Methodological development on the topic: mathematical research in mathematics lessons. Mathematical methods for operations research

Plan:
1. Research of methods of mathematical statistics in pedagogical research.
1. Research of methods of mathematical statistics in pedagogical research.
IN Lately serious steps are being taken aimed at introducing mathematical methods of assessment and measurement into pedagogy pedagogical phenomena and establishing quantitative relationships between them. Mathematical methods allow us to approach the solution of one of the most difficult problems of pedagogy - the quantitative assessment of pedagogical phenomena. Only the processing of quantitative data and the conclusions obtained can objectively prove or disprove the hypothesis put forward.
A number of methods are proposed in the pedagogical literature statistical processing data from a pedagogical experiment (L. B. Itelson, Yu. V. Pavlov, etc.). When using the methods of mathematical statistics, it should be borne in mind that statistics itself does not reveal the essence of the phenomenon and cannot explain the reasons for the differences that arise between individual aspects of the phenomenon. For example, an analysis of the results of the study shows that the teaching method used gave more good results compared to previously recorded ones. However, these calculations cannot answer the question of why the new method is better than the previous one.
The most common mathematical methods used in pedagogy are:
1. Registration is a method of identifying the presence of a certain quality in each group member and a general count of the number of those who have or do not have this quality (for example, the number of children who attended classes without skipping and allowed absences, etc.).
2. Ranking (or the method of ranking assessment) involves arranging the collected data in a certain sequence, usually in descending or increasing order of some indicators and, accordingly, determining the place in this series of each of the studied (for example, compiling a list of children depending on the number of missing classes, etc.).
3. Scaling as a quantitative research method makes it possible to introduce digital indicators into the assessment of individual aspects of pedagogical phenomena. For this purpose, subjects are asked questions, answering which they must indicate the degree or form of assessment selected from among the given assessments, numbered in a certain order (for example, a question about playing sports with a choice of answers: a) I am interested in, b) I do regularly, c) I don’t exercise regularly, d) I don’t do any sports).
Correlating the results obtained with the norm (for given indicators) involves identifying deviations from the norm and correlating these deviations with acceptable intervals (for example, with programmed training, 85-90% of correct answers is often considered the norm; if there are fewer correct answers, this means that the program is too difficult , if more, it means it is too light).
The penetration of mathematical methods into the most diverse spheres of human activity actualizes the problem of modeling, with the help of which the correspondence of a real object to a mathematical model is established. Any model is a homomorphic image of a certain system in another system (homomorphism is a one-to-one correspondence between systems that preserves basic relationships and basic operations). Mathematical models in relation to the simulated objects there are analogues at the level of structures.
The specificity of statistical processing of the results of psychological and pedagogical research is that the analyzed database is characterized by a large number of indicators various types, their high variability under the influence of uncontrolled random factors, the complexity of correlations between sample variables, the need to take into account objective and subjective factors influencing diagnostic results, especially when deciding on the representativeness of the sample and assessing hypotheses regarding the general population. Research data according to their type can be divided into groups:
The first group is nominal variables (gender, personal data, etc.). Arithmetic operations on such quantities are meaningless, so the results descriptive statistics(average, variance) are not applicable to such quantities. The classic way to analyze them is to divide them into contingency classes with respect to certain nominal characteristics and check for significant differences between classes.
The second group of data has a quantitative scale of measurement, but this scale is ordinal (ordinal). When analyzing ordinal variables, both subsampling and rank technologies are used. Parametric methods are also applicable with some restrictions.
The third group - quantitative variables that reflect the degree of expression of the measured indicator - these are Cattell tests, academic performance and other assessment tests. When working with variables in this group, all standard types of analysis are applicable, and with a sufficient sample size their distribution is usually close to normal. Thus, the variety of variable types requires a wide range of mathematical methods to be used.
The analysis procedure can be divided into the following stages:
Preparing the database for analysis. This stage includes converting the data into electronic format, checking it for outliers, and choosing a method for working with missing values.
Descriptive statistics (calculation of means, variances, etc.). The results of descriptive statistics determine the characteristics of the parameters of the analyzed sample or subsamples specified by one or another partition.
Exploratory analysis. The task this stage is a meaningful study of various groups of sample indicators, their relationships, identifying the main explicit and hidden (latent) factors influencing the data, tracking changes in indicators, their relationships and the significance of factors when dividing the database into groups, etc. The research tools are various methods and technologies of correlation, factor and cluster analysis. The purpose of the analysis is to formulate hypotheses concerning both the given sample and the general population.
Detailed analysis of the results obtained and statistical testing of the hypotheses put forward. At this stage, hypotheses are tested regarding the types of distribution functions of random variables, the significance of differences in means and variances in subsamples, etc. When generalizing the research results, the issue of sample representativeness is resolved.
It should be noted that this sequence of actions is not, strictly speaking, chronological, with the exception of the first stage. As the results of descriptive statistics are obtained and certain patterns are identified, the need arises to test emerging hypotheses and immediately move on to their detailed analysis. But in any case, when testing hypotheses, it is recommended to analyze them using various mathematical means that adequately correspond to the model, and a hypothesis should be accepted at a particular level of significance only when it is confirmed by several different methods.
When organizing any measurement, a correlation (comparison) of what is being measured with a meter (standard) is always assumed. After the correlation (comparison) procedure, the measurement result is assessed. If in technology, as a rule, material standards are used as measuring instruments, then in social measurements, including pedagogical and psychological measurements, measuring instruments can be ideal. Indeed, in order to determine whether a child has formed or not formed a specific mental action, it is necessary to compare the actual with the necessary. In this case, what is necessary is ideal model, existing in the head of the teacher.
It should be noted that only some pedagogical phenomena can be measured. Most pedagogical phenomena cannot be measured, since there are no standards of pedagogical phenomena, without which measurement cannot be carried out.
As for such phenomena as activity, vigor, passivity, fatigue, skills, etc., it is not yet possible to measure them, since there are no standards for activity, passivity, vigor, etc. Due to the extreme complexity and, for the most part, practical impossibility of measuring pedagogical phenomena, special methods for approximate quantitative assessment of these phenomena are currently used.
Currently, it is customary to divide all psychological and pedagogical phenomena into two large categories: objective material phenomena (phenomena that exist outside and independently of our consciousness) and subjective intangible phenomena (phenomena characteristic of a given person).
Objective material phenomena include: chemical and biological processes, movements performed by a person, sounds he makes, actions he performs, etc.
Subjective intangible phenomena and processes include: sensations, perceptions and ideas, fantasies and thinking, feelings, drives and desires, motivation, knowledge, skills, etc.
All signs of objective material phenomena and processes are observable and can, in principle, always be measured, although modern science is sometimes unable to do this. Any property or sign can be measured directly. This means that through physical operations it can always be compared with some real value taken as the standard measure of the corresponding property or attribute.
Subjective intangible phenomena cannot be measured, since there are no and cannot be material standards for them. Therefore, approximate methods for assessing phenomena are used here - various indirect indicators.
The essence of using indirect indicators is that the measured property or sign of the phenomenon being studied is associated with certain material properties, and the value of these material properties is taken as an indicator of the corresponding intangible phenomena. For example, the effectiveness of a new teaching method is assessed by the performance of students, the quality of a student’s work by the number of mistakes made, the difficulty of the material being studied by the amount of time spent, the development of mental or moral traits by the number of corresponding actions or misdeeds, etc.
For all that great interest, which researchers usually show to methods quantitative analysis experimental data and bulk material obtained using different methods, the essential stage of processing is their qualitative analysis. By using quantitative methods it is possible, with varying degrees of reliability, to identify the advantage of a particular method or to discover a general trend, to prove that the scientific assumption being tested was justified, etc. However, a qualitative analysis should answer the question of why this happened, what favored it, and what served as an obstacle, and how significant the influence of these interferences is, whether the experimental conditions were too specific in order to this technique could be recommended for use in other conditions, etc. At this stage, it is also important to analyze the reasons that prompted individual respondents to give a negative answer, and to identify the reasons for certain typical and even random errors in the works of individual children, etc. The use of all these methods of analyzing the collected data helps to more accurately evaluate the results of the experiment, increases the reliability of conclusions about them and provides more grounds for further theoretical generalizations.
Statistical methods in pedagogy are used only for quantitative characteristics of phenomena. In order to draw conclusions and conclusions, qualitative analysis is necessary. Thus, in pedagogical research, methods of mathematical statistics should be used carefully, taking into account the characteristics of pedagogical phenomena.
Thus, most of the numerical characteristics in mathematical statistics are used in the case when the property or phenomenon being studied has a normal distribution, which is characterized by a symmetrical arrangement of the values ​​of the elements of the population relative to the average value. Unfortunately, due to insufficient knowledge of pedagogical phenomena, the laws of distribution in relation to them are, as a rule, unknown. Further, to evaluate the results of a study, rank values ​​are often taken, which are not the results of quantitative measurements. Therefore, arithmetic operations cannot be performed with them, and therefore numerical characteristics cannot be calculated for them.
Each statistical series and its graphical representation represent grouped and clearly presented material that should be subjected to statistical processing.
Statistical processing methods make it possible to obtain a number of numerical characteristics that allow us to make a forecast of the development of the process of interest to us. These characteristics, in particular, make it possible to compare different series of numbers obtained in pedagogical research and make appropriate pedagogical conclusions and recommendations.
All variation series may differ from each other in the following ways:
1. In scope, i.e. its upper and lower boundaries, which are usually called limits.
2. The value of the attribute around which the majority of the variant is concentrated. This attribute value reflects the central tendency of the series, i.e. typical for the series.
3. Variations around the central tendency of the series.
In accordance with this, all statistical indicators of the variation series are divided into two groups:
-indicators that characterize the central tendency or level of the series;
-indicators characterizing the level of variation around the central tendency.
The first group includes various characteristics of the average value: median, arithmetic mean, geometric mean, etc. To the second - variation range (limits), average absolute deviation, average standard deviation, dispersion, asymmetry and variation coefficients. There are other indicators, but we will not consider them, because... they are not used in educational statistics.
Currently, the concept of “model” is used in different meanings, the simplest of them is the designation of a sample, a standard. In this case, the model of the thing does not carry any new information and does not serve the purposes of scientific knowledge. The term “model” is not used in science in this sense. IN in a broad sense A model is understood as a mentally or practically created structure that reproduces part of reality in a simplified and visual form. In a narrower sense, the term “model” is used to depict a certain area of ​​phenomena using another, more studied, easily understood one. In pedagogical sciences, this concept is used in a broad sense as a specific image of the object being studied, which displays real or expected properties, structure, etc. In educational subjects, modeling is widely used as an analogy that can exist between systems at the following levels: the results that the compared systems produce; functions that determine these results; structures ensuring the implementation of these functions; elements that make up structures.
V. M. Tarabaev points out that the technique of the so-called multifactor experiment is currently used. In a multifactorial experiment, researchers approach the problem empirically - they vary with a large number of factors on which, as they believe, the course of the process depends. This variation by various factors is carried out using modern methods of mathematical statistics.
A multifactorial experiment is based on statistical analysis and using a systematic approach to the subject of research. It is assumed that the system has an input and output that can be controlled, and it is also assumed that this system can be controlled in order to achieve a certain output result. In a multifactorial experiment, the entire system is studied without an internal picture of its complex mechanism. This type of experiment opens up great opportunities for pedagogy.
Literature:
1. Zagvyazinsky, V.I. Methodology and methods of psychological and pedagogical research: textbook. aid for students higher ped. textbook institutions / Zagvyazinsky V.I., Atakhanov R. - M.: Academy, 2005.
2. Gadelshina, T. G. Methodology and methods psychological research: textbook method. manual / Gadelshina T. G. - Tomsk, 2002.
3. Kornilova, T. V. Experimental psychology: theory and methods: textbook for universities / Kornilova T. V. - M.: Aspect Press, 2003.
4. Kuzin, F. A. Candidate's dissertation: writing methods, rules of execution and order of defense / Kuzin F. A. - M., 2000.

In the history of mathematics, we can roughly distinguish two main periods: elementary and modern mathematics. The milestone from which it is customary to count the era of new (sometimes called higher) mathematics was the 17th century - the century of the appearance of mathematical analysis. By the end of the 17th century. I. Newton, G. Leibniz and their predecessors created a new apparatus differential calculus and integral calculus, which forms the basis mathematical analysis and even, perhaps, the mathematical basis of all modern natural science.

Mathematical analysis is a vast area of ​​mathematics with a characteristic object of study (variable quantity), a unique research method (analysis by means of infinitesimals or by means of passages to limits), a certain system of basic concepts (function, limit, derivative, differential, integral, series) and constantly improving and a developing apparatus, the basis of which is differential and integral calculus.

Let's try to give an idea of ​​what kind of mathematical revolution occurred in the 17th century, what characterizes the transition associated with the birth of mathematical analysis from elementary mathematics to what is now the subject of research in mathematical analysis, and what explains its fundamental role in the entire modern system of theoretical and applied knowledge .

Imagine that in front of you is a beautifully executed color photograph of a storm rushing onto the shore. ocean wave: a powerful stooped back, a steep but slightly sunken chest, a head already tilted forward and ready to fall with a gray mane tormented by the wind. You stopped the moment, you managed to catch the wave, and you can now carefully study it in every detail without haste. A wave can be measured, and using the tools of elementary mathematics, you can draw many important conclusions about this wave, and therefore all its ocean sisters. But by stopping the wave, you deprived it of movement and life. Its origin, development, running, the force with which it hits the shore - all this turned out to be outside your field of vision, because you do not yet have either a language or a mathematical apparatus suitable for describing and studying not static, but developing, dynamic processes, variables and their relationships.

“Mathematical analysis is no less comprehensive than nature itself: it determines all tangible relationships, measures times, spaces, forces, temperatures.” J. Fourier

Movement, variables and their relationships surround us everywhere. Various types of motion and their patterns constitute the main object of study of specific sciences: physics, geology, biology, sociology, etc. Therefore, precise language and corresponding mathematical methods for describing and studying variable quantities turned out to be necessary in all areas of knowledge to approximately the same extent as numbers and arithmetic are necessary when describing quantitative relationships. So, mathematical analysis forms the basis of the language and mathematical methods for describing variables and their relationships. Nowadays, without mathematical analysis it is impossible not only to calculate space trajectories, work nuclear reactors, the running of the ocean wave and the patterns of development of the cyclone, but also to economically manage production, resource distribution, organization technological processes, predict the course of chemical reactions or changes in the numbers of various interconnected species of animals and plants in nature, because all of these are dynamic processes.

Elementary mathematics was mainly the mathematics of constant quantities; it studied mainly the relationships between the elements of geometric figures, the arithmetic properties of numbers and algebraic equations. Its attitude to reality can to some extent be compared with an attentive, even thorough and complete study of each fixed frame of a film that captures the changing, developing living world in its movement, which, however, is not visible in a separate frame and which can only be observed by looking the tape as a whole. But just as cinema is unthinkable without photography, so modern mathematics is impossible without that part of it that we conventionally call elementary, without the ideas and achievements of many outstanding scientists, sometimes separated by tens of centuries.

Mathematics is united, and the “higher” part of it is connected with the “elementary” part in much the same way as the next floor of a house under construction is connected with the previous one, and the width of the horizons that mathematics opens to us in the world, depends on which floor of this building we managed to climb to. Born in the 17th century. mathematical analysis has opened up opportunities for us to scientifically describe, quantitatively and qualitatively study variables and motion in the broad sense of the word.

What are the prerequisites for the emergence of mathematical analysis?

By the end of the 17th century. The following situation has arisen. Firstly, within the framework of mathematics itself, over many years, some important classes of problems of the same type have accumulated (for example, problems of measuring areas and volumes of non-standard figures, problems of drawing tangents to curves) and methods for solving them in various special cases have appeared. Secondly, it turned out that these problems are closely related to the problems of describing arbitrary (not necessarily uniform) mechanical motion, and in particular with the calculation of its instantaneous characteristics (speed, acceleration at any time), as well as with finding the distance traveled for movement occurring at a given variable speed. The solution to these problems was necessary for the development of physics, astronomy, and technology.

Finally, thirdly, by the middle of the 17th century. the works of R. Descartes and P. Fermat laid the foundations of the analytical method of coordinates (the so-called analytical geometry), which made it possible to formulate geometric and physical tasks in the general (analytical) language of numbers and numerical dependencies, or, as we now say, numerical functions.

NIKOLAY NIKOLAEVICH LUZIN (1883-1950) N. N. Luzin - Soviet mathematician, founder of the Soviet school of function theory, academician (1929). Luzin was born in Tomsk and studied at the Tomsk gymnasium. The formalism of the gymnasium mathematics course alienated the talented young man, and only a capable tutor was able to reveal to him the beauty and greatness of mathematical science. In 1901, Luzin entered the mathematics department of the Faculty of Physics and Mathematics of Moscow University. From the first years of his studies, issues related to infinity fell into his circle of interests. At the end of the 19th century. German scientist G. Cantor created the general theory of infinite sets, which received numerous applications in research discontinuous functions. Luzin began to study this theory, but his studies were interrupted in 1905. The student, who took part in revolutionary activities, had to leave for France for a while. There he listened to lectures by the most prominent French mathematicians that time. Upon returning to Russia, Luzin graduated from the university and was left to prepare for a professorship. Soon he again left for Paris, and then to Göttingen, where he became close to many scientists and wrote his first scientific works. The main problem that interested the scientist was the question of whether there could be sets containing more elements than the set of natural numbers, but less than the set of points on a segment (the continuum problem). For any infinite set that could be obtained from segments using the operations of union and intersection of countable collections of sets, this hypothesis was satisfied, and in order to solve the problem, it was necessary to find out what other ways there were to construct sets. At the same time, Luzin studied the question of whether it is possible to represent any periodic function, even one with infinitely many discontinuity points, as a sum of a trigonometric series, i.e. sums of an infinite set harmonic vibrations. Luzin obtained a number of significant results on these issues and in 1915 defended his dissertation “Integral and Trigonometric Series,” for which he was immediately awarded the academic degree of Doctor of Pure Mathematics, bypassing the intermediate master’s degree that existed at that time. In 1917 Luzin became a professor at Moscow University. A talented teacher, he attracted the most capable students and young mathematicians. Luzin's school reached its peak in the first post-revolutionary years. Luzin’s students formed a creative team, which they jokingly called “Lusitania.” Many of them received first-class scientific results while still a student. For example, P. S. Aleksandrov and M. Ya. Suslin (1894-1919) discovered a new method for constructing sets, which served as the beginning of the development of a new direction - descriptive set theory. Research in this area carried out by Luzin and his students showed that the usual methods of set theory are not enough to solve many of the problems that arise in it. Luzin's scientific predictions were fully confirmed in the 60s. XX century Many of N. N. Luzin’s students later became academicians and corresponding members of the USSR Academy of Sciences. Among them is P. S. Alexandrov. A. N. Kolmogorov. M. A. Lavrentyev, L. A. Lyusternik, D. E. Menshov, P. S. Novikov. L. G. Shnirelman and others. Modern Soviet and foreign mathematicians in their works develop the ideas of N. N. Luzin.

The confluence of these circumstances led to the fact that at the end of the 17th century. two scientists - I. Newton and G. Leibniz - independently of each other managed to create a mathematical apparatus for solving these problems, summing up and generalizing individual results of their predecessors, including the ancient scientist Archimedes and contemporaries of Newton and Leibniz - B. Cavalieri, B. Pascal , D. Gregory, I. Barrow. This apparatus formed the basis of mathematical analysis - a new branch of mathematics that studies various developing processes, i.e. relationships between variables, which in mathematics are called functional dependencies or, in other words, functions. By the way, the term “function” itself was required and naturally arose precisely in the 17th century, and by now it has acquired not only general mathematical, but also general scientific significance.

Initial information about the basic concepts and mathematical apparatus of analysis is given in the articles “Differential calculus” and “Integral calculus”.

In conclusion, I would like to dwell on only one principle of mathematical abstraction, common to all mathematics and characteristic of analysis, and in this regard explain in what form mathematical analysis studies variables and what is the secret of such universality of its methods for studying all kinds of specific developing processes and their interrelations .

Let's look at a few illustrative examples and analogies.

Sometimes we no longer realize that, for example, a mathematical relation written not for apples, chairs or elephants, but in an abstract form abstracted from specific objects, is an outstanding scientific achievement. This is a mathematical law that, as experience shows, is applicable to various specific objects. So, studying in mathematics general properties abstract, abstract numbers, we thereby study quantitative relationships real world.

For example, from a school mathematics course it is known that, therefore, in a specific situation you could say: “If they don’t give me two six-ton ​​dump trucks to transport 12 tons of soil, then I can ask for three four-ton dump trucks and the work will be done, and if they give me only one four-ton dump truck, then she will have to make three flights.” Thus, the abstract numbers and numerical patterns that are now familiar to us are associated with their specific manifestations and applications.

The laws of change in specific variables and developing processes of nature are related in approximately the same way to the abstract, abstract form-function in which they appear and are studied in mathematical analysis.

For example, an abstract ratio may reflect the dependence of a cinema's box office on the number of tickets sold, if 20 is 20 kopecks - the price of one ticket. But if we are riding a bicycle on a highway, traveling 20 km per hour, then this same ratio can be interpreted as the relationship between the time (hours) of our cycling trip and the distance covered during this time (kilometers). You can always say that, for example, a change of several times leads to a proportional (i.e., the same number of times) change in the value of , and if , then the opposite conclusion is also true. This means, in particular, to double the box office of a movie theater, you will have to attract twice as many spectators, and in order to travel twice as far on a bicycle at the same speed, you will have to ride twice as long.

Mathematics studies both the simplest dependence and other, much more complex dependences in a general, abstract form, abstracted from a particular interpretation. The properties of a function or methods for studying these properties identified in such a study will be of the nature of general mathematical techniques, conclusions, laws and conclusions applicable to each specific phenomenon in which the function studied in abstract form occurs, regardless of what area of ​​knowledge this phenomenon belongs to .

So, mathematical analysis as a branch of mathematics took shape at the end of the 17th century. The subject of study in mathematical analysis (as it appears from modern positions) are functions, or, in other words, dependencies between variable quantities.

With the advent of mathematical analysis, mathematics became accessible to the study and reflection of developing processes in the real world; mathematics included variables and motion.

FEDERAL AGENCY FOR EDUCATION

State educational institution higher professional education "Ural State University them. »

History department

Department of Documentation and Information Support of Management

Mathematical methods in scientific research

Course program

Standard 350800 “Documentation and documentation support management"

Standard 020800 “Historical and archival studies”

Ekaterinburg

I approve

Vice-Rector

(signature)

The program of the discipline “Mathematical methods in scientific research” is compiled in accordance with the requirements university component to the mandatory minimum content and level of training:

certified specialist by specialty

Documentation and documentation support for management (350800),

Historical and archival studies (020800),

in the cycle “General humanitarian and socio-economic disciplines” of the state educational standard higher vocational education.

Semester III

According to the curriculum of specialty No. 000 – Documentation and documentation support for management:

Total labor intensity of the discipline: 100 hours,

including lectures 36 hours

According to the curriculum of specialty No. 000 – Historical and Archival Studies

Total labor intensity of the discipline: 50 hours,

including lectures 36 hours

Control activities:

Tests 2 people/hour

Compiled by: , Ph.D. ist. Sciences, Associate Professor of the Department of Documentation and information support Department of the Ural State University

Department of Documentation and Information Support of Management

dated 01.01.01 No. 1.

Agreed:

Deputy chairman

Humanitarian Council

_________________

(signature)

(C) Ural State University

(WITH) , 2006

INTRODUCTION

The course “Mathematical methods in socio-economic research” is designed to familiarize students with the basic techniques and methods of processing quantitative information developed by statistics. Its main task is to expand the methodological scientific apparatus of researchers, to teach them to apply in practical and research activities in addition to traditional methods, based on logical analysis, mathematical methods that help quantitatively characterize historical phenomena and facts.

Currently, mathematical apparatus and mathematical methods are used in almost all areas of science. This natural process, it is often called the mathematization of science. In philosophy, mathematization is usually understood as the application of mathematics in various sciences. Mathematical methods have long been firmly established in the arsenal of research methods of scientists; they are used to summarize data, identify trends and patterns in the development of social phenomena and processes, typology and modeling.

Knowledge of statistics is necessary to correctly characterize and analyze the processes occurring in the economy and society. To do this, you need to master the sampling method, summarize and group data, be able to calculate average and relative values, indicators of variation, and correlation coefficients. An element of information culture is the skills of correctly formatting tables and constructing graphs, which are an important tool for systematizing primary socio-economic data and visually presenting quantitative information. To assess temporary changes, it is necessary to have an idea of ​​the system of dynamic indicators.

Using sampling research techniques allows you to study large amounts of information presented by mass sources, save time and labor, while obtaining scientifically significant results.

Mathematics -statistical methods occupy auxiliary positions, complementing and enriching traditional methods of socio-economic analysis; their development is a necessary component of the qualifications of a modern specialist - a document specialist, a historian-archivist.

Currently, mathematical and statistical methods are actively used in marketing and sociological research, in collecting operational management information, drawing up reports and analyzing document flows.

Quantitative analysis skills are required to prepare qualification works, abstracts and other research projects.

Experience in using mathematical methods shows that their use must be carried out in compliance with following principles to obtain reliable and representative results:

1) the determining role is played by the general methodology and theory of scientific knowledge;

2) a clear and correct formulation of the research problem is necessary;

3) selection of quantitatively and qualitatively representative socio-economic data;

4) correct application of mathematical methods, i.e. they must correspond to the research problem and the nature of the data being processed;

5) a meaningful interpretation and analysis of the results obtained is required, as well as mandatory additional verification of the results obtained mathematical processing information.

Mathematical methods help improve the technology of scientific research: increase its efficiency; they provide great time savings, especially when processing large amounts of information, and allow you to identify hidden information stored in the source.

In addition, mathematical methods are closely related to such areas of scientific information activities as the creation of historical data banks and archives of machine-readable data. The achievements of the era cannot be ignored, and information technology is becoming one of the most important factors in the development of all spheres of society.

COURSE PROGRAM

Topic 1. INTRODUCTION. MATHEMATIZATION OF HISTORICAL SCIENCE

Purpose and objectives of the course. Objective need for improvement historical methods through the use of mathematics.

Mathematization of science, main content. Prerequisites for mathematization: natural science prerequisites; socio-technical prerequisites. Boundaries of mathematization of science. Levels of mathematization for natural, technical, economic and human sciences. The main laws of mathematization of science: the impossibility of fully covering the areas of research of other sciences by means of mathematics; correspondence of the applied mathematical methods to the content of the science being mathematized. The emergence and development of new applied mathematical disciplines.

Mathematization historical science. Main stages and their features. Prerequisites for the mathematization of historical science. The significance of the development of statistical methods for the development of historical knowledge.

Socio-economic research using mathematical methods in pre-revolutionary and Soviet historiography of the 20s (, etc.)

Mathematical and statistical methods in the works of historians of the 60-90s. Computerization of science and dissemination of mathematical methods. Creation of databases and prospects for the development of information support for historical research. The most important results of the application of mathematical methods in socio-economic and historical and cultural research (, etc.).

Correlation of mathematical methods with other methods historical research: historical-comparative, historical-typological, structural, systemic, historical-genetic methods. Basic methodological principles of application of mathematical and statistical methods in historical research.

Topic 2. STATISTICAL INDICATORS

Basic techniques and methods statistical study social phenomena: statistical observation, reliability of statistical data. Basic forms of statistical observation, purpose of observation, object and unit of observation. Statistical document as a historical source.

Statistical indicators (indicators of volume, level and ratio), its main functions. Quantitative and qualitative side of a statistical indicator. Varieties of statistical indicators (volumetric and qualitative; individual and generalizing; interval and moment).

Basic requirements for the calculation of statistical indicators, ensuring their reliability.

Interrelation of statistical indicators. System of indicators. Summary indicators.

Absolute values, definition. Types of absolute statistical quantities, their meaning and methods of obtaining. Absolute values ​​as a direct result of a summary of statistical observation data.

Units of measurement, their choice depending on the essence of the phenomenon being studied. Natural, cost and labor units of measurement.

Relative values. The main content of the relative indicator, the forms of their expression (coefficient, percentage, ppm, decimille). Dependence of the form and content of the relative indicator.

Base of comparison, choice of base when calculating relative values. Basic principles for calculating relative indicators, ensuring comparability and reliability of absolute indicators (by territory, range of objects, etc.).

Relative values ​​of structure, dynamics, comparison, coordination and intensity. Methods for calculating them.

The relationship between absolute and relative values. The need for their complex use.

Topic 3. DATA GROUPING. TABLES.

Summary indicators and grouping in historical research. Problems solved by these methods in scientific research: systematization, generalization, analysis, ease of perception. Statistical population, units of observation.

Objectives and main contents of the summary. Summary - second stage statistical research. Varieties of summary indicators (simple, auxiliary). The main stages of calculating summary indicators.

Grouping is the main method of processing quantitative data. Grouping tasks and their significance in scientific research. Types of groups. The role of groupings in the analysis of social phenomena and processes.

The main stages of constructing a grouping: determination of the population being studied; selection of a grouping characteristic (quantitative and qualitative characteristics; alternative and non-alternative; factorial and effective); distribution of the population into groups depending on the type of grouping (determining the number of groups and the size of intervals), scale of measurement of characteristics (nominal, ordinal, interval); selecting the form of presentation of grouped data (text, table, graph).

Typological grouping, definition, main tasks, principles of construction. The role of typological grouping in the study of socio-economic types.

Structural grouping, definition, main tasks, principles of construction. The role of structural grouping in the study of the structure of social phenomena

Analytical (factorial) grouping, definition, main tasks, principles of construction, The role of analytical grouping in the analysis of the interrelations of social phenomena. The need for integrated use and study of groupings for the analysis of social phenomena.

General requirements for the construction and design of tables. Table layout development. Table details (numbering, title, names of columns and rows, symbols, designation of numbers). Methodology for filling out table information.

Topic 4. GRAPHICAL METHODS FOR ANALYSIS OF SOCIO-ECONOMIC

INFORMATION

The role of schedules and graphic image in scientific research. Objectives of graphical methods: providing clarity of perception of quantitative data; analytical tasks; characterization of the properties of signs.

Statistical graph, definition. The main elements of a graph: graph field, graphic image, spatial reference points, scale reference points, graph explication.

Types of statistical graphs: line diagram, features of its construction, graphic images; bar chart (histogram), definition of the rule for constructing histograms in the case of equal and unequal intervals; pie chart, definition, methods of construction.

Characteristic distribution polygon. Normal distribution sign and its graphic representation. Features of the distribution of features characterizing social phenomena: skewed, asymmetric, moderately asymmetric distribution.

Linear dependence between characteristics, features of a graphical representation of a linear relationship. Features of linear dependence in characterizing social phenomena and processes.

The concept of a trend in a time series. Trend identification using graphical methods.

Topic 5. AVERAGE VALUES

Average values ​​in scientific research and statistics, their essence and definition. Basic properties of average values ​​as a generalizing characteristic. The relationship between the method of averages and groupings. General and group averages. Conditions for the typicality of averages. Basic research problems that solve averages.

Methods for calculating averages. Arithmetic mean - simple, weighted. Basic properties of the arithmetic mean. Features of calculating the average for discrete and interval distribution series. The dependence of the method of calculating the arithmetic mean depending on the nature of the source data. Features of the interpretation of the arithmetic average.

Median - average aggregate structures, definition, basic properties. Determination of the median indicator for a ranked quantitative series. Calculate the median for a measure represented by interval grouping.

Fashion is an average indicator of the structure of a population, basic properties and content. Determination of mode for discrete and interval series. Features of the historical interpretation of fashion.

The relationship between the arithmetic mean, median and mode, the need for them integrated use, checking the typicality of the arithmetic mean.

Topic 6. INDICATORS OF VARIATION

Study of variability (variability) of attribute values. The main content of measures of trait dispersion, and their use in research activities.

Absolute and average variations. Variational scope, main content, calculation methods. Average linear deviation. Standard deviation, main content, calculation methods for discrete and interval quantitative series. The concept of trait dispersion.

Relative measures of variation. Oscillation coefficient, main content, calculation methods. Coefficient of variation, main content, methods of calculation. The significance and specificity of the use of each indicator of variation in the study of socio-economic characteristics and phenomena.

Topic 7.

The study of changes in social phenomena over time is one of the most important tasks socio-economic analysis.

The concept of a time series. Moment and interval time series. Requirements for constructing time series. Comparability in dynamics series.

Indicators of changes in dynamics series. The main content of the indicators of the dynamics series. Row level. Basic and chain indicators. Absolute increase in the level of dynamics, basic and chain absolute increases, calculation methods.

Growth rate indicators. Basic and chain growth rates. Features of their interpretation. Growth rate indicators, main content, methods for calculating basic and chain growth rates.

Average level of a series of dynamics, basic content. Techniques for calculating the arithmetic mean for moment series with equal and unequal intervals and for interval series with equal intervals. Average absolute increase. Average growth rate. Average growth rate.

Comprehensive analysis interconnected series speakers. Identification of the general development trend - trend: moving average method, enlargement of intervals, analytical techniques for processing dynamics series. The concept of interpolation and extrapolation of time series.

Topic 8.

The need to identify and explain relationships to study socio-economic phenomena. Types and forms of relationships studied by statistical methods. The concept of functional and correlation connection. The main content of the correlation method and the problems solved with its help in scientific research. Main stages correlation analysis. Peculiarities of interpretation of correlation coefficients.

Linear correlation coefficient, properties of features for which the linear correlation coefficient can be calculated. Methods for calculating the linear correlation coefficient for grouped and ungrouped data. Regression coefficient, main content, calculation methods, interpretation features. Determination coefficient and its meaningful interpretation.

The limits of application of the main types of correlation coefficients depending on the content and form of presentation of the source data. Correlation coefficient. Coefficient rank correlation. Association and contingency coefficients for alternative qualitative signs. Approximate methods for determining the relationship between characteristics: the Fechner coefficient. Autocorrelation coefficient. Information coefficients.

Methods for ordering correlation coefficients: correlation matrix, pleiad method.

Methods of multivariate statistical analysis: factor analysis, component analysis, regression analysis, cluster analysis. Modeling prospects historical processes to study social phenomena.

Topic 9. SAMPLING RESEARCH

Reasons and conditions for conducting a sample study. The need for historians to use methods for partial study of social objects.

Main types of partial survey: monographic, main array method, sample study.

Definition of the sampling method, basic properties of the sample. Sample representativeness and sampling error.

Stages of conducting a sample study. Determining the sample size, basic techniques and methods for finding the sample size (mathematical methods, table of large numbers). The practice of determining sample size in statistics and sociology.

Methods of forming a sample population: proper random sampling, mechanical sampling, typical and cluster sampling. Methodology for organizing sample population censuses, budget surveys of families of workers and peasants.

Methodology for proving the representativeness of the sample. Random, systematic sampling and observation errors. The role of traditional methods in determining the reliability of sampling results. Mathematical methods for calculating sampling error. Dependence of error on sample size and type.

Features of interpretation of sample results and distribution of sample population indicators to the general population.

Natural sampling, main content, features of formation. The problem of representativeness of natural sampling. The main stages of proving the representativeness of a natural sample: the use of traditional and formal methods. The method of sign criterion, the method of series - as methods of proving the property of random sampling.

The concept of a small sample. Basic principles of using it in scientific research

Topic 11. METHODS FOR FORMALIZING INFORMATION FROM MASS SOURCES

The need to formalize information from mass sources to obtain hidden information. The problem of measuring information. Quantitative and qualitative characteristics. Scales for measuring quantitative and qualitative characteristics: nominal, ordinal, interval. The main stages of measuring source information.

Types of mass sources, features of their measurement. Methodology for constructing a unified questionnaire based on materials from a structured, semi-structured historical source.

Features of measuring information from an unstructured narrative source. Content analysis, its content and prospects for use. Types of content analysis. Content analysis in sociological and historical research.

The relationship between mathematical and statistical methods of information processing and methods for formalizing source information. Computerization of research. Databases and data banks. Database technology in socio-economic research.

Tasks for independent work

To consolidate the lecture material, students are offered assignments for independent work on the following course topics:

Relative indicators Average indicators Grouping method Graphical methods Dynamics indicators

Completion of assignments is controlled by the teacher and is a prerequisite for admission to the test.

Sample list of questions for testing

1. Mathematization of science, essence, prerequisites, levels of mathematization

2. Main stages and features of mathematization of historical science

3. Prerequisites for the use of mathematical methods in historical research

4. Statistical indicator, essence, functions, varieties

3. Methodological principles application of statistical indicators in historical research

6. Absolute values

7. Relative quantities, content, forms of expression, basic principles of calculation.

8. Types of relative quantities

9. Objectives and main content of the data summary

10. Grouping, main content and objectives in the study

11. The main stages of building a group

12. The concept of a grouping characteristic and its gradations

13. Types of grouping

14. Rules for constructing and designing tables

15. Time series, requirements for constructing a time series

16. Statistical graph, definition, structure, tasks to be solved

17. Types of statistical graphs

18. Polygon distribution of the characteristic. Normal distribution of the trait.

19. Linear dependence between characteristics, methods for determining linearity.

20. The concept of a trend in a time series, methods for determining it

21. Average values ​​in scientific research, their essence and basic properties. Conditions for the typicality of averages.

22. Types of population averages. Interrelation of average indicators.

23. Statistical indicators of dynamics, general characteristics, kinds

24. Absolute indicators of changes in dynamics series

25. Relative indicators of changes in dynamics series (growth rates, growth rates)

26. Average indicators of the dynamic series

27. Indicators of variation, main content and tasks to be solved, types

28. Types of partial observation

29. Selective research, main content and tasks to be solved

30. Sample and general population, basic properties of the sample

31. Stages of conducting a sample study, general characteristics

32. Determining sample size

33. Methods for forming a sample population

34. Sampling error and methods for determining it

35. Representativeness of the sample, factors influencing representativeness

36. Natural sampling, the problem of representativeness of natural sampling

37. Main stages of proving the representativeness of a natural sample

38. Correlation method, essence, main tasks. Features of interpretation of correlation coefficients

39. Statistical observation as a method of collecting information, the main types of statistical observation.

40. Types of correlation coefficients, general characteristics

41. Linear correlation coefficient

42. Autocorrelation coefficient

43. Methods of formalization historical sources: unified questionnaire method

44. Methods for formalizing historical sources: content analysis method

III.Distribution of course hours by topics and types of work:

according to the specialty curriculum (No. 000 – document management and documentation support for management)

Name

sections and topics

Auditory lessons

Independent work

including

Introduction. Mathematization of science

Statistical indicators

Grouping data. Tables

Average values

Variation indicators

Statistical indicators of dynamics

Methods multivariate analysis. Correlation coefficients

Sample study

Methods for formalizing information

Distribution of course hours by topics and types of work

according to the curriculum of specialty No. 000 – historical and archival studies

Name

sections and topics

Auditory lessons

Independent work

including

Practical (seminars, laboratory work)

Introduction. Mathematization of science

Statistical indicators

Grouping data. Tables

Graphic methods for analyzing socio-economic information

Average values

Variation indicators

Statistical indicators of dynamics

Multivariate analysis methods. Correlation coefficients

Sample study

Methods for formalizing information

IV. Final control form - test

V. Educational and methodological support of the course

Slavko methods in historical research. Textbook. Ekaterinburg, 1995

Mazur methods in historical research. Guidelines. Ekaterinburg, 1998

additional literature

Andersen T. Statistical analysis of time series. M., 1976.

Borodkin statistical analysis in historical research. M., 1986

Borodkin informatics: stages of development // New and recent history. 1996. № 1.

Tikhonov for humanists. M., 1997

Garskova and data banks in historical research. Göttingen, 1994

Gerchuk methods in statistics. M., 1968

Druzhinin method and its application in socio-economic research. M., 1970

Jessen R. Methods of statistical surveys. M., 1985

Ginny K. Average values. M., 1970

Yuzbashev theory of statistics. M., 1995.

Rumyantsev theory of statistics. M., 1998

Shmoilov study of the main trend and relationship in the dynamics series. Tomsk, 1985

Yates F. Sampling method in censuses and surveys / trans. from English . M., 1976

Historical information science. M., 1996.

Kovalchenko historical research. M., 1987

Computer in economic history. Barnaul, 1997

Circle of ideas: models and technologies of historical informatics. M., 1996

Circle of ideas: traditions and trends of historical informatics. M., 1997

Circle of ideas: macro and micro approaches in historical information science. M., 1998

Circle of ideas: historical computer science on the threshold of the 21st century. Cheboksary, 1999

Circle of ideas: historical computer science in the information society. M., 2001

General theory of statistics: Textbook / ed. And. M., 1994.

Workshop on the theory of statistics: Proc. allowance M., 2000

Eliseeva statistics. M., 1990

Slavko-statistical methods in historical and research M., 1981

Slavko's methods in studying the history of the Soviet working class. M., 1991

Statistical Dictionary / ed. . M., 1989

Theory of statistics: Textbook / ed. , M., 2000

Ursul Society. Introduction to social informatics. M., 1990

Schwartz G. Selective method / trans. with him. . M., 1978

Mathematical methods for operations research

regression analysis model software

Introduction

Description subject area and statement of the research problem

Practical part

Conclusion

Bibliography

Introduction

In economics, the basis of almost any activity is a forecast. Based on the forecast, a plan of actions and measures is drawn up. Thus, we can say that the forecast of macroeconomic variables is a fundamental component of the plans of all subjects economic activity. Forecasting can be carried out both on the basis of qualitative (expert) and quantitative methods. The latter themselves can do nothing without qualitative analysis, as well as expert assessments must be supported by reasonable calculations.

Now forecasts, even at the macroeconomic level, are of a scenario nature and are developed according to the principle: what happens if… , - and are often a preliminary stage and justification for large national economic programs. Macroeconomic forecasts are usually carried out with a lead period of one year. Modern practice of economic functioning requires short-term forecasts (six months, months, ten days, weeks). Designed for the task of providing advanced information to individual participants in the economy.

With changes in forecasting objects and tasks, the list of forecasting methods has changed. Adaptive methods of short-term forecasting have received rapid development.

Modern economic forecasting requires developers to have versatile specialization, knowledge from various areas science and practice. The tasks of a forecaster include knowledge of the scientific (usually mathematical) forecasting apparatus, the theoretical foundations of the forecasted process, information flows, software, and interpretation of forecasting results.

The main function of a forecast is to justify the possible state of an object in the future or determine alternative paths.

The importance of gasoline as the main type of fuel today is difficult to overestimate. And it is equally difficult to overestimate the impact of its price on the economy of any country. The development of the country's economy as a whole depends on the dynamics of fuel prices. An increase in gasoline prices causes an increase in prices for industrial goods, leading to increased inflationary costs in the economy and a decrease in the profitability of energy-intensive industries. The cost of petroleum products is one of the components prices of consumer goods, and transportation costs influence the price structure of all consumer goods and services without exception.

Special meaning The issue of the cost of gasoline is becoming increasingly important in the developing Ukrainian economy, where any change in prices causes an immediate reaction in all its sectors. However, the influence of this factor is not limited only to the economic sphere; the consequences of its fluctuations can also include many political and social processes.

Thus, research and forecasting of the dynamics of this indicator acquires special significance.

The purpose of this work is to forecast fuel prices for the near future.

1. Description of the subject area and statement of the research problem

The Ukrainian gasoline market can hardly be called constant or predictable. And there are many reasons for this, starting with the fact that the raw material for the production of fuel is oil, the prices and volume of production of which are determined not only by supply and demand in the domestic and foreign markets, but also by state policy, as well as special agreements of manufacturing companies. Given the highly dependent Ukrainian economy, it is dependent on the export of steel and chemicals, and prices for these products are constantly changing. And speaking of gasoline prices, one cannot fail to note their upward trend. Despite the government's restrictive policy, growth is what most consumers are accustomed to. Prices for petroleum products in Ukraine today change daily. Mainly depend on the price of oil on the world market ($/barrel) and the level of tax burden.

The study of gasoline prices is very relevant at the present time, since the prices of other goods and services depend on these prices.

This paper will examine the dependence of gasoline prices on time and factors such as:

ü oil prices, US dollar per barrel

ü official dollar exchange rate (NBU), hryvnia per US dollar

ü consumer price index

The price of gasoline, which is a product of petroleum refining, is directly related to the price of the specified natural resource and the volume of its production. The dollar exchange rate has a significant impact on the entire Ukrainian economy, in particular on the formation of prices in its domestic markets. The direct connection of this parameter with gasoline prices directly depends on the US dollar exchange rate. CPI reflects overall change prices within the country, and since it is economically proven that changes in prices for some goods in the vast majority of cases (in conditions of free competition) lead to an increase in prices for other goods, it is reasonable to assume that changes in prices of goods throughout the country affect the indicator studied in the work.

Description of the mathematical apparatus used when carrying out calculations

Regression analysis

Regression analysis is a method of modeling measured data and studying their properties. The data consists of pairs of values ​​of the dependent variable (response variable) and the independent variable (explanatory variable). Regression model<#"19" src="doc_zip1.jpg" />. Regression analysis is the search for a function that describes this dependence. Regression can be presented as the sum of non-random and random components. where is the regression function, and is an additive random variable with zero expected value. The assumption about the nature of the distribution of this quantity is called the data generation hypothesis<#"8" src="doc_zip6.jpg" />has a Gaussian distribution<#"20" src="doc_zip7.jpg" />.

The problem of finding a regression model of several free variables is posed as follows. Sample set<#"24" src="doc_zip8.jpg" />values ​​of free variables and the set of corresponding values ​​of the dependent variable. These sets are denoted as the set of initial data.

Given regression model- a parametric family of functions depending on parameters and free variables. You need to find the most probable parameters:

The probability function depends on the data generation hypothesis and is given by Bayesian inference<#"justify">Method least squares

Least squares method - method of finding optimal parameters linear regression, such that the sum of squared errors ( regression residuals) is minimal. The method consists in minimizing the Euclidean distance between two vectors - the vector of reconstructed values ​​of the dependent variable and the vector of actual values ​​of the dependent variable.

The task of the least squares method is to select a vector that minimizes the error. This error is the distance from vector to vector. The vector lies in the space of the columns of the matrix, since there is a linear combination of the columns of this matrix with the coefficients. Finding a solution using the least squares method is equivalent to the problem of finding the point that lies closest to and is located in the space of the matrix columns.

Thus, the vector must be a projection onto the column space and the residual vector must be orthogonal to this space. Orthogonality is that each vector in the column space is a linear combination of columns with some coefficients, that is, it is a vector. For everyone in space, these vectors must be perpendicular to the residual:

Since this equality must be true for an arbitrary vector, then

The least squares solution to an inconsistent system consisting of equations with unknowns is the equation

which is called a normal equation. If the columns of the matrix are linearly independent, then the matrix is ​​invertible and the only solution is

The projection of the vector onto the matrix column space has the form

The matrix is ​​called the projection matrix of the vector onto the space of the columns of the matrix. This matrix has two main properties: it is idempotent and symmetric. The converse is also true: a matrix that has these two properties is a projection matrix onto its column space.

Let us have statistical data about the parameter y depending on x. We present these data in the form

xx1 X2 …..Xi…..Xny *y 1*y 2*......y i* …..y n *

The least squares method allows, for a given type of dependence y= ?(x) choose its numerical parameters so that the curve y= ?(x) the best way displayed experimental data according to a given criterion. Let us consider the justification from the point of view of probability theory for mathematical definition parameters included in ? (x).

