1, Toshboyeva Nargiza Yoldashevna2, Kuljonov Nadir Jonadil ugli3, Abdukodirova Patmakhon Tursunboevna

INFORMATICS, METHODS OF TEACHING MATHEMATICS
Makhmudova Dilfuza Meliyevna¹, Toshboyeva Nargiza Yoldashevna², Kuljonov Nadir Jonadil ugli³, Abdukodirova Patmakhon Tursunboevna⁴,
¹Chirchik State Pedagogical University. Head of the Mathematics Teaching Methodology Department, E-mail: [e-mail]
²Chirchik State Pedagogical University. Teacher of mathematics teaching methodology and geometry department, E-mail: [e-mail]
³Chirchik State Pedagogical University. Teacher of mathematics teaching methodology and geometry department, E-mail. [e-mail]
⁴senior lecturer of Andijan State Pedagogical Institute
ANNOTATION
Informatics teaching methodology Science for students of higher education in general secondary schools, academic Lyceum and vocational teaching Informatics in their colleges, deepening the structure and content of Science from a scientific and psychological-pedagogical point of view is a course that provides their learning.
It is known that mathematics is the most difficult for auxiliary schoolchildren is considered one of the sciences. This, on the one hand, is the science of mathematics abstraction of Dream consonants; on the other hand, auxiliary school on the face of the desire of his students to master mathematical knowledge with originality, the dream is called.
Key words: originality, the dream, rapidly entering all spheres
By the present time, the main of modern information technology the represented computers are rapidly entering all spheres of our life. Therefore, special attention is required to the study of computer technology. In this place is the teaching of Computer Science the importance of the Computer Science Teaching Methodology course, which deals with various techniques and methods, is very great.
Informatics teaching methodology course independent study of the problems of teaching in future Informatics teachers, training aimed at solving such issues as the ability to use modern methods, the formation of independent teaching skills, and the upbringing of methodological creativity.
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In one way or another, mathematical concepts have always depended on the calculation methods and writing methods: the etymology of the word informatics comes from the contraction of the words information and mathematics, cognate with the French word informatique which comes from information and automatique (automated information). The aim of our study is to have a perspective over some aspects related to the manner in which computers and informatics have affected Mathematics and mathematical research and to provide insights on the impact of computer science over the way mathematics is taught nowadays. Over the last decades, information has grown exponentially, research and problems have become more complex, requiring an analysis of a high volume of data, which the traditional hand - calculation has not coped with. Moreover, in the field of computer science it was observed that a deep analysis of a mathematical approach can lead to achieving more efficient algorithmic methods and on the other hand, mathematicians can find the same in-depth analysis of the approach to the algorithmics. Consequently, there will be given as much importance to the algorithmic and mathematical methods, and this has led to the emergence of new branches in Applied Mathematics, branches that are intertwined with Computer Science Informatics and Information Technology. Critical thinking, logical reasoning and mathematics were united and became a pattern to other sciences.
Certain branches of Mathematics have always been open to experimentation, but the appearance of computers means the scope for experimentation in Mathematics has been greatly increased. Among the long-established branches of pure Mathematics on which computers have had a major impact, there are Group Theory, Combinatory and Number Theory. An important part of Applied Mathematics, developed in the early 1950s, with high implications in Computer Science / Informatics, is Computational Mathematics which includes among other the following:
- numerical methods used in scientific computation, for example, numerical linear algebra and numerical solution of partial differential equations;
- stochastic methods, such as Monte Carlo methods and other representations of uncertainty;
- computational algebraic geometry; - computational group theory; - computational geometry;
- computational number theory; - computational topology; - computational statistics;
- algorithmic information theory;
- algorithmic game theory;
Computational Mathematics provides not only the main theoretical data, but also the theoretical instructions on how to program a numerical method or algorithm.
Programming in itself is beneficial for a student due to a number of reasons. First of all, programming teaches students logical-algorithmic thinking, and algorithmization is a fundamental requirement in solving any problem using a computer. Nowadays, problems whose solutions (demonstration process) are the result of combining purely mathematical methods with algorithmic (computational) methods occur more frequently. Sometimes, the computational results can provide inspiration for the mathematical methods.
same time, they are themselves the source of new areas of research. It is clear for us that computers have, and will continue to have, a significant impact on the research directions of Mathematics and on the manner in which mathematicians carry out their research. [4] Computers might be used in mathematical proofs; they might, initially, suggest what is true and, equally important, what is not true; they might be used for computations which are required in a proof.
Even before the availability of the modern computing technology, experimental sampling and Monte Carlo methods have played a role in Statistics for studying the performance of statistical techniques under the assumption of probability models. Mathematical modeling also plays an important role in the modern scientific research and the mathematical notions and mathematical symbols are the construction bricks for these models. Practically, every mathematical concept, every mathematical object, starting from the concept of number is a mathematical model. The scientific knowledge is based on the property schematization/mapping of the parts of the material systems or of any other kind, until obtaining some attributes between which simple relations can be established, which aims at setting some general laws. This representation is called modeling, and the schematic structures resulted are called models. A mathematical model of a particular phenomenon or object from the surrounding environment is a rough description of it (i.e. phenomenon or object) performed with mathematical concepts, objects and symbols. Mathematical models are commonly used in various fields, especially due to their ability to condense the essence rigorously and also due to their ability to be programmed using electronic computers. The use of mathematical concepts and mathematical software in the simulation and study of the economic processes has gained a significant development, by solving a growing number of problems. For example, the optimization of a production process (this may be accomplished with the lowest possible cost and with higher revenue potential) requires first to determine the mathematical model and then to use the mathematical instrument in order to obtain the appropriate solution for the respective purpose. In this respect, the use of specialized software for simulating and solving such mathematical models is a tool which is accepted and increasingly used. The real problems using the mathematical modeling can sometimes require a large volume of information to be processed. Computer science students have understood the need to develop logical, mathematical, and algorithmic thinking through the study of mathematics while the students specializing in Mathematics, who are faced with such problems, have understood the need for specialized knowledge of numerical computing software which is indispensable for solving nonlinear problems. The Approximation Theory represents an important chapter in Mathematics and its specific methods can be found in various problems of numerical structure studied by disciplines such as Numerical Methods, Numerical Calculus or Numerical Analysis. These disciplines are found in most university curricula, belonging to the domain of Applied Mathematics and they will be based on the calculation performed by using computing systems taking us away from the historical hand-calculating techniques. The nowadays’ numerical methods, whether classical or new, are only used by means of a computer. Given the complexity of the problems, a user must study the cases where you have to decide on what calculation system will be adequate to the problem, but, at the same time, to “intuitively perceive” the approach to the reasoning to be implemented in order to solve the problem. Lately, both in research as well as in the teaching area, applied mathematics has been gaining ground, so that disciplines that appear in the specialized curricula are also found in the branches of Mathematics pertaining to this area. The Approximation Theory is often found in the study of the phenomena around us, a study characterized by the functions of one or more real variables, but for which there is no possibility to determine them completely, but only to find certain particular values of the arguments. Knowing these values implies an approximation either of the respective functions on other new points within their definition domain, or of certain numerical features, such as the values on data points of the derivatives or integrals of certain orders in particular domains. In Numerical Analysis, a discipline on the border between Mathematics and Information Technology, students will recall concepts such as algorithm, time complexity of an algorithm or the representation of numbers in the internal memory of a computer system. Therefore, numerical methods should be chosen taking into account the convergence, the stability, the error propagation, but also the ease for implementation of the associated algorithm. The most effective approximation methods in case of large complex problems are represented by the iterative methods. Not even the simple equations admit solutions that could be determined by rational expressions or radicals. For this reason, it would be impossible to calculate the solutions to equations by means of a finite number of arithmetic operations. Therefore, the iterative methods are required, i.e. a
procedure that generates an infinite sequence of approximations Nnnx∈}{ so lxnn=∞→lim , where l is a root of the equation. In case of solving large systems of linear equations (n > 50) , the iterative methods are also agreed as order of complexity. The methods for solving systems of linear equations are traditionally divided into (i) direct and (ii) indirect or iterative methods. The direct methods include Gaussian elimination or the Crammer method and the iterative methods include the Gauss-Seidel and the Jacoby methods. The direct methods have the advantages (a) that they will always produce the solution provided that it exists, is unique and that sufficient accuracy is retained at every stage, and (b) that the solution is found after a known number of operations. They have the disadvantage that very sparse systems of equations, especially arising in finite difference approximations to differential equations, may become rapidly less sparse as the elimination process proceeds, therefore, raising the storage requirement from a multiple of n (for n equations) to something like n2. The iterative methods, on the other hand, may fail to converge to a solution and, if they do converge, it is not obvious how many operations they will require to produce the desired accuracy. Solving linear and quadratic equations, third and fourth-degree equations, large systems of linear equations, simplification of rational expressions with “towers” of double fractions, division and simplification of polynomials can all be done by using symbolic algebra, often integrated with the direct processing of very large integers. Where exact methods fall, approximations are possible [5]. As an example, in Newton’s method for the solution of transcendental equations, the tangent of a function could be taken instead of the function itself. Or the tangents with the same constant slope as the first tangent (the method converges in many cases) could be taken. Alterations of this kind are generally impossible with the acquired programs, which seldom allow such open didactic processing. Naturally, for these purposes, the teacher needs a simple and transparent computer language with natural key-words and sufficient mathematical operators as well as a compiler which can understand the language just as humans can. Maths Teachers also need a good cooperation with Computer Science / Informatics teachers and pupils, who can create good programs according to their desires.
In the numerical problem solving and analysis, a special attention is given to the complexity of the associated algorithm which will need to take into account the ease of implementation, while saving internal memory or computing time. Accordingly, algorithms which are ideal on a single processor may be highly inefficient, or even fail entirely, on parallel processors. The search for algorithms for the efficient solution of mathematical problems on parallel computer systems is a major area of research and, thus, many conferences on this topic are held regularly. When solving nonlinear equations, before applying an iterative method, it is often suggested to students to locate the solution on an interval, by applying the graphical solving method. This method becomes highly efficient and simple to use in case of using a software that easily allows the plotting/representation of a function, e.g. Matlab, Maple, or GeoGebra. The deployment of algorithms has represented the focus of exercises, homework and control of achievement and subsequently, pupils have been educated as if they were little computers. Related to this secondary role of algorithms, there is the fact that several thousand years of history of Mathematics have not produced a uniform language for the description of algorithms. The algorithmic problem represents a completely defined function:FIP→:, where I is the set of original information (problem input) and F is the set of final information . Suppose I and F are at most countable. Out of the multitude of the existing algorithms for solving a problem, the most “efficient” will be chosen, namely to be understandable, codified, editable, debugged and to use the resources effectively. The efficiency of an algorithm is also found in the time consumed which means the time spent for a number of operations (within the floating point). Therefore, in this case, the focus is not only on finding a solution, but that solution should be optimal in terms of accuracy or complexity of the algorithm. There are countless situations and real problems using the mathematical instrument, but which appeal to the knowledge of programming environments necessary for implementing the associated algorithms. Further on, there are some examples of this type.
• The evaluation of a functional RFl→: such as, for example, when calculating the value of a function )(xf, derivatives),(''),(' xfxf (digital derivation) of the integrals defined ∫badttf )( (numerical integration) and of the rules pfin the situation where this functional has a complex analytical expression; when solving algebraic equations: the determination of the values of some unknown variables in algebraic relations by solving some systems of linear or nonlinear equations.
• When solving some analytical equations: by determining the functions (or function values) as solutions of an operation equation, such as ordinary differential equations or with partial derivates, integral equations, etc.
• For optimization problems: by determining some numerical values specific to some functions which optimize (minimize or maximize) an objective function restrictively or unrestrictively.
The constructive mathematical problems, which numerical problems come from can be regarded as an abstract application YXF→:, between two normed linear spaces. Depending on which of the quantities y, x or F is unknown in the equation yFx=, we are dealing with a direct problem, a reverse problem or an identification problem. In probability and mathematical statistics teaching, the use of software, such as Matlab, that has function library specialized in this field can contribute significantly to the understanding of concepts and solve problems easily. An example of using the probability functions implemented in Matlab is where the connection between the hypergeometric law and the binomial law will be examined. The following example uses probability functions implemented in Matlab, to examine the link between hypergeometric law and binomial law. It will be proven by the following program that if mba=+ becomes large, then lim =∞→, namely the hypergeometric distribution acts as binomial distribution:
n=input( ‘Enter the parameter n=’); a=input( ‘Enter the parameter a=’); b=input(‘Enter the parameter b=’); x = 0 : n; f=pdf(‘hyge’, x, a+b, a, n); p = a/(a + b); ff=pdf( ‘bino’, x, n, p); bar(x’, [f’, ff’]); legend(‘hypergeometric distribution’,’ binomial distribution’); For 9=n and 11=+ba, the hypergeometric and the binomial distribution are represented by bars in Figure 1. 0 1 2 3 4 5 6 7 8 900.10.20.30.40.50.60.7hypergeometric distributionbinomial distribution Figure 1. The following graph for 9=n and 110=+ba shows that the hypergeometric distribution acts as a binomial distribution: