary school. Elements of algebra in elementary school

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INTRODUCTION

CONCLUSION

BIBLIOGRAPHY

Introduction

At any modern system 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, there appears to be a significant flaw in this statement. The language of mathematics is so widespread and so often effective precisely because mathematics cannot be reduced to it.

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 concentrates the results of the exact thinking of many people. Using mathematics, you can connect one reasoning with another The obvious complexities of nature with its strange laws and rules, each of which allows for a very different detailed explanation, are in fact closely related. However, if you do not want to use mathematics, then in this huge variety of facts you will not see that logic allows you to move from one to another."

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

What is the influence of mathematics in general and school mathematics in particular on education? 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. Necessity research activities develops interest in patterns, teaches to see the beauty and harmony of human thought. All this is, in our opinion, the most important element general culture. The mathematics course has an important influence on the formation various forms thinking: logical, spatial-geometric, algorithmic. Any creative process begins with the formulation of a hypothesis. Mathematics, with the appropriate organization of education, being a good school for constructing and testing hypotheses, teaches you to compare different hypotheses, find the best option, pose new problems, and look for ways to solve them. Among other things, she also develops the habit of methodical work, without which no creative process is conceivable. By maximizing the possibilities of human thinking, mathematics is its highest achievement. It helps a person to understand himself and form his character. This is a little of big list reasons why mathematical knowledge should become integral part general culture and an obligatory element in the upbringing and education of a child. The mathematics course (without geometry) in our 10-year school is actually divided into three main parts: arithmetic (grades I - V), algebra (grades VI - VIII) and elements of analysis (grades IX - X). What is the basis for such a division? Of course, each of these parts has its own special “technology”.

So, in arithmetic it is associated, for example, with calculations performed on multi-digit numbers, in algebra - with identical transformations, logarithms, in analysis - with differentiation, etc. But what are the deeper reasons associated with the conceptual content of each part? The next question concerns the basis for distinguishing school arithmetic and algebra (i.e., the first and second parts of the course). Arithmetic includes the study of natural numbers (positive integers) and fractions (prime and decimal). However, a special analysis shows that combining these types of numbers in one school subject is unlawful.

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

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

A.N. Kolmogorov considered justified both from the point of view of the history of the development of mathematics and essentially the proposal of A. Lebesgue to move in teaching after natural numbers directly to the origin and logical nature of real numbers. At the same time, as noted by A.N. Kolmogorov, “the approach to the construction of rational and real numbers from the point of view of measuring quantities is no less scientific than, for example, the introduction of rational numbers in the form of “pairs”. For the school it has an undoubted advantage” (.

So there is real opportunity on the basis of natural (integer) numbers, 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 20 years ago, are still relevant today.

1. General theoretical aspects of studying algebraic material in primary school

algebraic school comparison mathematics

1.1 Experience of introducing algebra elements in primary school

Content academic subject, as is known, depends on many factors - on life’s demands on students’ knowledge, on the level of relevant sciences, on the mental and physical age capabilities of children, etc. Correct consideration of these factors is an essential condition for the most effective learning schoolchildren, expanding their cognitive capabilities. But sometimes this condition is not met for one reason or another. In this case, teaching does not give the desired effect both in terms of children’s acquisition of the range of necessary knowledge and in terms of the development of their intelligence.

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

The foundation of mathematical knowledge is laid in elementary school. But, unfortunately, both mathematicians themselves, methodologists and psychologists pay very little attention to the content elementary mathematics. Suffice it to say that the mathematics curriculum in primary school (grades I - IV) 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 characteristics state standard in mathematics in elementary school. Its main content is integers and operations on them, studied in a certain sequence. First, four operations are studied in the limit of 10 and 20, then - oral calculations in the limit of 100, oral and written calculations in the limit of 1000 and, finally, in the limit of millions and billions. In grade IV, some relationships between data and the results of arithmetic operations, as well as simple fractions, are studied. Along with this, the program involves the study of metric measures and measures of time, mastering the ability to use them for measurement, knowledge of some elements of visual geometry - drawing a rectangle and square, measuring segments, areas of a rectangle and square, 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 actions, understand various cases of their application, establish relationships between quantities, and acquire basic skills of analysis and synthesis.

From grades I to IV, 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 by two differences, calculating the arithmetic mean and some other types of problems.

WITH different types children encounter dependencies of quantities when solving problems. But it is very typical that students begin problems after and as they study numbers; the main thing required when solving is to find a numerical answer. Children with with great difficulty identify the properties of quantitative relations in specific, particular situations, which are considered to be 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 a system 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 vary 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 (III grade) more difficult than problems on difference and multiple comparison (II grade)? The methodology does not provide a convincing and logical answer to this question.

Thus, students primary classes They 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 the technique of calculations, or when solving problems, because the latter do not have the appropriate form and do not have the required system. Attempts by methodologists to improve teaching methods, although they lead to partial success, do not change general position cases, 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 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 relationships are quite expressible without numbers and before numbers, for example, in segments, volumes, etc. (relationship “more”, “less”, “equal”). The presentation of the original general mathematical concepts in modern manuals is carried out in such symbolism that does not necessarily imply the expression of objects by numbers. So, in the book by E.G. Gonin's "Theoretical Arithmetic" the basic 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 unique existing form expressions. Further, it is noteworthy that many illustrations of individual mathematical definitions are given in graphical form, through the ratio of segments, areas. 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), however, this does not change their essential function in the child’s general orientation in the properties of things. And sometimes it happens that the depth of these supposedly “pre-mathematical formations” is more significant for the development of a child’s own mathematical thinking than knowledge of the intricacies of computer technology and the ability to find purely numerical dependencies. It is noteworthy that 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. It is only necessary to properly apply this simple idea to the solution of the problem that at first glance it seems inaccessible."

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

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 the methods of mathematical thinking, and not just calculation skills: this involves building 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 (in particular, electronic) means.

The meaning of these requirements is clear: in elementary school it is quite possible to teach mathematics as a science about the 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 late 1960s, now allow us to talk about the possibility of introducing a systematic mathematics course in school starting from the first grade, providing knowledge about quantitative relationships and dependencies of quantities in algebraic form .

1.2 The problem of the origin of algebraic concepts and its significance for the construction of an educational subject

Separation school course mathematics for algebra and arithmetic, of course, conditionally. The transition from one to another occurs gradually. In school practice, the meaning of this transition is masked by the fact that the study of fractions actually occurs without extensive support for measuring quantities - fractions are given as ratios of pairs of numbers (although formally the importance of measuring quantities is recognized in methodological manuals). An extensive introduction of fractional numbers based on the measurement of quantities inevitably leads to the concept of a real number. But the latter usually does not happen, since students are kept working with rational numbers for a long time, and thereby their transition to “algebra” is delayed.

In other words, school algebra begins precisely when the conditions are created for the transition from integers to real numbers, to expressing the result of a measurement as a fraction (simple and decimal - finite, and then infinite). Moreover, the initial step may be familiarity with the measurement operation, obtaining the final decimals and studying actions on them. If students already know this form of writing the result of a measurement, then this serves as a prerequisite for “abandoning” the idea that a number can also be expressed as an infinite fraction. And it is advisable to create this prerequisite already within the limits primary school.

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

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

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

The fact is that, despite all the differences between these types of numbers, they refer specifically to numbers, i.e. to a special form of displaying quantitative relationships. The fact that integer and real numbers belong to “numbers” serves as the basis for the assumption of the genetic derivatives of the very differences between counting and measurement: they have a special and single source corresponding to the very form of the number.

Knowledge of the features of this unified basis of counting and measurement will make it possible to more clearly imagine the conditions of their origin, on the one hand, and the relationship, on the other.

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

So, what is “quantity” and what interest is it in constructing the initial sections of school mathematics? In general use, the term “magnitude” is associated with the concepts “equal”, “more”, “less”, which describe a variety of qualities (length and density, temperature and whiteness). V.F. Kagan raises the question of what common properties these concepts have. It shows that they relate to aggregates - sets of homogeneous objects, the comparison of elements of which allows us to apply the terms “more”, “equal”, “less” (for example, to aggregates of all straight segments, weights, velocities, etc.).

A set of objects is only transformed into magnitude 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 ratios: A=B, A>B, A<В. Эти предложения составляют полную дизъюнкцию (по крайней мере, одно имеет место, но каждое исключает все остальные).

V.F. Kagan 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<В.

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

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

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

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

6) If A<В и В<С, то А<С.

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

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

The first three sentences characterize the disjunction of the basic relations "=", ">", "<". Предложения 4 - 6 - их транзитивность при любых

three elements A, B and C. The following sentences 7 - 8 characterize only equality - its reversibility and recurrence (or reflexivity). V.F. Kagan calls these eight basic provisions postulates of comparison, on the basis of which a number of other properties of quantity can be derived.

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

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

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

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

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

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

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

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

VIII. If there is equality or inequality A=B, or A>B, or A<В, то оно не нарушится, когда мы один из его элементов заменим равным ему элементом (здесь имеет место соотношение типа: если А=В и А=С, то С=В; если А>B and A=C, then C>B, etc.).

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

The properties specified in postulates and theorems can characterize not only those immediate features of objects that we are accustomed to associate with “equal”, “more”, “less”, but also with many other features (for example, they can characterize the relation “ancestor - descendant"). This allows us to take a general point of view when describing them and consider, for example, from the point of view of these postulates and theorems any three types of relations “alpha”, “beta”, “gamma” (in this case it is possible to establish whether these relations satisfy the postulates and theorems and under what conditions).

From this point of view, one can, for example, consider such a property of things as hardness (harder, softer, equal hardness), the sequence of events in time (following, preceding, simultaneous), etc. In all these cases, the ratios “alpha”, “beta”, “gamma” receive their own specific interpretation. The task associated with the selection of such a set of bodies that would have these relationships, as well as the identification of signs by which one could characterize “alpha”, “beta”, “gamma” - this is the task of determining comparison criteria in a given set of bodies (in practice, in some cases it is not easy to solve). “By establishing comparison criteria, we transform multitude into magnitude,” wrote V.F. Kagan. Real objects can be viewed from the perspective of different criteria. Thus, a group of people can be considered according to such a criterion as the sequence of moments of birth of each of its members. Another criterion is the relative position that the heads of these people will take if they are placed side by side on the same horizontal plane. In each case, the group will be transformed into a quantity that has a corresponding name - age, height. In practice, a quantity usually denotes not the set of elements itself, but a new concept introduced to distinguish comparison criteria (the name of the quantity). This is how the concepts of “volume”, “weight”, “electrical voltage”, 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. Kagan.

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 (they occupy the same place, follows ..., precedes), this series satisfies the postulates and therefore represents a quantity. According to the corresponding comparison criteria, a set of fractions is also converted into a quantity. This is according to V.F. Kagan, the content of the theory of quantity, which plays a crucial role in the foundation of all mathematics.

Working with quantities (it is advisable to record their individual values ​​in letters), you can perform a complex system of transformations, establishing the dependencies of their properties, moving from equality to inequality, performing addition (and subtraction), and when adding you can be guided by commutative and associative properties. So, if the relation A=B is given, then when “solving” problems you can be guided by the relation B=A. In another case, if there are relations A>B, B=C, we can conclude that A>C. Since for a>b there is a c such that a=b+c, ​​then we can find the difference between a and b (a-b=c), etc.

All these transformations can be performed on physical bodies and other objects by establishing comparison criteria and the correspondence of the selected relations to the postulates of comparison.

The above materials allow us to conclude that both natural and real numbers are equally strongly associated with quantities and some of their essential features. Is it possible to make these and other properties the subject of 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 the number and its different types, in particular for propaedeutics of fractions, concepts of coordinates, functions and other concepts already in the junior 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. This content needs to be developed into a relatively detailed teaching program and, most importantly, linked to those actions of the child through which he can master this content (of course, in the appropriate form). At the same time, it is necessary to experimentally establish whether 7-year-old children can master this program, and what is the feasibility of its introduction for subsequent teaching of mathematics in primary school in the direction of bringing arithmetic and elementary algebra closer together.

So far our reasoning has been theoretical nature and were aimed at clarifying the mathematical prerequisites for constructing such an initial section of the course that would introduce children to basic algebraic concepts (before the special introduction of numbers). The main properties characterizing quantities were described above. Naturally, it makes no sense for 7-year-old children to give “lectures” regarding these properties.

It was necessary to find such a form of work for children with didactic material, through which they could, on the one hand, identify these properties in the things around them, on the other hand, they would learn to fix them with certain symbolism and carry out an elementary mathematical analysis of the identified relationships.

In this regard, the program should contain, firstly, an indication of those properties of the subject that are to be mastered, secondly, a description of didactic materials, thirdly - and this is the main thing from a psychological point of view - the characteristics of those actions through which the child identifies certain properties subjects and masters them. These “components” form the teaching program in the proper sense of the word. It makes sense to present the specific features of this hypothetical program and its “components” when describing the learning process itself and its results.

Here is the outline of this program and its key topics.

Topic I. Leveling and completing objects (by length, volume, weight, composition of parts and other parameters).

Practical tasks on leveling and acquisition. Identification of characteristics (criteria) by which the same objects can be equalized or completed. Verbal designation of these characteristics (“by length”, by weight”, etc.).

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

Choosing the “same” item,

Reproduction (construction) of the “same” object according to a selected (specified) parameter.

Topic II. Comparing objects and fixing its results using the equality-inequality formula.

1. Tasks on comparing objects and symbolically designating the results of this action.

2. Verbal recording of comparison results (terms “more”, “less”, “equal”). Written characters ">", "<", "=".

3. Indication of the comparison result with a drawing (“copying” and then “abstract” - lines).

4. Designation of compared objects with letters. Recording the comparison result using the formulas: A=B; A<Б, А>B. A letter as a sign that fixes a directly given, particular value of an object according to a selected parameter (by weight, by volume, etc.).

5. Impossibility of fixing the comparison result using different formulas. Selecting a specific formula for a given result (complete disjunction of the relations greater - less - equal).

Topic III. Properties of equality and inequality.

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

2. The connection between the relations “more” and “less” in inequalities during “permutations” of the compared parties (if A>B, then B<А и т.п.).

3. Transitivity as a property of equality and inequality:

if A=B, if A>B, if A<Б,

a B=B, a B>B, a B<В,

then A=B; then A>B; then A<В.

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

Topic IV. Addition (subtraction) operation.

1. Observations of changes in objects according to one or another parameter (by volume, by weight, by duration, etc.). Illustration of increasing and decreasing with "+" and "-" (plus and minus) signs.

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

if A=B, if A=B,

then A+K>B; then A-K<Б.

3. Methods of transition to new equality (its “restoration” according to the principle:

adding "equal" to "equal" gives "equal").

Working with formulas like:

then A+K>B, but A+K=B+K.

4. Solving various problems that require the use of addition (subtraction) when moving from equality to inequality and back.

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

1. Tasks requiring such a transition. The need to determine the value of the quantity by which the compared objects differ. The ability to write equality when the specific value of this quantity is unknown. Method of using x (x).

Writing formulas like:

if A<Б, если А>B,

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

2. Determining the value of x. Substituting this value into the formula (introduction to parentheses). Type formulas

3. Solving problems (including “plot-textual”) that require performing the specified operations.

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

1. Addition-subtraction of equalities-inequalities:

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

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

2. The ability to represent the value of a quantity as the sum of several values. Type substitution:

3. Solving various problems that require taking into account the properties of relationships that children became familiar with in the process of work (many tasks require simultaneous consideration of several properties, intelligence in assessing the meaning of formulas; descriptions of problems and solutions are given below).

This is a program designed for 3.5 - 4 months. first half of the year. As the experience of experimental teaching shows, with proper planning of lessons, improvement of teaching methods and a successful choice of didactic aids, all the material presented in the program can be fully absorbed by children in a shorter period of time (in 3 months). How is our program going forward? First of all, children become familiar with the method of obtaining a number that expresses the relationship of an object as a whole (the same quantity represented by a continuous or discrete object) to its part. This ratio itself and its specific meaning is depicted by the formula A/K = n, where n is any integer, most often expressing the ratio to the nearest “unit” (only with a special selection of material or by counting only “qualitatively” individual things can one obtain absolutely exact integer). From the very beginning, children are “forced” to keep in mind that when measuring or counting, a remainder may result, the presence of which must be specially stipulated. This is the first step to subsequent work with fractions. With this form of obtaining a number, it is not difficult to lead children to describe an object with a formula like A = 5k (if the ratio was equal to “5”). Together with the first formula, it opens up opportunities for a special study of the dependencies between the object, the base (measure) and the result of counting (measurement), which also serves as a propaedeutic for the transition to fractional numbers (in particular, for understanding the basic property of a fraction). Another line of program development, implemented already in the first grade, is the transfer to numbers (integers) of the basic properties of quantity (disjunction of equality-inequality, transitivity, invertibility) and the operation of addition (commutativity, associativity, monotonicity, the possibility of subtraction). In particular, by working on the number line, children can quickly convert sequences of numbers into magnitudes (for example, clearly assess their transitivity by doing type 3 notations<5<8, одновременно связывая отношения "меньше-больше": 5<8, но 5<3, и т.д.) .

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

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

2.1 Teaching in primary school in relation to the needs of secondary school

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

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 her. 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 subsequent experiments convincingly prove that material enters long-term memory only if it is remembered as a result of work corresponding to this material.

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

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

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

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

Another example of incorrect concept formation is working with the literal equality relation. For example, the rules for multiplying a number by one and a number by zero in all textbooks are given in letter form: a x 1 = a, a x 0 = 0. The equality relation, as is known, is symmetrical, and therefore, such a notation provides not only that when multiplied by 1, the same number is obtained, but also that any number can be represented as the product of this number and one. However, the verbal formulation proposed in textbooks after the letter entry speaks only of the first possibility.

Exercises on this topic are also aimed only at practicing replacing the product of a number and one with this number. All this leads not only to the fact that a very important point does not become the subject of children’s consciousness: any number can be written in the form of a product, which in algebra will cause corresponding difficulties when working with polynomials, but also to the fact that children, in principle, do not know how to correctly work with the relation of equality. For example, when working with the difference of squares formula, children, as a rule, cope with the task of factoring the difference of squares. However, those tasks where the opposite action is required cause difficulties in many cases. Another striking illustration of this idea is the work with the distributive law of multiplication relative to addition. Here, too, despite the letter writing of the law, both its verbal formulation and the system of exercises only train the ability to open brackets. As a result, putting the common factor out of brackets will cause significant difficulties in the future.

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

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

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

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

When studying 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 do without such visibility in elementary 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 mathematics methods for grades 1-3 M.I. Moreau says that children have to do division by arranging objects into piles or making a drawing for 30 lessons. Such actions lose the essence of the division operation as the inverse action of multiplication. As a result, division is learned with the greatest difficulty and is much worse than other arithmetic operations.

When teaching mathematics in elementary school, there is no talk of 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 rules for dividing a number by 1, zero by a number, and a number by itself. Children are quite capable of proving them using the definition of division and the corresponding multiplication rules.

The elementary school material also allows for algebra propaedeutics - working 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 accustom them 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 into a letter expression already in elementary school. This was done, for example, in the textbook by L.G. Peterson.

Speaking about the shortcomings of teaching mathematics in elementary school, which interfere with further learning, it is necessary to especially emphasize the fact that often the material in textbooks is presented without a look at how it will work in the future. A very striking example of this is the organization of learning multiplication by 10, 100, 1000, etc. In all the textbooks reviewed, the presentation of this material is structured in such a way that it inevitably leads to the formation in the minds of children of the rule: “To multiply a number by 10, 100, 1000, etc., you need to add as many zeros to the right side as there are in 10, 100, 1000, etc." This rule is one of those that is learned very well in elementary school. And this leads to a large number of errors when multiplying decimal fractions by whole digit units. Even after remembering a new rule, children often automatically add zero to the right side of the decimal when multiplying by 10.

In addition, it should be noted that when multiplying a natural number and when multiplying a decimal fraction by whole digit units, essentially the same thing happens: each digit of the number is shifted to the right by the corresponding number of digits. Therefore, there is no point in teaching children two separate and completely formal rules. It is much more useful to teach them a general way of proceeding when solving similar problems.

2.2 Comparison (contrast) of concepts in mathematics lessons

The current program provides for the study in the first grade of only two operations of the first level - addition and subtraction. Limiting the first year of study to only two operations is, in essence, a departure from what was already achieved in the textbooks that preceded the current ones: not a single teacher then ever complained 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 arithmetic.

Mathematics relies, first of all, 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.

To be fair, it should be noted that in the first versions of M.I.Moro’s textbooks for grade I, multiplication and division were provided. However, an accident got in the way: the authors of the new programs persistently clung to one “new thing” - coverage in the first grade of all cases of addition and subtraction within 100 (37+58 and 95-58, etc.). But, since there was not enough time to study such an expanded amount of information, it was decided to shift multiplication and division completely to the next year of study.

So, the fascination with the linearity of the program, i.e., a purely quantitative expansion of knowledge (the same actions, but with larger numbers), took up the time that was previously allocated to the qualitative deepening of knowledge (studying all four actions within two dozen). Studying multiplication and division already in the first 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 for adding single-digit numbers is developed 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 the “Twenty” as a second integral 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 into 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 in the next topic actions within 10 are considered. In the experimental textbook by P.M. Erdnieva, in contrast to this, carried out a joint study of numbering, the composition of numbers and operations (addition and subtraction) within 10 at once in one section. With this approach, a monographic study of numbers is used, namely: within the number under consideration (for example, 3), all “cash mathematics” is immediately comprehended: 1 + 2 = 3; 2 + 1 = 3; 3 - 1 = 2; 3 - 2 = 1.

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

The question of classifying tasks and the names of their types requires special attention in the methodology of initial training. Generations of methodologists worked to streamline the system of school tasks, to create their effective types and varieties, right down to the selection of successful terms for the names of tasks intended for study in school. It is known that at least half of the teaching time in mathematics lessons is devoted to solving them. School tasks certainly need systematization and classification. What type (type) of problems to study, when to study, what type of problems to study in connection with the passage of this or that section is a legitimate object of study of the methodology and the central content of the programs. The significance of this circumstance is clear from the history of mathematics methodology.

Conclusion

Currently, quite favorable conditions have arisen for a radical improvement in the organization of mathematics education in primary school:

1) the primary school was transformed from a three-year to a four-year school;

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The main goals of studying algebraic material in the elementary grades are for primary schoolchildren to obtain initial information about equalities and inequalities, about a variable, about equalities and inequalities with a variable, about mathematical expressions (numeric and alphabetic), about calculating their values, about simple equations and inequalities, training schoolchildren on ways to solve them, as well as solving problems algebraically. The study of algebraic material in the elementary grades contributes to the generalization of concepts about numbers, arithmetic operations and their properties, and is a preparation for the study of algebra in high school.

Children get their first ideas about equalities and inequalities when comparing sets and numbers. Their study is associated with the study of numbering, arithmetic operations and quantities. Next, the idea of ​​true and false equalities and inequalities, equalities and inequalities with a variable is introduced.

The equation is treated as an equality with a variable. Solving an equation means choosing a value of a variable such that, when substituted into the equation, it turns into a correct numerical equality. This is the basis for the method of solving equations by selection. In the elementary grades, equations are also solved on the basis of the relationship between the components and the results of arithmetic operations, on the basis of the application of the basic properties of equalities (L.V. Zankov’s system), as well as with the help of graphs (UMK “Primary School of the 21st Century”). The solution to inequalities is limited by the selection method. Equations and inequalities are used in solving problems, however, the algebraic method of solving problems is limited in the elementary grades to the level of familiarization.

Concepts about the simplest expressions are formed in connection with the study of arithmetic operations, then complex expressions and expressions with a variable are introduced. Younger students learn to calculate the values ​​of complex numerical expressions using rules of order. They also learn to find the meaning of expressions with a variable given the values ​​of the letters.

Letter symbols are used to generalize the recording of laws and properties of arithmetic operations, as well as formulas for calculating the areas of rectangles, triangles, polygons, volumes, velocities, etc.

Currently, there are two radically opposite trends in determining the volume of algebraic material in a primary school mathematics course. One trend is associated with the early algebraization of elementary school mathematics courses. Representatives of this trend are I.I. Arginskaya, E.I. Aleksandrova, L.G. Peterson, V.N. Rudnitskaya and others. Another trend is associated with the introduction of algebraic material into the primary school mathematics course at its final stage, at the end of 4 class (N.B. Istomina) The textbook of the traditional school (M.I. Moro and others) is a representative of the “middle” views.

Questions and tasks for independent work

1. Name the geometric concepts that are studied in elementary school. Why are they the subject of study?

2. Does geometric material constitute an independent section in the initial mathematics course? Why?

3. Describe the methodology for forming geometric concepts among students: line segment, triangle, angle, rectangle.

4. What opportunities does the study of geometric material provide for the development of students’ logical thinking? Give examples.

5. What relationships do students become familiar with when studying geometric material?

6. What function do construction tasks serve in elementary school?

7. Give examples of construction problems typical for elementary school.

8. What are the stages of solving construction problems? Show to what extent the general scheme for solving construction problems can be used in elementary grades.

Lecture 14. Methods for studying algebraic material

1. Basic concepts of mathematics.

2. General questions of methods for studying algebraic material in primary school mathematics courses.

3. Numerical expressions. Studying the rules for the order of performing arithmetic operations.

4. Expressions with a variable.

5. Methods for studying equations.

6. Methodology for studying numerical equalities and numerical inequalities.

7. Introducing students to functional dependence.

References: (1) Chapter 4; (2) § 27, 37, 52; (5) - (12).

Basic concepts of mathematics

A numerical expression in general can be defined as follows:

1) Each number is a numerical expression.

2) If A and B are numerical expressions, then (A) + (B), (A) - (B), (A) (B), (A): (B); (A)⁽ⁿ⁾ and f(A), where f (x) is some numerical function, are also numerical expressions.

If all the actions specified in it can be performed in a numerical expression, then the resulting real number is called the numerical value of this numerical expression, and the numerical expression is said to have meaning. Sometimes a numeric expression does not have a numeric value because not all actions specified in it are feasible; such a numerical expression is said to have no meaning. So, the following numerical expressions (5 - 3): (2 – 8:4); √7 – 2 · 6 and (7 – 7)° do not make sense.

Thus, any numerical expression either has one numerical value or is meaningless. -

The following procedure is adopted when calculating the value of a numerical expression:

1. All operations inside the parentheses are performed first. If there are multiple pairs of parentheses, calculations begin with the innermost ones.

2. Inside the parentheses, the order of calculations is determined by the priority of the operations: function values ​​are calculated first, then exponentiation is performed, then multiplication or division is performed, and addition and subtraction are performed last.

3. If there are several operations of the same priority, calculations are performed sequentially from left to right.

Numerical equality- two numeric expressions A and B, connected by an equal sign ("=").

Numerical inequality- two numerical expressions A and B, connected by an inequality sign (“<", ">", "≤" or "≥").

An expression containing a variable and which becomes a number when the variable is replaced by its value is called expression with variable or numerical form.

Equation with one variable(with one unknown) – a predicate of the form f₁(x) = f₂(x), where x ∊X, where f₁(x) and f₂(x) are expressions with variable x defined on the set X.

Any value of a variable x from the set X for which the equation turns into a true numerical equality is called root(solving the equation). Solve the equation- this means finding all its roots or proving that they do not exist. The set of all roots of the equation (or the truth set T of the predicate f₁(x) = f₂(x)) is called the set of solutions to the equation

The set of values ​​at which both sides of the equation are defined is called the region of permissible values ​​(ADV) of the variable x and the region of definition of the equation.

2. General questions of methods for studying algebraic material

The initial course of mathematics, along with basic arithmetic material, also includes elements of algebra, represented by the following concepts:

Numeric expressions;

Expressions with a variable;

Numerical equalities and inequalities;

Equations.

The purpose of including algebra elements in a primary school mathematics course is:

Consider arithmetic material more fully and deeply;

Bring students' generalizations to a higher level;

Create the prerequisites for more successful study of algebra in middle and high school.

Algebraic material is not highlighted as a separate topic in the program. It is distributed throughout the primary school mathematics course with individual questions. These questions are studied starting from grade 1, in parallel with the study of basic arithmetic material. The sequence of consideration of the questions proposed by the program is determined by the textbook.

Mastering the studied algebraic concepts in the elementary grades involves introducing appropriate terminology and performing simple operations without constructing formal logical definitions.