Math modeling. Determination of the dominant classification features and development of a mathematical model of facial expression images

Mathematical model- an approximate description of the modeling object, expressed using mathematical symbols.

Mathematical models appeared along with mathematics many centuries ago. The advent of computers gave a huge impetus to the development of mathematical modeling. The use of computers has made it possible to analyze and apply in practice many mathematical models that were previously not amenable to analytical research. Computer-implemented mathematical model called computer mathematical model, A carrying out targeted calculations using computer model called computational experiment.

The stages of computer mathematical modeling are shown in the figure. First stage- determination of modeling goals. These goals can be different:

1) a model is needed in order to understand how a specific object is structured, what its structure is, its basic properties, the laws of development and interaction with the outside world (understanding);

2) a model is needed in order to learn how to control an object (or process) and determine the best ways management with given goals and criteria (management);

3) the model is needed in order to predict the direct and indirect consequences of the implementation of given methods and forms of influence on the object (forecasting).

Let's explain with examples. Let the object of study be the interaction of a flow of liquid or gas with a body that is an obstacle to this flow. Experience shows that the force of resistance to flow on the part of the body increases with increasing flow speed, but at some sufficiently high speed this force decreases abruptly so that with a further increase in speed it increases again. What caused the decrease in resistance force? Mathematical modeling allows us to obtain a clear answer: at the moment of an abrupt decrease in resistance, the vortices formed in the flow of liquid or gas behind the streamlined body begin to break away from it and are carried away by the flow.

An example from a completely different area: populations of two species of individuals that had peacefully coexisted with stable numbers and had a common food supply, “suddenly” begin to sharply change their numbers. And here mathematical modeling allows (with a certain degree of reliability) to establish the cause (or at least refute a certain hypothesis).

Developing a concept for managing an object is another possible goal of modeling. Which aircraft flight mode should I choose to ensure that the flight is safe and most economically profitable? How to schedule hundreds of types of work on the construction of a large facility so that it is completed as quickly as possible short term? Many such problems systematically arise before economists, designers, and scientists.

Finally, predicting the consequences of certain impacts on an object can be a relatively simple matter in simple physical systems, and extremely complex - on the verge of feasibility - in biological, economic, and social systems. While it is relatively easy to answer the question about changes in the mode of heat distribution in a thin rod due to changes in its constituent alloy, it is incomparably more difficult to trace (predict) the environmental and climatic consequences of the construction of a large hydroelectric power station or the social consequences of changes in tax legislation. Perhaps here, too, mathematical modeling methods will provide more significant assistance in the future.

Second phase: determination of input and output parameters of the model; division of input parameters according to the degree of importance of the influence of their changes on the output. This process is called ranking, or separation by rank (see . “Formalization and modeling”).

Third stage: construction of a mathematical model. At this stage, there is a transition from an abstract formulation of the model to a formulation that has a specific mathematical representation.

Mathematical model- these are equations, systems of equations, systems of inequalities, differential equations or systems of such equations, etc.

Fourth stage: choosing a method for studying a mathematical model. Most often, numerical methods are used here, which lend themselves well to programming. As a rule, several methods are suitable for solving the same problem, differing in accuracy, stability, etc. From the right choice method often depends on the success of the entire modeling process.

Fifth stage: development of an algorithm, compilation and debugging of a computer program is a difficult process to formalize. Among the programming languages, many professionals prefer FORTRAN for mathematical modeling: both due to traditions and due to the unsurpassed efficiency of compilers (for calculation work) and the availability of huge, carefully debugged and optimized libraries of standard programs for mathematical methods written in it. Languages ​​such as PASCAL, BASIC, C are also in use, depending on the nature of the task and the inclinations of the programmer.

Sixth stage: program testing. The operation of the program is tested on a test problem with a previously known answer. This is just the beginning of a testing procedure that is difficult to describe in a formally comprehensive manner. Typically, testing ends when the user, based on his professional characteristics, considers the program correct.

Seventh stage: the actual computational experiment, during which it is determined whether the model corresponds to a real object (process). The model is sufficiently adequate to the real process if some characteristics of the process obtained on a computer coincide with the experimentally obtained characteristics with a given degree of accuracy. If the model does not correspond to the real process, we return to one of the previous stages.

Classification mathematical models

The classification of mathematical models can be based on various principles. Models can be classified according to branches of science (mathematical models in physics, biology, sociology, etc.). Can be classified according to the mathematical apparatus used (models based on the use of ordinary differential equations, partial differential equations, stochastic methods, discrete algebraic transformations etc.). Finally, based on common tasks modeling in different sciences, regardless of the mathematical apparatus, the most natural classification is:

· descriptive (descriptive) models;

· optimization models;

· multicriteria models;

· game models.

Let's explain this with examples.

Descriptive (descriptive) models. For example, modeling the motion of a comet that has invaded the solar system is carried out to predict its flight path, the distance at which it will pass from the Earth, etc. In this case, the modeling goals are descriptive in nature, since there is no way to influence the movement of the comet or change anything in it.

Optimization models are used to describe processes that can be influenced in an attempt to achieve a given goal. In this case, the model includes one or more parameters that can be influenced. For example, when changing the thermal regime in a granary, you can set the goal of choosing a regime that will achieve maximum grain safety, i.e. optimize the storage process.

Multicriteria models. It is often necessary to optimize a process along several parameters simultaneously, and the goals can be quite contradictory. For example, knowing food prices and a person’s need for food, you need to organize meals large groups people (in the army, children's summer camp, etc.) is physiologically correct and, at the same time, as cheap as possible. It is clear that these goals do not coincide at all, i.e. When modeling, several criteria will be used, between which a balance must be sought.

Game models may relate not only to computer games, but also to very serious things. For example, before a battle, a commander, if there is incomplete information about the opposing army, must develop a plan: in what order to introduce certain units into battle, etc., taking into account the possible reaction of the enemy. There is a special branch of modern mathematics - game theory - that studies methods of decision-making under conditions of incomplete information.

In the school computer science course, students receive an initial understanding of computer mathematical modeling as part of the basic course. In high school, mathematical modeling can be studied in depth in general education course for classes in physics and mathematics, as well as within the framework of a specialized elective course.

The main forms of teaching computer mathematical modeling in high school are lectures, laboratory and test classes. Typically, the work of creating and preparing to study each new model takes 3-4 lessons. During the presentation of the material, problems are set that must be solved by students independently in the future, and ways to solve them are outlined in general terms. Questions are formulated, the answers to which must be obtained when completing tasks. Indicated additional literature, which allows you to obtain auxiliary information for more successful completion of tasks.

The form of organization of classes when studying new material is usually a lecture. After completing the discussion of the next model, students have at their disposal the necessary theoretical information and a set of tasks for further work. In preparation for completing the task, students choose suitable method solutions, using some well-known private solution to test the developed program. In case of quite possible difficulties when completing tasks, consultation is given, and a proposal is made to study these sections in more detail in literary sources.

Most relevant to the practical part of training computer modeling is the project method. The task is formulated for the student in the form of an educational project and is carried out over several lessons, and the main organizational form is computer laboratory works. Training in modeling using the method educational projects can be implemented at different levels.
First- a problematic presentation of the process of completing the project, led by the teacher.
Second- implementation of the project by students under the guidance of the teacher.
Third- independent implementation by students of an educational research project.

The results of the work must be presented in numerical form, in the form of graphs and diagrams. If possible, the process is presented on the computer screen in dynamics. Upon completion of the calculations and receipt of the results, they are analyzed and compared with known facts from the theory, reliability is confirmed and meaningful interpretation is carried out, which is subsequently reflected in a written report.

If the results satisfy the student and teacher, then the work is considered completed, and its final stage is the preparation of a report. The report includes brief theoretical information on the topic under study, a mathematical formulation of the problem, a solution algorithm and its justification, a computer program, the results of the program, analysis of the results and conclusions, and a list of references.

When all the reports have been compiled, students present their short messages about the work done, defend their project. This is effective form report of the group carrying out the project to the class, including setting the problem, building a formal model, choosing methods for working with the model, implementing the model on a computer, working with the finished model, interpreting the results, forecasting. As a result, students can receive two grades: the first - for the elaboration of the project and the success of its defense, the second - for the program, the optimality of its algorithm, interface, etc. Students also receive grades during theory quizzes.

An essential question is what tools to use in a school computer science course for mathematical modeling? Computer implementation of models can be carried out:

· using a spreadsheet processor (usually MS Excel);

· by creating programs in traditional programming languages ​​(Pascal, BASIC, etc.), as well as in their modern versions (Delphi, Visual Basic for Application, etc.);

· using special application packages for solving mathematical problems (MathCAD, etc.).

At the basic school level, the first method seems to be more preferable. However, in high school, when programming is, along with modeling, a key topic in computer science, it is desirable to use it as a modeling tool. During the programming process, details of mathematical procedures become available to students; Moreover, they are simply forced to master them, and this also contributes to mathematical education. As for the use of special software packages, this is appropriate in a specialized computer science course as a supplement to other tools.

INTRODUCTION

Objects material world complex and diverse. Reflecting all their properties in the images created, studied and used is very difficult, and not necessary. It is important that the image of the object contains the features that are most important for its use. The modeling method is the replacement of the original object with a substitute object that has a certain similarity to the original, in order to obtain new information about the original. A model is a substitute object for the original object, designed to obtain information about the original.

Mathematical models belong to symbolic models and represent a description of objects in the form mathematical symbols, formulas, expressions. If you have a sufficiently accurate mathematical model, you can use mathematical calculations to predict the results of an object’s functioning under various conditions, and select from a variety of possible options the one that gives the best results.

This paper provides types of classification of mathematical modeling methods and describes some methods:

Linear programming is a method of mathematical modeling that is used to find the optimal distribution of limited resources between competing jobs.

Simulation modeling. The purpose of simulation modeling is to reproduce the behavior of the system under study based on the results of the analysis of the most significant relationships between its elements or, in other words, to develop a simulator of the system under study. subject area for carrying out various experiments.

Classification of mathematical modeling methods

Due to the variety of mathematical models used, they general classification difficult. In the literature, classifications are usually given based on different approaches and principles.

According to hierarchical level mathematical models are divided into micro-level, macro-level, and meta-level models. Mathematical models at the micro-level of the process reflect physical processes occurring, for example, when cutting metals. They describe processes at the transition (passage) level.

Mathematical models at the macro level of the process describe technological processes.

Mathematical models at the meta-level of the process describe technological systems (sections, workshops, the enterprise as a whole).

By the nature of the object properties displayed models can be classified into structural and functional

A structural model is if it can be represented by a data structure or data structures and relationships between them. In turn, a structural model can be hierarchical or network.

The model is hierarchical (tree-like), – if it can be represented by some hierarchical structure (tree); for example, to solve the problem of finding a route in a search tree, you can build a tree model shown in Figure 1.

Figure 1 - Model of hierarchical structure.

The model is network - if it is represented by some network structure. For example, the construction of a new house includes various operations that can be represented in the form of a network model shown in Figure 2.

Figure 2 - Network structure model.

A model is functional if it is representable in the form of a system of functional relationships. For example, Newton's law and the model of production of goods are functional.

By way of representing object properties models are divided into analytical, numerical, algorithmic and simulation.

Analytical mathematical models are explicit mathematical expressions of output parameters as functions of input and internal parameters and have unique solutions for any initial conditions. For example, the cutting (turning) process from the point of view of the acting forces is an analytical model. Also, a quadratic equation that has one or more solutions will be an analytical model. The model will be numerical if it has solutions under specific initial conditions (differential, integral equations).

A model is algorithmic if it is described by some algorithm or set of algorithms that determines its functioning and development. Introduction of this type models (indeed, it seems that any model can be represented by an algorithm for its study) is quite justified, since not all models can be studied or implemented algorithmically. For example, a model for calculating the sum of an infinite decreasing series of numbers can be a calculation algorithm final amount series to some given degree of accuracy. An algorithmic model of the square root of a number X can be an algorithm for calculating its approximate, arbitrarily accurate value using a known recurrent formula.

A simulation model is if it is intended to test or study possible paths of development and behavior of an object by varying some or all parameters of the model, for example a model economic system production of two types of goods. Such a model can be used as a simulation model to determine and vary the total cost depending on certain values ​​of the volume of goods produced.

By method of receipt models are divided into theoretical and empirical. Theoretical mathematical models are created as a result of studying objects (processes) at a theoretical level. For example, there are expressions for cutting forces obtained based on the generalization physical laws. But they are unacceptable for practical use, because they are very cumbersome and not entirely adapted to real processes. Empirical mathematical models are created as a result of conducting experiments (studying the external manifestations of the properties of an object by measuring its parameters at the input and output) and processing their results using methods mathematical statistics.

According to the form of representation of object properties models are divided into logical, set-theoretic and graph. A model is logical if it can be represented by predicates and logical functions; for example, a set of two logical functions can serve as a mathematical model of a one-bit adder. A model is set-theoretic if it is representable using certain sets and relations of membership to them and between them. A graph model is if it can be represented by a graph or graphs and the relationships between them.

According to the degree of stability. models can be divided into stable and unstable. A stable system is one that, having been removed from its initial state, tends to it. It can oscillate for some time around the starting point, like an ordinary pendulum set in motion, but the disturbances in it fade over time and disappear. In an unstable system, which is initially at rest, the resulting disturbance intensifies, causing an increase in the values ​​of the corresponding variables or their oscillations with increasing amplitude

In relation to external factors models can be divided into open and closed. A closed model is a model that operates without connection with external (exogenous) variables. In a closed model, changes in the values ​​of variables over time are determined by the internal interaction of the variables themselves. A closed-loop model can reveal the behavior of a system without introducing an external variable. Example: information systems with feedback are closed systems. They are self-tuning systems, and their characteristics arise from the internal structure and interactions that reflect the input external information. A model associated with external (exogenous) variables is called open.

In relation to the time factor models are divided into dynamic and static. A model is called static if there is no time parameter among the parameters involved in its description. A model is called a dynamic model if among its parameters there is a time parameter, i.e. it displays the system (processes in the system) in time. simultaneously.

Linear programming

Among the tasks mathematical programming the simplest (and best studied) are the so-called problems linear programming. What is characteristic of them is that:

a) efficiency indicator (objective function) W linearly depends on the solution elements x 1, x 2, ....., x n and

b) restrictions imposed on the elements of the solution take the form of linear equalities or inequalities with respect to x 1, x 2, ..., x n

Such problems are encountered quite often in practice, for example, when solving problems related to the distribution of resources, production planning, organization of transport, etc. This is natural, since in many practical problems “expenses” and “income” linearly depend on the number of purchased or disposed of goods (for example, the total cost of a consignment of goods depends linearly on the number of purchased units; payment for transportation is made in proportion to the weight of the goods transported, etc.).

Any linear programming problem can be reduced to a standard form, the so-called “basic linear programming problem” (OBLP), which is formulated as follows: find non-negative values ​​of the variables x 1, x 2, ..., x n that would satisfy the equality conditions ( 1).

The case when f must be turned not to a maximum, but to c. the minimum can easily be reduced to the previous one if we simply change the sign of f to the opposite (maximize not f, but f" = - f). In addition, from any inequality conditions one can move to equality conditions at the cost of introducing new additional variables.

Depending on the type of objective function and restrictions, several types of linear programming problems or linear models can be distinguished: general linear problem, transport problem, assignment problem.

Transport task(Monge-Kantorovich problem) is a mathematical linear programming problem of a special type about finding the optimal distribution of homogeneous objects from the accumulator to the receivers while minimizing the cost of movement. For ease of understanding, it is considered as a problem about the optimal plan for transporting goods from points of departure to points of consumption, with minimal transportation costs.

The assignment problem is formulated as follows:

There is a certain number of works and a certain number of performers. Any performer can be assigned to perform any (but only one) job, but at unequal costs. It is necessary to distribute the work so as to complete the work with minimal costs. If the number of jobs and performers coincides, then the problem is called a linear assignment problem.

There are several ways to solve a linear programming problem, in particular the graphical method and the simplex method. The graphical method is based on the geometric interpretation of a linear programming problem and is used to solve problems in two-dimensional space. Problems of three-dimensional space are solved very rarely, because constructing their solution is inconvenient and lacks clarity. Let's consider the method using the example of a two-dimensional problem.

Find a solution X = (x 1,x 2) that satisfies the system of inequalities (3)

at which the value of the objective function F = 2x 1 x 2 reaches its maximum.

Let us construct a region on the plane in the Cartesian rectangular coordinate system x 1 Ox 2 admissible solutions tasks.

Each of the constructed straight lines divides the plane into two half-planes. The coordinates of the points of one half-plane satisfy the original inequality, but the other does not. To determine the desired half-plane, you need to take some point belonging to one of the half-planes and check whether its coordinates satisfy this inequality. If the coordinates of a taken point satisfy this inequality, then the desired half-plane is the half-plane to which this point belongs. Otherwise, another half-plane.

Let's find the half-plane defined by the inequality x 1 -x 2 ≥-3. To do this, having constructed a straight line (I) x 1 -x 2 =-3, we take some point belonging to one of the two resulting half-planes, for example, point O(0,0). The coordinates of this point satisfy the inequality x 1 -x 2 ≥-3. This means that the half-plane to which the point O(0,0) belongs is determined by the inequality x 1 -x 2 ≥-3.

Now let's find the half-plane defined by the inequality 6x1+7x 2 ≤42.

We build line II 6x 1 +7x 2 =42. The coordinates of the point O(0,0) satisfy the inequality 6x 1 + 7x 2 ≤42, which means that the required half-plane will be the second one.

Now we are looking for a half-plane for the inequality 2 x 1 -3 x 2 ≤6. The coordinates of the point O(0,0) satisfy the inequalities 2 x 1 -3 x 2 ≤6. Consequently, the half-plane to which the point O(0,0) belongs is determined by the inequality 2 x 1 -3 x 2 ≤6 (Line III).

And the half-plane for the inequality x 1 + x 2 ≥4. The coordinates of the point O(0,0) satisfy the inequality x 1 + x 2 ≥4 (Straight IV). Hence the straight line x 1 + x 2 =4 is determined by the first half-plane.

The inequalities x 1 ≥0 and x 2 ≥0 mean that the solution region will be located to the right of the ordinate axis and above the abscissa axis. Thus, the shaded region ABCD in Figure 3 will be the region of feasible solutions determined by the constraints of the problem. The objective function takes its maximum value at one of the vertices of the figure ABCD. To determine this vertex, we construct a vector C (2; -1) and a straight line 2x 1 -x 2 =p, where p is some constant such that the straight line 2x 1 -x 2 =p has common points with the solution polygon. Let us put, for example, p=1/2 and construct a straight line 2 x 1 -x 2 =1/2. Next, we will move the constructed line in the direction of the vector until it passes through its last common point with the solution polygon. The coordinates of the specified point determine the optimal plan for this task.

Figure 3 shows that the last common point of the straight line 2x 1 -x 2 =p with the solution polygon is point A. This point is the intersection of straight lines II and III, so its coordinates are found as a solution to the system of equations defining these straight lines:

In this case, the value of the objective function F = 2 x 1 -x 2 = 2* 5.25 – 1 *1.5 = 9.

Point B will be the optimal solution to the problem X opt = (x 1 opt, x 2 opt) and its coordinates will be equal to x 1 opt = 5.25, x 2 opt = 1.5.

Figure 3 - Range of feasible solutions to the problem

Simplex - method

This method is a method of purposeful enumeration of reference solutions to a linear programming problem. He allows for final number steps to either find an optimal solution or establish that there is no optimal solution.

1) Indicate a method for finding the optimal reference solution.

2) Indicate the method of transition from one reference solution to another, at which the value of the objective function will be closer to the optimal one, i.e. indicate a way to improve the reference solution.

3) Set criteria that allow you to promptly stop searching for support solutions at the optimal solution or make a conclusion about the absence of an optimal solution.

In order to solve a problem using the simplex method, you must do the following:

1) Bring the problem to canonical form.

2) Find the initial support solution with a “unit basis” (if there is no support solution, then the problem has no solution due to the incompatibility of the system of constraints).

3) Calculate estimates of vector decompositions based on the reference solution and fill out the table of the simplex method.

4) If the criterion for the uniqueness of the optimal solution is satisfied, then the solution of the problem ends. If the condition for the existence of a set of optimal solutions is met, then all optimal solutions are found by simple enumeration.

The computational efficiency of mathematical methods is usually assessed using two parameters:

1) The number of iterations required to obtain a solution;

2) Computer time consumption.

As a result of numerical experiments, the following results were obtained for the simplex method:

1) The number of iterations when solving linear programming problems in standard form with constraints and variables is between and . Average number of iterations. The upper bound on the number of iterations is .

2) The required machine time is proportional to .

The number of constraints has a greater impact on computational efficiency than the number of variables, therefore, when formulating linear programming problems, one should strive to reduce the number of constraints, even by increasing the number of variables.

Basic concepts of the simulation method.

The term “simulation modeling” (“simulation model”) usually means calculating the values ​​of some characteristics of a process developing over time by reproducing the flow of this process on a computer using its mathematical model, and it is either impossible or extremely difficult to obtain the required results by other methods. Reproducing the flow of a process on a computer using a mathematical model is usually called a simulation experiment.

Simulation models belong to the class of models that are a system of relationships between the characteristics of the process being described. These characteristics are divided into internal (“endogenous”, “phase variables”) and external (“exogenous”, “parameters”). Approximately internal characteristics- these are those whose values ​​are intended to be determined using mathematical modeling tools; external - those on which the internal characteristics significantly depend, but the inverse dependence (with practically acceptable accuracy) does not occur.

A model capable of predicting the values ​​of internal characteristics must be closed (“closed model”), in the sense that its relations allow one to calculate internal characteristics given known external ones. The procedure for determining the external characteristics of a model is called its identification, or calibration. Mathematical models of the described class (these include simulation models) define a mapping that allows one to obtain known values external characteristics and internal values. In what follows, this mapping will be called the mapping associated with the model.

The models of the class under consideration are based on the postulate about the independence of external characteristics from internal ones, and the relations of the model are a form of recording the mapping associated with it. As shown in Figure 4, during the simulation process, the researcher deals with four main elements:

Real system;

Logical-mathematical model of the simulated object;

Simulation (machine) model;

The computer on which the simulation is carried out is a directed computational experiment.

The researcher studies a real system, develops a logical-mathematical model of a real system. The simulation nature of the research presupposes the presence of logical or logical-mathematical models that describe the process being studied. Above, a real system was defined as a set of interacting elements operating over time. Composite character complex system describes the representation of its model in the form of three sets: A, S, T, where
A is a set of elements (including external environment);
S – set of admissible connections between elements (model structure);
T is the set of time points under consideration.

Figure 4 Simulation process

A feature of simulation modeling is that the simulation model allows you to reproduce the simulated objects:

While maintaining their logical structure;

With the preservation of behavioral properties (sequence of alternation in time of events occurring in the system), i.e. dynamics of interactions.

In simulation modeling, the structure of the simulated system is adequately displayed in the model, and the processes of its functioning are played out (simulated) on the constructed model. Therefore, the construction of a simulation model consists of describing the structure and functioning processes of the modeled object or system.

There are simulation models:

Continuous;

Discrete;

Continuous-discrete.

In continuous simulation models, variables change continuously, the state of the simulated system changes as a continuous function of time, and, as a rule, this change is described by systems of differential equations. Accordingly, the advancement of model time depends on numerical methods for solving differential equations. In discrete simulation models, variables change discretely at certain moments of simulation time (the occurrence of events).

Dynamics discrete models represents the process of transition from the moment of occurrence of the next event to the moment of the occurrence of the next event. Since in real systems continuous and discrete processes often cannot be separated, continuous-discrete models have been developed that combine the time advancement mechanisms characteristic of these two processes.

The simulation method allows you to solve problems high complexity, provides simulation of complex and diverse processes with a large number of elements. Individual functional dependencies in such models can be described by cumbersome mathematical relationships. Therefore, simulation modeling is effectively used in problems of studying systems with complex structure in order to solve specific problems. The simulation model contains elements of continuous and discrete action, therefore it is used to study dynamic systems when analysis is required bottlenecks, study of the dynamics of functioning, when it is desirable to observe the progress of a process on a simulation model over a certain time.

Simulation modeling is an effective tool for studying stochastic systems, when the system under study can be influenced by numerous random factors of a complex nature. It is possible to conduct research under conditions of uncertainty, with incomplete and inaccurate data. Simulation modeling is an important factor in decision support systems because... allows you to explore big number alternatives (solution options), play various scenarios for any input data.

The main advantage of simulation modeling is that the researcher can test new strategies and make decisions when studying possible situations, can always get an answer to the question “What happens if?” A simulation model makes it possible to make predictions when it comes to the system being designed or when development processes are being studied (that is, in cases where the real system does not yet exist). The simulation model can provide various high level details of the simulated processes. In this case, the model is created step by step, evolutionarily.

BIBLIOGRAPHY

1. Blinov, Yu.F. Methods of mathematical modeling [Text]: Electronic tutorial/ Yu.F. Blinov, V.V. Ivantsov, P.V. Serbian – Taganrog: TTI SFU, 2012. – 42 p.

2. Ventzel, E.S. Operations research. Objectives, principles, methodology. [Text]: Textbook / E.S. Ventzel - M.: KNORUS, 2010. - 192 p.

3. Getmanchuk, A. V. Economics mathematical methods and models [Text]: Textbook for bachelors. / A.V. Getmanchuk - M.: Publishing and trading corporation "Dashkov and Co", 2013. -188 p.

4. Zamyatina, O.M. Systems modeling. [Text]: Training manual. / O.M. Zamyatin - Tomsk: TPU Publishing House, 2009. - 204 p.

5. Pavlovsky, Yu.N. Simulation modeling. [Text]: textbook for university students / Yu.N. Pavlovsky, N.V. Belotelov, Yu.I. Brodsky - M.: Publishing Center "Academy", 2008. - 236 p.

Yalta educational complex “School-Lyceum No. 9”

Deputy Director for HRRomanova A.N.

“Modeling in mathematics lessons in primary school»

Practical seminar

Mathematics should be taught in school

Also, set your goal so that knowledge

who get here would be

sufficient for ordinary

needs in life.

M. Lobachevsky

Report plan

New guidelines in mathematics education.

Methodological foundations of modeling. Mathematical model.

Using the modeling method in mathematics lessons in elementary school.

Introducing students to the techniques of mathematical modeling.

Application of modeling in solving equations.

Modeling while solving word problems.

Using modeling to study numbering, adding and subtracting numbers, and working on units of length.

New guidelines in mathematics education. (5 minutes)

It is well known that models are the language of mathematics, and modeling is their speech. The success of mastering mathematics is determined, first of all, by how well the child has learned to “speak” their language. This is determined not only by the student’s academic success in solving scientific and cognitive tasks, but to a greater extent success in life personalities - thanksability to apply mathematical methods for solving practical, real tasks who require it. Agree, this is also a good result of learning mathematics at school.

Do we teach our students mathematical language? Or maybe we think so difficult task for elementary school? Or do we simply hope that in the course of daily solving examples and problems, children themselves will gradually learn to use it?

According to monitoring data in schools in Kiev, as well as all-Ukrainian monitoring data, indicate that the majority of students (60% and 53%, respectively) do not know how to work with ready-made graphic models, perform creative tasks, or apply acquired knowledge in new situations to solve problems .

This state of mathematics education has caused the need for a significant revision state requirements in teaching mathematics to schoolchildren. The new edition of the “Sovereign Standard...”, which came into force this year. From the position of a personality-oriented and competence-based approach, it actually reorients the teacher’s activities.Competence - the presence of knowledge and experience necessary for effective activity in a given subject area . Let's compare . In yetcurrent The state standard states: “Studying mathematics in primary school ensures that students acquire the knowledge, skills and abilities necessary for further study of mathematics and other subjects... Studying mathematics contributes to the development cognitive abilities younger schoolchildren - memory, logical and creative thinking, imagination, mathematical speech."In the new edition of the state standard the goal in the educational field “Mathematics” has already been defined as “the formation of subject mathematical and key competencies necessary for students’ self-realization in a rapidly changing world.” Subject mathematical competence is considered as “ personal education, which characterizes the student’s ability to create mathematical models of processes in the surrounding world, to apply the experience of mathematical activity while solving educational, cognitive and practically oriented problems.”

Therefore, mastering mathematical speech—the ability to build mathematical models—becomes the main goal of teaching mathematics, which is realized through the formation in students of “the ability to use mathematical terminology, symbolic and graphic information.”

The positive experience of teaching students modeling (and not only in mathematics lessons) accumulated by the system of developmental education by D.B. Elkonina - V.V. Davydov, aimed at developing students’ full-fledged educational activities, one of which is modeling.

Forming the ability to model in students is one of the goals of developmental education, and the models that children create and use are, first of all, one of the ways to develop learning skills (and not just a method of clarity).

The purpose of our seminar today is to understand the issues of modeling, to show how models can be used to teach primary schoolchildren to solve equations and problems, mathematical properties, techniques for adding and subtracting numbers.

2. Methodological foundations of modeling. (8 min)

Modeling is one of the means of understanding reality. The model is used to study any objects (phenomena, processes), to solve various problems and obtain new information. Consequently, a model is a certain object (system), the use of which serves to obtain knowledge about another object (original).

The use of modeling is considered in two aspects:

firstly, modeling serves as the content that should be learned by children as a result pedagogical process;

secondly, modeling is that educational action and means without which full-fledged learning is impossible.

The visibility of models is based on the following important pattern: the creation of a model is carried out on the basis of the preliminary creation of a mental model - visual images of the objects being modeled, that is, the subject creates a mental image of this object, and then (together with children) builds a material or figurative model (visual). Mental models are created by adults and can be transformed into visual ones with the help of certain practical actions (in which children can also participate); children can also work with already created visual models.

When working with children, you can use substitution of objects: symbols and signs, planar models (plans, maps, drawings, diagrams, graphs), three-dimensional models, layouts.

Using the modeling method helps solve the complex very important tasks:

development of children's productive creativity;

development of higher forms imaginative thinking;

application of previously acquired knowledge in solving practical problems;

consolidation of mathematical knowledge acquired by children previously;

creating conditions for business cooperation;

activation of children's mathematical vocabulary;

development fine motor skills hands;

acquiring new ideas and skills in the process of work;

children's deepest understanding of the principles of operation and structure of originals with the help of models.

A model gives us not just the opportunity to create a visual image of the modeled object, it allows us to create an image of its most essential properties reflected in the model. All other unimportant properties are discarded when developing the model. Thus, we create a generalized visual image of the modeled object.

The scientific basis of modeling is the theory of analogy, in which the main concept is the concept of analogy - the similarity of objects according to their qualitative and quantitative characteristics. All these types are united by the concept of a generalized analogy - abstraction. Analogy expresses a special kind of correspondence between compared objects, between the model and the original.

Modeling is multifunctional, that is, it is used in a variety of ways for different purposes at different levels (stages) of research or transformation. In this regard, the centuries-old practice of using models has given rise to an abundance of forms and types of models.

Let's consider the classification proposed by L.M. Friedman. From the point of view of the degree of clarity, he divides all models into two classes:

step 1. 1-2

· material (real, real);

· perfect.

To the material Models include those that are built from any material objects.

Step 2

Material models, in turn, can be divided intostatic (stationary) anddynamic (current).

Step 3

The next type of dynamic models areanalog and simulation , which reproduce this or that phenomenon with the help of another, in some sense more convenient. For example, such a model - an artificial kidney - functions in the same way as a natural (living) kidney, removing toxins and other metabolic products from the body, but, of course, it is designed completely differently than a living kidney.

Ideal Models are usually divided into three types:

Step 4

· figurative (iconic);

· iconic (sign-symbolic);

· mental (mental).

Models can be classified according to various signs:

1) by the nature of the models (that is, by the modeling tools);

2) by the nature of the objects being modeled;

3) by area of ​​application of modeling (modeling in technology, physical sciences, chemistry, modeling of living processes, modeling of the psyche, etc.)

4) by levels (“depth”) of modeling.

The most famous isclassification according to the nature of the models .

Step 5.

According to it, the following are distinguished:types of modeling :

Step 6.

1. Subject modeling , in which the model reproduces the geometric, physical, dynamic or functional characteristics of an object. For example, a model of a bridge, a dam, a model of an airplane wing, etc.

Step 7

2. Analog Modeling , in which the model and the original are described by a single mathematical relationship. An example is electrical models used to study mechanical, hydrodynamic and acoustic phenomena.

Step 8

3. Iconic modeling , in which the models are iconic formations any type: diagrams, graphs, drawings, formulas, graphs, words and sentences.

Step 9

4. Closely related to the iconicmental simulation , in which the models acquire a mentally visual character.

Step 10

5. Simulated experiment – a special type of modeling where not the object itself is used, but its model.

The main purpose of modeling is to highlight and record the most common relationships in a subject for its study.

The modeling method is a complex, integrative education. According to the classification of didactic methods by N.G. Kazansky and T.S. Nazarova, the modeling method has a three-component structure

Step 11(see diagram). Thus, in the structure of the modeling methodouter side - This is a specific form of interaction between teacher and students.Inner side – this is a set of general educational techniques (analysis, synthesis, generalization, etc.) and methods of educational work.Technological side – this is a set of specific techniques of this method (preliminary analysis, building a model, working with it, transferring information from the model to the desired object - the original).

Outer side

Inner side

Technological side

Shapes:

presentation

conversation

independent work

Psychological essence:

dogmatic way of educational work;

heuristic way of educational work

research method of educational work

Logical entity:

analytical;

synthetic;

inductive;

deductive;

analytical-synthetic

Techniques for constructing a model;

model transformation techniques;

methods for specifying the model

Mathematical model. Math modeling.

A mathematical model is an approximate description of a class of phenomena outside world using mathematical symbolism. For example, the relationship between elements A, B, C is expressed by the formula A+B=C - a mathematical model.

The process of mathematical modeling, i.e. studying phenomena using mathematical models can be divided into four stages.

Step 12

First stage – isolating essential features object.

13.

Second phase – building a model.

14 .

Third stage – study of the model.

15 .

Fourth stage – transfer of information obtained from models to the object being studied.

The peculiarity of modeling is that visibility is not a simple demonstration of natural objects, but stimulates independent practical activities children. Students’ ability to work with the model, its transformation for study general properties concepts being taught constitutes one of the main objectives of learning in all subject areas.

Various models are used for modelingmathematical objects: numerical formulas, numeric tables, letter formulas, functions, algebraic equations, series, geometric figures, various graph diagrams, Euler-Venn diagrams, graphs.

3. Using the modeling method in mathematics lessons in elementary school. (1.5 min)

The need for junior schoolchildren to master the modeling method as a method of cognition in the learning process can be justified from different positions.

Step 16

Firstly , this contributes to the formation of a dialectical-materialistic worldview.

17.

Secondly , as experiments show, the introduction of the concepts of model and simulation into the content of teaching significantly changes the attitude of students to the educational subject, makes their educational activities more meaningful and more productive.

18.

Third , targeted and systematic training in the modeling method brings younger schoolchildren closer to the methods scientific knowledge, ensures their intellectual development. In order to “equip” students with modeling as a way of cognition, it is not enough for a teacher to just show them different scientific models and show them the process of modeling individual phenomena. It is necessary that schoolchildren build models themselves, study any objects or phenomena themselves using modeling. When students, solving a practical mathematical (plot) problem, understand that it represents a symbolic model of some real situation, compose a sequence of its various models, then study (solve) these models and, finally, translate the resulting solution into the language of the original problem, then schoolchildren thereby master the modeling method.

Introducing students to the techniques of mathematical modeling. (10 min)

The famous psychologist P. Halperin and his colleagues developed a theory of the step-by-step formation of mental actions. According to this theory, the learning process is considered as the child’s mastery of a system of mental actions, which occurs in the process of internalization (transition inward) in response to external practical activity.

The child performs practical actions with objects (first with real ones, and then with imaginary ones) - objective actions. From them he, relying first on the copy drawing, and then on subject models, moves on to graphical models. After introducing mathematical symbols and letters to denote quantities, the student uses formulas to describe actions, i.e. symbol-letter models, and then verbal models (definitions, rules).

For example, children are given a specific practical task that requires them to find two vessels of the same volume (different in shape).Photo step 19

After this, the children (and not the teacher) perform practical actions: pour water into one jar, pour it into another. If all the water from the first one enters another jar, then the volumes of these jars are equal. It is advisable to invite children to pick up these two strips, with the help of which they can communicate the relationships between volumes and shapes - whether they are the same or different. If the volumes of the cans are the same, children must lift two strips of the same length, and if different, then different in length.Photo

step 20

To lead children to use a graphic model, it is again necessary to set a specific practical task: using a drawing, show that the volume of one can is larger than the other. Experience shows that children begin to draw the shape of cans, i.e. make a copy drawing, or draw stripes, with the help of which they show the ratio of the volumes of the cans.

After discussing the drawings, we conclude: drawing cans is an unsuccessful way (inaccurate drawings, the ratio of the volumes of cans is not depicted, the work takes a lot of time). But children’s stripes are also different in width and length, and this also takes a lot of time.

As a result, we come to the conclusion that it is more convenient not to draw the width of the strip at all, but to draw only the length of the strip (i.e., segments). If the quantities (length, area, mass, volume, etc.) are identical, then they have segments of the same length, and if they are unequal, then their lengths should be different.Photo in notebook. step 21.

In this way, the image of quantities is introduced using segments. Children learn to schematically designate quantities and then build graphical (linear) models.

It is also advisable to introduce in the 1st grade the concepts of “whole” and “part” and to develop students’ skills to establish relationships between these concepts. How can we write in mathematical language that, for example, an apple consists of separate parts? If the apple is whole, we denote it by a circle, and the heaps of apples are denoted by triangles, and we get the following graphical model.

Step 22Slide 7

+ + + =

Let's simplify and we will have basic model:

step 23. + =

The whole and the parts are relative concepts. The main properties of this relation (on the set natural numbers): the whole cannot be less than the part, and the part cannot be greater than the whole; the whole is equal to the sum of the parts, and the part is equal to the difference between the whole and the other part

Step 24 = -

Everyone is well aware of the rays that are traditionally used to depict the composition of numbers.Step 25Slide 8

So the relationship between the parts and the whole can be shown using a sign graphic notation:

WITHstep 26

A |____________|_____________|

B A B

The diagram that describes the action of addition also describes the reverse action - subtraction:

Step 27slide 9

The concepts of part and whole make it possible to introduce the commutative and associative properties of adding quantities.Slide 10, 11 (2 steps), 12

Step 28, 29, 30

Just like learning addition and subtraction, simulations can also be used to learn multiplication and division.

Traditionally, multiplication is viewed as adding identical terms. Let the value A be added B times:slide 13.

step 31.A+A+A+A+A = AxB

The formula A x B reads like this: “take B times from A” or “take B times from A”,

Step 32where A is the part (measurement) that was designated by a triangle.

B – the number of equal parts (number of measurements), we can denote by a square.

To designate the whole we use the same icon - a circle.

The whole is characterized as a result arithmetic action multiplying numbers A and B.

X = A x B = C Scheme that describes this action:

|____|_A___|_____________|

It is clear that when we consider division as an objective action aimed at dividing according to content or into equal parts, it will be possible to establish a connection between multiplication and division. Now, in addition to the multiplication formulaStep 33Ax B = C, we get two inverses of divisionstep 34.C: A = B andstep 35. C: B = A (with geometric shapes). This means that the multiplication circuit is a division circuit.

Application of modeling in solving equations. (10 min)

To correctly choose a method for solving equations, you must be able to find the relationship between the whole and the part. When this concept is formed, children acquire the ability to express the whole through parts and parts through the whole. Establishing connections between addition and subtraction of quantities based on the concept of part and whole makes it possible to compare the whole with the sum and the minuend, the parts with the addends or the subtrahend and the difference and see that different actions: A+B=C, C-A=B, or C-B=A – characterize the same relationships between quantities.

Finding the unknown when solving equations helps not only the rules, but also the relationships between the parts and the whole, presented in the form of a graphical model.Slide 14 step 36.

The algorithm for teaching how to solve equations is as follows:

Let's draw a diagram of the equation. X +5 = 12step 37.

We find the whole and parts first in the diagram, then in the equation (we underline)

We name the unknown component. Let's find out what it is: a whole or a part.

We analyze how we will find the unknown quantity.

We findX. step 38, 39

The constructed circuit can be used to solve subtraction equations. 12 – x = 5, since the circuit that describes the action of addition is also a circuit for subtraction. Examples of photos from the notebook

Slides 15,16 (+1 step ), 17, 18.

Step, 40, 41, 41-a, 42,43

The task is to divide these equations into diagrams and create an expression

slide 19 step 44, 45. 44-a, 45-b

Modeling is used similarly when solving equations to find unknown multiplier, divisor and dividend.

Slide 20( 8 steps ) step 46.

When establishing the connection between multiplication and division, it is advisable to introduce the concept of area, the formula for finding the area of ​​a rectangle, and finding the unknown side.Slide 21 (1 step)

Example equation. Slide 22 ( 4 steps)

Algorithm for solving the equationSlide 23 .

Since the multiplication scheme is a division scheme, two division equations can be made from one equation. The area is a whole, and the sides length and width are parts.

In addition, modeling makes it possible to diversify creative work over the equations. So, the teacher can offer the following types of tasks:

Slide 24

Using the diagram, create and solve an equation.Step 48

Slide 25 ( decide with the guests )

(several equations and a diagram are given) Which equation will this diagram fit to? Find and decide.Step 49

Slide 26, 27. 28, 29.

Solve equations while counting mentally. Step 50, 51, 52,53

Slide 30 (10 steps), 31

Drawing up the conditions of the problem according to the equation diagram.

New presentation. (Seminar 2)

Modeling while solving word problems (18 min)

Slide 1

One cannot but agree with the opinion that modern education is the student’s ability to look at the real, life situation from the position of a physicist, chemist, historian, geographer, not at all in order to become a researcher in this field, but in order to subsequently find a solution in specific life situations.

A junior student can become a real researcher by solving word problems when teaching mathematics to junior schoolchildren.

One One of these approaches is to develop in students the ability to solve problems of a certain type (for example, solving difference comparison problems, etc., when practicing certain type tasks).Another is based on the use of semantic and mathematical analysis of text problems, when the problem is analyzed from data to the goal (synthetic method) and from goal to data (analytical).Third approach based on solution method educational tasks. The formation of modeling action presupposes a qualitatively different formation of the ability to solve word problems.

Arithmetic and algebraic problems in literature they are also called plot plots, because they always contain a verbal description of some event, phenomenon, action, process. The text of any plot problem can be recreated in a different way (subject-wise, graphically, using tables, formulas, etc.), and this is the transition from verbal modeling to other forms of modeling. Therefore, when working on problems, we pay great attention to the construction of schematic and symbolic models, as well as the ability to work with segments, graphically model a text problem with their help, pose a question, determine an algorithm for solving and finding an answer. The younger schoolchild, as is known, does not have a sufficient level abstract thinking. And our task is precisely to progressively teach him to represent specific objects in the form of a symbolic model, to help him learn to translate a text problem into mathematical language. We believe that it is the graphical modeling of a word problem that, most importantly, gives real opportunity clearly see and determine the algorithm for solving it, and carry out independent reflection on the completed task.

But not every record will be a task model. To build a model, for its further transformation, it is necessary to select in the problemgoal, given quantities, all relationships, so that, based on this model, it is possible to continue the analysis, allowing us to move forward in the solution and search for optimal solutions. Solving any problem using an arithmetic method is associated with the choice of an arithmetic operation, as a result of which one can answer the question posed. To facilitate the search for a mathematical model, it is necessary to use an auxiliary model.Slide 2 (familiarity with the components in 1st grade).

To recreate the situation in the task conditions, you can use a schematic drawing, which would provide a transition from the text of the problem to the correlation of a certain arithmetic operation on numbers, which contributes to the formation of a conscious and strong assimilation of the general method of working on the task. This model allows the student to develop the ability to explain how he received the answer to the question of the problem. But a schematic model is effective only if it is understandable to every student and the ability to translate a verbal model into the language of a diagram has been developed. When learning to solve simple addition and subtraction problems, the following concepts are introduced: whole, part and their relationship.Slide 3. (2 steps)

To find a part, you need to subtract another part from the whole.

To find the whole you need to add the parts.

When learning to solve simple multiplication and division problems, a diagram and corresponding rules are proposed:

To find the whole, you need to multiply the measure by the number of measures.

To find a measure, you need to divide the whole number by the number of measures.

To find the number of measures, you need to divide the whole by the measure.

Slide 4. (3 steps)

This approach to teaching allows us to move away from the old classification of simple tasks. It is important to depict the data and what is being sought in such a way that the relationships between the quantities are clear enough. Considered in the problem, and their relationships.

As an example, I will give several text problems and how to solve them using graphic models.

Problem 1Slide 5. (5 steps)

There are 4 large and 5 small fish in the aquarium. How many fish are there in total in the aquarium?

Exercises for composing problems and expressions from pictures (inverse problems)Slide 6. ( 8 steps)Slide 7.

Problem 2Slide 8

Lena has 5 pears. And Misha has 4 more than Lena. How many pears does Misha have?

An example of a task for composing problems based on a picture and writing down the solution.Slide 9.

Problem 3Slide 10. (5 steps)

Lena has 10 pears. This is 3 more than peaches. How many peaches does Lena have?

Task 4.Slide 11 (4 steps).

Sasha bought 5 notebooks for 8 UAH and a sketchbook for 33 UAH. How much money did Sasha pay for the purchase?

The price of one notebook is 8 UAH – this is unit segment(measurement). The number of unit segments (5) indicates the number of notebooks. The second part of the segment reflects the price (33 UAH) and quantity (1) of albums.

Task 5.Slide 12 (7 steps).Two ways to create a diagram. Two solutions

The plant needs 90 workers: 50 turners, 10 mechanics, the rest are loaders. How many movers are needed?

Slide 13 (3 steps)compilation inverse problem. STOP

Techniques for working on tasks.

At the familiarization stage I use the following techniques:

Explanation of each component part of the model.

Instructions for building a model.

Modeling using guiding questions and step-by-step implementation of the scheme.

At the stage of understanding a schematic drawing, I use the following techniques:

Formulating the text of the problem according to the proposed plot and segmental diagram.

Correlation between a diagram and a numerical expression.

Filling out the template with task data.

Finding errors in filling out the diagram.

Selecting a scheme for the problem.

Selecting a task for the diagram.

Addition of task conditions.

Changing the scheme.

Changing the conditions of the problem.

Changing the task text.

The result of learning to construct and comprehend a schematic drawing is students’ independent modeling of problems.

When solving word problems, we work to develop the action of modeling, and vice versa, the better the child masters the action of modeling, the easier it is for him to solve problems.

Students should be introduced to various methods of solving word problems: arithmetic, algebraic, geometric, logical and practical; With various types mathematical models underlying each method; as well as with various solutions within the chosen method. Solving word problems provides rich material for the development and education of students. Brief notes conditions of word problems - examples of models used in initial course mathematics. The method of mathematical modeling allows you to teach schoolchildren:

a) analysis (at the stage of perceiving the problem and choosing the path to implement the solution);

b) establishing relationships between the objects of the problem, constructing the most appropriate solution scheme;

c) interpretation of the obtained solution for the original problem;

d) drawing up tasks using ready-made models, etc.

Presentation working on tasksSlides15-22 .

Combinatorics on models from 1st grade

2nd grade

Arrange the numbers 4, 6, 8 in different ways:

In grades 3-4

"Tree" (36 lunches)

Photo from notebook

Using simulations to teach numbering, adding and subtracting numbers, and working on units of length (5 min)

The ability to convert numbers into units of account and measurement units most often causes some difficulties. And here it is advisable to use the modeling method to help. By studying the “Tens” concentration, children learn to schematically represent units using dots.Slide 25. Learn to add and subtract using models.Slide 26. (7 steps)Slide 27.

While studying “The Hundred,” children depict tens using small triangles. They learn to convert numbers into units of counting (dec. and units) and at the same time, children become familiar with the centimeter and decimeter. This allows us to draw an analogy in the conversion of units of length. They also teach addition techniques. double digit numbers on numerical diagrams.Slide 28

While studying “A Thousand,” children will learn that we will conventionally represent 10 triangles (tens) by one large triangle (one hundred). At the same time, children are learning a new unit of length - the meter. When converting numbers into units of counting, we do similar work with units of length.Slide 29 example for number 342Slide 30 (5 steps)

Example for the number 320Slide 31 (6 steps)

Example for the number 302Slide 32 (8 steps)

Algorithms.Slides 33 and 34(7 steps)

Recommendations for using the modeling method in mathematics lessons (3 min)

It is necessary to understand that modeling in teaching is not desirable, but necessary, since it creates conditions for students to fully and firmly master methods of cognition and methods of educational activity.

The main objectives of modeling in the lesson are:

building a model as a way to construct a new way of action.

training in building a model based on an analysis of the principles and methods of its construction.

Remember that the first lessons are related to modeling, in fact, they are lessons in setting up an educational and practical task. The problem that children have is that they do not have enough ways to display general attitudes. Every time a new practical situation appears, children define new relationships - and again the question arises of how to convey it graphically.

Such “abstract tasks” as drawing a diagram using a formula, establishing a relationship between quantities that are part of several formulas, etc. offer when relationships are explored, informed and displayed in signs and diagrams repeatedly. Behind the model, each child should have actions with real objects, which he is now able to perform in his imagination (mental actions).

The place of the model for the child is determined depending on the task

The action may be accompanied by a model. For example, if it is easier to construct a method on a model, as a stage of working on a text problem (the relationships between quantities during reading are displayed schematically).

The model is built after the actions are completed. In order to understand the action performed, it is necessary to construct a diagram of a separate relationship. The construction of a diagram is motivated by questions like: “How did you do it?”, “How would you teach others to perform such tasks?

And a few more tips.

You need to start by studying specialized literature. For example, this is a method of teaching mathematics in primary school and textbooks by E. Alexandrova, L. Peterson.

On parent meetings Be sure to introduce parents to the method of teaching their children. Your advice and instructions may be useful to them.

Take every opportunity to take part in master classes on mathematical modeling.

Where I invite you.

Tasks

Subject

To solve this problem, it is necessary, first of all, to set the system the features of modifying the task of detecting a face with a video camera, and then carry out localization of lip movements.

As for the first task, two types of them should be distinguished:
Face localization;
Face tracking.
Since we are faced with the task of developing a facial expression recognition algorithm, it is logical to assume that this system will be used by one user who will not move his head too much. Therefore, to implement lip motion recognition technology, it is necessary to take as a basis a simplified version of the detection problem, where there is one and only one face in the image.

This means that face search can be carried out relatively rarely (about 10 frames/sec. or even less). At the same time, the movements of the speaker’s lips during a conversation are quite active, and, therefore, the assessment of their contour should be carried out with greater intensity.

The task of finding a face in an image can be solved using existing tools. Today there are several methods for detecting and localizing a face in an image, which can be divided into 2 categories:
1. Empirical recognition;
2. Facial image modeling. .

The first category includes top-down recognition methods based on invariant features of facial images, based on the assumption that there are some signs of the presence of faces in the image that are invariant with respect to shooting conditions. These methods can be divided into 2 subcategories:
1.1. Detection of elements and features that are characteristic of a facial image (edges, brightness, color, characteristic shape facial features, etc.) .;
1.2. Analysis of detected features, making a decision on the number and location of persons (empirical algorithm, statistics relative position signs, process modeling visual images, the use of rigid and deformable templates, etc.) , .

For the algorithm to work correctly, it is necessary to create a database of facial features with subsequent testing. For a more accurate implementation empirical methods models can be used that allow taking into account the possibilities of facial transformation, and, therefore, have either an expanded set of basic data for recognition, or a mechanism that allows modeling the transformation on basic elements. Difficulties in building a classifier database targeting a wide range of users with individual characteristics, facial features, and so on, helps to reduce the recognition accuracy of this method.

The second category includes methods of mathematical statistics and machine learning. Methods in this category are based on image recognition tools, considering the task of face detection as a special case of the recognition task. The image is assigned a certain feature vector, which is used to classify images into two classes: face/non-face. The most common way to obtain a feature vector is to use the image itself: each pixel becomes a component of the vector, turning the n×m image into a vector in R^(n×m) space, where n and m are positive integers. . The disadvantage of this representation is the extremely high dimension of the feature space. The advantage of this method is that it excludes the construction of a human participation classifier from the entire procedure, as well as the possibility of training the system itself for a specific user. Therefore, the use of image modeling methods to build a mathematical model of face localization is optimal for solving our problem.

As for segmenting a face profile and tracking the position of lip points over a sequence of frames, mathematical modeling methods should also be used to solve this problem. There are several ways to determine the movement of facial expressions, the most famous of which are the use of a mathematical model based on active contour models:

Localization of the facial expression area based on the mathematical model of active contour models

To adapt the active contour to the image of facial expressions, it is necessary to carry out the appropriate binarization of the object under study, that is, its transformation into a type of digital raster images, and then an appropriate assessment of the parameters of the active contour and calculation of the feature vector should be carried out.

The active contour model is defined as:
Set of points N;
Internal elastic energy term;
External edge based energy term.

To improve the quality of recognition, two color classes are distinguished: skin and lips. The color class membership function has a value ranging from 0 to 1.

The equation of the active contour model (snake) is represented by the formula v(s) as:

Where E is the energy of the snake (active contour model). The first two terms describe the regularity energy of the active contour model (snake). In our polar coordinate system v(s) = , s from 0 to 1. The third term is the energy related to external force, obtained from the image, the fourth - with the pressure force.

The external force is determined based on the characteristics described above. It is capable of shifting control points to a certain intensity value. It is calculated as:

The gradient multiplier (derivative) is calculated at the points of the snake along the corresponding radial line. The force increases if the gradient is negative and decreases otherwise. The coefficient before the gradient is a weighting factor that depends on the topology of the image. The compressive force is simply a constant, using ½ of the minimum weight factor. Best form snakes is obtained by minimizing the energy functional after a certain number of iterations.

Let's look at the basic image processing operations in more detail. For simplicity, let's assume that we have already somehow selected the area of ​​the speaker's mouth. In this case, the main operations for processing the resulting image that we need to perform are presented in Fig. 3.

Conclusion

List of sources used

To be continued