Let us assume that the true dependence of y on x is exactly expressed by the formula y= ?(x). The experimental points presented in Table 2 deviate from this dependence as a result of measurement errors. Measurement errors obey the normal law according to Lyapunov's theorem. Consider some value of the argument x i . The result of the experiment is a random variable y i , distributed according to the normal law with mathematical expectation ?(x i ) and with standard deviation ?i , characterizing the measurement error. Let the measurement accuracy at all points x=(x 1, X 2, …, X n ) is the same, i.e. ?1=?2=…=?n =?. Then the normal distribution law Yi has the form:

As a result of a series of measurements, the following event occurred: random variables(y 1*, y 2*, …, yn *).

Description of the selected software product

Mathcad is a computer algebra system from the class of computer-aided design systems<#"justify">4. Practical part

The objective of the study is to forecast gasoline prices. Background information is a time series with a dimension of 36 weeks - from May 2012 to December 2012.

The statistical data (36 weeks) is presented in matrix Y. Next, we will create matrix H, which will be needed to find vector A.

Let's present the initial data and values ​​calculated using the model:

To assess the quality of the model, we use the coefficient of determination.

First, let's find the average value of Xs:

The portion of variance that is due to regression is total variance indicator Y is characterized by the coefficient of determination R2.

The coefficient of determination takes values ​​from -1 to +1. The closer its coefficient value in absolute value is to 1, the closer the connection between the effective attribute Y and the studied factors X.

The value of the coefficient of determination serves as an important criterion for assessing the quality of linear and nonlinear models. The greater the proportion of explained variation, the less the role of other factors, which means that the regression model well approximates the original data and such a regression model can be used to predict the values ​​of the performance indicator. We obtained the coefficient of determination R2 = 0.78, therefore, the regression equation explains 78% of the variance of the effective attribute, and other factors account for 22% of its variance (i.e. residual variance).

Therefore, we conclude that the model is adequate.

Based on the data obtained, it is possible to make a forecast of fuel prices for the 37th week of 2013. The formula for calculation is as follows:

Calculated forecast using this model: the price of gasoline is 10.434 UAH.

Conclusion

This paper demonstrated the possibility of conducting regression analysis to predict gasoline prices for future periods. Purpose course work were consolidating knowledge in the course “Mathematical methods of operations research” and acquiring development skills software, which allows you to automate operations research in a given subject area.

The forecast regarding the future price of gasoline is, of course, not unambiguous, which is due to the peculiarities of the initial data and the developed models. However, based on the information received, it is reasonable to assume that gasoline prices, of course, will not decrease in the near future, but, most likely, will remain at the same level or will grow slightly. Of course, factors related to consumer expectations, customs duty policies and many other factors are not taken into account here, but I would like to note that they are largely mutually extinguishable . And it is quite reasonable to note that the sharp rise in gasoline prices at the moment is indeed extremely doubtful, which is, first of all, due to the policy pursued by the government.

Bibliography

1.Byul A., Zöfel P. SPSS: the art of information processing. Analysis of statistical data and restoration of hidden patterns. - St. Petersburg: DiaSoftYUP LLC, 2001. - 608 p.

2. Internet resources http://www.ukrstat.gov.ua/

3. Internet resources http://index.minfin.com.ua/

Internet resources http://fx-commodities.ru/category/oil/

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A project method that has enormous potential for creating universal educational activities, is becoming increasingly widespread in the school education system. But it is quite difficult to “fit” the project method into the classroom system. I include mini studies in the regular lesson. This form of work opens up great opportunities for the formation cognitive activity and provides accounting individual characteristics students, prepares the ground for developing skills on large projects.

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“If a student at school has not learned to create anything himself, then in life he will only imitate and copy, since there are few who, having learned to copy, would be able to independently apply this information.” L.N. Tolstoy.

Characteristic feature modern education is a sharp increase in the amount of information that students need to learn. A student’s degree of development is measured and assessed by his ability to independently acquire new knowledge and use it in educational and practical activities. The modern pedagogical process requires the use innovative technologies in teaching.

The new generation Federal State Educational Standard requires use in educational process activity-type technologies, design and research methods are defined as one of the conditions for the implementation of the main educational program.

A special role is given to such activities in mathematics lessons, and this is not accidental. Mathematics is the key to understanding the world, the basis of scientific and technological progress and an important component personality development. It is designed to cultivate in a person the ability to understand the meaning of the task assigned to him, the ability to reason logically, and to acquire algorithmic thinking skills.

Place the project method in class-lesson system hard enough. I try to judiciously combine the traditional and learner-centered systems by incorporating elements of inquiry into the regular lesson. I will give a number of examples.

So, when studying the topic “Circle,” we conduct the following research with students.

Mathematical study "Circle".

1. Think about how to build a circle, what tools are needed for this. Circle symbol.
2. In order to define a circle, let's look at what properties this circle has. geometric figure. Connect the center of the circle with a point belonging to the circle. Let's measure the length of this segment. Let's repeat the experiment three times. Let's draw a conclusion.
3. The segment connecting the center of the circle with any point on it is called the radius of the circle. This is the definition of radius. Radius designation. Using this definition, construct a circle with a radius of 2cm5mm.
4. Construct a circle of arbitrary radius. Construct a radius and measure it. Record your measurements. Construct three more different radii. How many radii can be drawn in a circle?
5. Let us try, knowing the property of the points of a circle, to give its definition.
6. Construct a circle of arbitrary radius. Connect two points on the circle so that this segment passes through the center of the circle. This segment is called the diameter. Let's define diameter. Diameter designation. Construct three more diameters. How many diameters does a circle have?
7. Construct a circle of arbitrary radius. Measure the diameter and radius. Compare them. Repeat the experiment three more times with different circles. Draw a conclusion.
8. Connect any two points on the circle. The resulting segment is called a chord. Let's define a chord. Build three more chords. How many chords does a circle have?
9. Is the radius a chord? Prove it.
10. Is the diameter a chord? Prove it.

Research works may be of a propaedeutic nature. Having examined the circle, you can consider a number of interesting properties that students can formulate at the level of a hypothesis, and then prove this hypothesis. For example, the following study:

"Mathematical Research"

1. Construct a circle of radius 3 cm and draw its diameter. Connect the ends of the diameter to an arbitrary point on the circle and measure the angle formed by the chords. Carry out the same constructions for two more circles. What do you notice?
2. Repeat the experiment for a circle of arbitrary radius and formulate a hypothesis. Can it be considered proven using the constructions and measurements carried out.

When studying the topic “The relative position of lines on a plane”, mathematical research is carried out in groups.

Tasks for groups:

1. group.

1. In one coordinate system, construct graphs of the function

Y = 2x, y = 2x+7, y = 2x+3, y = 2x-4, y = 2x-6.

2.Answer the questions by filling out the table: