Solution Of Mathematical Problems Research Articles

Interaction of Students with an Intelligent System for Setting and Solving Combinatorial Tasks

Currently, there is a rapid development of artificial intelligence systems that can solve and explain the solution of mathematical problems in the same way as students do. The problem of organizing interaction of artificial and human intelligence which does not lead to the degradation of the student’s thinking skills arises. The article proposes one of the ways to solve this issue through the use of systems such as “smart problems” using the example of a support system for a combinatorics course. In this system, the student sets problems and solves them himself without the help of a computer, but the system can check the solutions to problems without simulating human reasoning. Such a system cannot replace a person, but it becomes an intelligent interlocutor when setting and solving problems. Thus, the task of formulating a problem shows the way to bypass the gap formed with the advent of artificial intelligence.

Mathematical understanding (hermeneutic aspects of educational activities)

The article discusses the issues of mathematical understanding as an educational practice of identifying the algorithmic meanings of mathematical (mathematized) educational texts. The hermeneutic orientation towards following the “guiding thread of language” (G.-G. Gadamer), as it is shown, cannot be narrowed to the attitude towards natural languages inherent in humanitarian educational practices. The “common place” of mathematical understanding in educational practices is: some of created languages (of scientific communication) being consolidated around the mathematical installation are not reducible to its sign universality or educational generalizations. But it often turns out to be outside the field of view of practicing teachers, in a didactically abstract position. The article substantiates the possibility of rehabilitating the hermeneutic attitude in mathematical education in connection with its development and strengthening in understanding the prerequisites for the algorithmic solution of mathematical problems (the need for additional information, non-standard solution techniques, transfer of educational goal setting to the area of interdisciplinary relationships or life experience, etc.). The author shows and comments on the effectiveness of the hermeneutic approach in mathematics education using examples of solving problems in algebra, geometry and in constructing mathematical models of various physical processes.

Olympiad on Lightning-Fast Solution of Mathematical Problems as a Psychological and Pedagogical Phenomenon

The article presents the author’s approach of designing an Olympiad on lightning-fast solution of mathematical problems for students of pedagogical universities in order to develop their quick thinking. The aim of the article is to analyse different models of organising Olympiads in mathematics for pupils and students, and to justify the expediency of including Olympiads on lightning-fast problem solving in the educational system of schools and universities. The research methodology is based on the system-activity approach and the requirements of educational standards of basic secondary and higher education, taking into account the labour functions of the professional standard of a teacher. Research results. One of the possible options for organising a mathematics Olympiad for students – future teachers of mathematics is the Olympiad «Dragonfly» for lightning-fast solution of problems of the school course of mathematics of different levels of complexity. When designing the content of such an Olympiad it is necessary to observe a number of principles: balance, brevity, accessibility, dynamism. On the basis of the Institute of Mathematics, Physics and Informatics of the V. P. Astafiev conducted a study to identify the impact of the Olympiad «Dragonfly» on the development of quick thinking in students – its participants. The study confirmed the hypothesis that if students are systematically involved in the process of lightning-fast solution of school mathematical problems of the Olympiad «Dragonfly», it will contribute to the development of their quick thinking. Conclusion. The experimental work has confirmed the expediency of including the Olympiads on lightning-fast problem solving in the educational system of school and university.

FEATURES OF THE APPLICATION OF THE SCILAB MATHEMATICAL PACKAGE IN THE PROCESS OF TEACHING THE DISCIPLINE «HIGHER MATHEMATICS»

The article notes that the choice of software is of great importance in teaching the discipline «Higher Mathematics». The complexity and type of tasks to be solved depend on the functionality of the selected program. It is emphasised that the methods of approximate, primarily numerical, solution are used. The methods of numerical solution of mathematical problems have always been an integral part of mathematics and have always been part of the content of natural-mathematical and engineering education. This article uses the Skilab math package as software. It is a multi-functional package designed for performing engineering and scientific calculations, which allows you to perform complex algebraic calculations, solve problems of differentiation and integration. All this contributes to the successful use of the Scilab mathematical package in the process of teaching the discipline «Higher Mathematic». The author draws attention to the fact that Skilab is a computer mathematics system designed to perform engineering and scientific calculations, such as: solving nonlinear equations and systems; solving linear algebra problems; solving optimization tasks; differentiation and integration; processing of experimental data (interpolation and approximation, method of least squares); solution of ordinary differential equations and systems. Skilab provides wide opportunities for creating and editing various types of graphs and surfaces. The Skilab system contains a sufficient number of built-in commands, operators and functions, and its distinctive feature is flexibility. The user can create any new command or function and then use it alongside the built-in ones. The article states that the use of modern mathematical packages significantly increases the intensity of cognitive activity, raises the level of mathematical preparation, improves the knowledge control system, and promotes learning motivation. The perspective of the development of the study of the use of mathematical packages is the development of a database on the issues of their use during the educational process, which should take into account the professional orientation of future activities. This approach provides a more effective application of interdisciplinary connections, which contributes to the in-depth study of the material and the expansion of opportunities for independent learning.

Optimization methods for multi-criteria decisions in pharmacy

Optimization methods for multi-criteria decisions in pharmacy
 In pharmaceutical technological research, the determination of the quantitative composition of granules is considered as a task of multi-criteria selection. Today, to solve this problem, the regression analysis and multi-criteria optimization methods are widely used; they are based on mathematical models obtained for the object under study.
 Aim. To identify a decision-making method in a multi-criteria space that is effective for use in pharmaceutical technology research with quantitative factors.
 Materials and methods. The study uses tools of the popular computer mathematics system Mathcad (MathSoft Ins., USA) to automate the solution of mathematical problems. To automatically search for the type and coefficients of regression equations, the MS Excel application was used, namely: the data analysis package (regression analysis). The MS Word processor was used to edit the code.
 Results. A variety of approaches to the formalization of the multi-criteria optimization task have been studied. The optimal quantitative content of excipients when developing the granule technology has been found using two different optimization criteria, which are formed according to different methodical approaches. The method proposed does not provide for the mandatory introduction of gradation of individual criteria or their weighting factors.
 Conclusions. As a result of the comparison of multi-criteria optimization methods, the effectiveness of the decision-making method in the multi-criteria space has been shown; it synthesizes a mathematical procedure related to the vector of criteria and is based on determining the ideal point and introducing the concept of a norm into the space of functionals; it has not been mathematically proven, but it is practically useful decision-making algorithm compared to the mathematical method of convolution of criteria. The optimization method proposed has advantages that are manifested in the possibility of using a relatively simple mathematical apparatus and simplified logic of obtaining a solution.

Developmental trajectories of number line estimations in math anxiety: Evidence from bounded and unbounded number line estimation

AbstractMathematical anxiety (MA) is a feeling of tension and anxiety that interferes with the manipulation of numbers and the solution of mathematical problems. Past studies have shown that high MA individuals experience difficulties at different levels and domains of mathematics, and that MA results from weakness in spatial abilities, subsequent to deficits in basic numerical abilities. Hence, the main goal of the present study is to examine developmental trajectories of number space associations and their role in MA. In the present study, 2nd, 3rd, and 5th grade children (n = 94) participated in number line estimation tasks, half with low MA (LMA) and half with high MA (HMA). They performed two number line estimation tasks (bounded‐ with numerical values for beginning and end of the number line, or unbounded‐ only the beginning of the number line is marked) in addition to examination of strategy during arithmetical problems. Results indicated that younger children with HMA showed non‐mature numerical estimations compared to participants with LMA. However, for older HMA children, estimations were normalized and group differences were eliminated. Moreover, we found that estimations predicted usage of advanced memory‐based strategies in simple addition operations. These results indicated that (1) non‐mature numerical estimations characterize younger participants with MA, and (2) that numerical estimations are associated with arithmetical performances.

HIGH-DIMENSIONAL STOCHASTIC DESIGN OPTIMIZATION UNDER DEPENDENT RANDOM VARIABLES BY A DIMENSIONALLY DECOMPOSED GENERALIZED POLYNOMIAL CHAOS EXPANSION

Newly restructured generalized polynomial chaos expansion (GPCE) methods for high-dimensional design optimization in the presence of input random variables with arbitrary, dependent probability distributions are reported. The methods feature a dimensionally decomposed GPCE (DD-GPCE) for statistical moment and reliability analyses associated with a high-dimensional stochastic response; a novel synthesis between the DD-GPCE approximation and score functions for estimating the first-order design sensitivities of the statistical moments and failure probability; and a standard gradient-based optimization algorithm, constructing the single-step DD-GPCE and multipoint single-step DD-GPCE (MPSS-DD-GPCE) methods. In these new design methods, the multivariate orthonormal basis functions are assembled consistent with the chosen degree of interaction between input variables and the polynomial order, thus facilitating to deflate the curse of dimensionality to the extent possible. In addition, when coupled with score functions, the DD-GPCE approximation leads to analytical formulae for calculating the design sensitivities. More importantly, the statistical moments, failure probability, and their design sensitivities are determined concurrently from a single stochastic analysis or simulation. Numerical results affirm that the proposed methods yield accurate and computationally efficient optimal solutions of mathematical problems and design solutions for simple mechanical systems. Finally, the success in conducting stochastic shape optimization of a bogie side frame with 41 random variables demonstrates the power of the MPSS-DD-GPCE method in solving industrial-scale engineering design problems.

Mathematical programming in Mathcad and Mathematica

An article generalizes the long-term work of authors with packages of applied mathematical programs. It discusses and demonstrates the features and methods of solution of mathematical tasks in mathematical package Mathcad and Mathematica: from the simplest ones, included in the set of typical problems of mathematical disciplines for training specialists for shipbuilding, to complex computational tasks and applied problems of professional orientation, which require the construction of a mathematical model and analysis of the results obtained. The examples show the solution of mathematical problems in symbolic form, mathematical studies in the Mathcad and Mathematica environment, and mathematical programming with these packages.

SIMULATION MODELING OF INVESTMENT PROJECTS IN THE SERVICE SECTOR

Scientific article is devoted to researching the possibility of developing a simulation model of an investment project for a travel agency. In recent years, the tourism industry in the world is experiencing rapid development. The development of the tourism industry requires new investments. The investor must assess the degree of risks, the likelihood of profit or loss. It is noted that the most important advantage of simulation modeling is that it makes it possible to study economic systems at the design stage. Due to this, simulation models can be used as a universal tool in making appropriate decisions under conditions of uncertainty and taking into account those factors that are difficult to predict and take into account, which is why simulation is so often used in the development of investment projects. When assessing the risks of investment projects, the collection of information requires significant financial expenses, is quite time-consuming, and sometimes impossible. The subject of the study is a set of the theoretical, methodological and organizational problems related to the management decisions regarding the investment of the projects in the field of tourism. The methodology is based on the stochastic simulation modeling of economic processes. The aim of the study is to develop a simulation model for an investment project in the tourism sector, which will allow the investor to appreciate the degree of the risk and the likelihood of return on the investment. The article reveals the features of computer simulation in the MathCAD system. It is proved that it is convenient to develop simulation models of investment projects in the MathCAD system. It has a powerful mathematical support, remains one of the systems in which the description of the solution of mathematical problems is given with the use of conventional mathematical formulas and symbols and does not require special training in programming. The simulation model of the investment project of the travel agency was developed in the MathCAD system. Statistical processing of the results of the experiments with the model has been carried out. The necessity of the detailed study of critical intervals of the histogram at the transition from loss-making to profitable NPV values has been substantiated. Based on the results of simulation experiments with the model, the errors of the model were estimated, the law of NPV random variables distribution was established. The developed simulation model allows the investor to estimate the risk factor, the probability of profit or loss from the investment, to assess the possible uncertainty of the results of their own decision to invest in the project. Conclusion: the simulation model of the investment project built and investigated in MathCAD system allows investor to estimate the risk factor, probability of profit or loss for choosing the optimal pricing policy and optimization of economic strategy of tourist agency.

DIDACTIC AND METHODOLOGICAL FEATURES OF SCHOOL TEXTBOOKS OF COMPUTER SCIENCE 80S OF ХХ CENTURY

Despite changes in education and society, the textbook remains one of the most important teaching aids. The school textbook on computer science has almost thirty-seven years of history during which has undergone significant changes in content and structure. The analysis of scientific works on school textbooks showed that the problem of creating textbooks on computer science in the 80s of the XX century remained out of the attention of scientists. Therefore, the article presents the results of a study of didactic and methodological features of school textbooks on computer science in the 80s of XX century. The structure of textbooks was clarified with the help of structural analysis, the educational text, control blocks, knowledge acquisition, practical mastering, illustrative material were analyzed. The study analyzes the textbooks that were introduced in Ukrainian schools in the period from 1985 to the late 80’s of XX century. Describes the chronology of the creation of trial textbooks in computer science for schools. Approximate curricula for the course “Fundamentals of Electronic Computer Science” are presented, which allowed the author to identify the dynamics of changes in content. Trends in the creation of educational literature on computer science have been studied. It was revealed that with the advent of electronic computers in educational institutions and their introduction into the educational process, variants of curricula for machine and machine-free computer science courses were developed. It was also found that the common features in the textbooks are the focus on programming and algorithmization, compliance with the requirements of the curriculum. Distinctive features are the presentation of the material, illustrative material, the sequence of studying topics, the apparatus of orientation. The general trend in the creation of educational literature on computer science in the 1980s of the XX century was to study algorithmization and programming, and control and verification units were aimed at the practical solution of problems in mathematics and physics with the help of algorithms.

Dependence of Mathematical Errors on Mathematical Thinking Style

The results of research of creative mathematical thinking have been analyzed and the expediency of studying its individual differences through the analysis of mental styles of solving mathematical problems has been stated. The aim of the article is to identify the reasons of students’ mathematical errors and to analyze the influence of mathematical thinking style on the reasons and sense of mathematical errors in the process of solving creative mathematical tasks. To identify and determine the essence of mathematical thinking styles, the method of analysis of search actions of subjects during the solution of mathematical problems of different classes was used. The results of the research. It is stated that mathematical thinking style is a holistic system of interconnected actions, by means of which the mental mathematical result is achieved. It has been found that mental errors occurred at all stages of the search process of students with different mathematical thinking styles. Differences in mathematical thinking styles become the basis of the content of errors. There are three groups of causes of mental errors: ignorance, low-quality analogy, irrelevance of links. Different influence of ignorance on the search actions of students with different styles of mathematical thinking is proved. It is stated that the introduction of low-quality analogy in different styles of mathematical thinking is different in content. It was found that the irrelevance of references and conclusions has a different nature in the thinking of students with different thinking mathematical styles. Conclusions. Thinking style is manifested throughout the process of solving a mathematical problem, provides a different content of the search process, through which a positive mathematical result is achieved or leads to mental errors.

COMPUTER DIAGNOSTICS OF THE LEVEL OF SELF-ESTEEM IN THE PROCESS OF SOLVING MATHEMATICAL PROBLEMS

The diagnostics of the dynamics of self-assessment of the level of independence of educational activity of students in teaching the solution of mathematical problems is considered. The groups of students with overestimated, average and low self-esteem were identified. It is shown that the self-assessment of algorithmic activity among male students is on average higher than among female students.

Solution of Space-Time-Fractional Problem by Shehu Variational Iteration Method

In this study, we deal with the problem of constructing semianalytical solution of mathematical problems including space-time-fractional linear and nonlinear differential equations. The method, called Shehu Variational Iteration Method (SVIM), applied in this study is a combination of Shehu transform (ST) and variational iteration method (VIM). First, ST is utilized to reduce the time-fractional differential equation with fractional derivative in Liouville-Caputo sense into an integer-order differential equation. Later, VIM is implemented to construct the solution of reduced differential equation. The convergence analysis of this method and illustrated examples confirm that the proposed method is one of best procedures to tackle space-time-fractional differential equations.

Thinking Styles of Understanding Creative Mathematical Problems in the Process of Solving Them

The results of research of creative mathematical thinking are analyzed and the expediency of studying its individual differences through the analysis of mental styles of solving mathematical problems is stated. The aim of the article is to identify thinking mathematical styles of students and to analyze the influence of style on the process of understanding the creative mathematical problem. For identifying and determining the essence of mathematical thinking styles, the method of analysis of search actions of subjects during the solution of mathematical problems of different classes was used. The experiment involved 100 students of a technical university, 1000 processes for solving mathematical problems were analyzed. The results of the research. It is stated that mental mathematical style is a holistic system of interconnected actions, by means of which the mental mathematical result is achieved, which distinguishes the activities of one person from another one. The authors identify three thinking styles in creative mathematical process of students: the differential, the integral and the differential-integral. It is established that the thinking style is manifested during all processes micro-stages of understanding. It is proved that the different thinking mathematical styles lead to the formation of different purports of the same problem. It is established that unconscious mental guesses have different meanings in the process of understanding a mathematical problem by students with different styles of mathematical thinking. Conclusions. Thinking mathematical style is manifested throughout the process of understanding a mathematical problem, provides a different content of the search process aimed at understanding a mathematical problem.

Методика применения эвристических подходов при решении математических задач

Taking into account the development of science and technology, the use of heuristic methods in the learning process is becoming popular for the development of thinking and logic. Heuristic methods in mathematics are not only something entertaining and fascinating, but also an effective tool for mastering the material and developing students ' logical thinking. The considered heuristic methods will simplify the solution of mathematical problems in the school mathematics course.

The difficulties that face Sixth Grade students in Riyadh in solving the mathematical problem and their causes from their point of view: الصعوبات التي تواجه تلاميذ الصف السادس الابتدائي بمدينة الرياض في حل المسألة الرياضية وأسبابها من وجهة نظرهم

The aim of this research is to identify the difficulties facing sixth- grade primary students in Riyadh in solving the mathematical problem and their causes. The researcher used the descriptive survey method, and the researcher applied two tools to find out that it is a diagnostic test and an interview where the test was applied to a sample of (125) students and then applied the interview to a random sample consisting of (20) students in the sixth grade in the city of Riyadh. Where the results reached indicated the following: Students in understanding, planning and implementing the mathematical problem got a percentage of (23.36%), i.e. an estimate (low), with regard to the percentage of their success in health from the problem solving the mathematical problem (24,24%), with an estimate (very low). Difficulties faced by sixth-graders in the city of Riyadh in solving the mathematical problem, as they were: Identify the unknown and required in the mathematical problem, then make sure of enough information given to solve the mathematical problem; They were followed by determining the strategies that will be used for the solution, then the student is not convinced of the importance of following the four steps to solve the mathematical problem, and finally, the mathematics teacher focuses on the step of implementing the mathematical problem solution only. According to the results, a number of recommendations and proposals were presented to overcome the difficulties facing students in solving the mathematical issue in Riyadh and the whole of the Kingdom.

Teaching how to solve mathematical problems: an analysis of multiple correspondences

The overall aim of this research is to analyze the typical classroom activities (TCA) in Mathematical Problem Solution classes in order to identify the type of actions linked to their teaching at the primary school level. Pedagogical aids, participation structures and collaborative work done at the beginning, development and closing moments were analyzed through a multiple correspondence analysis (ACM). The results indicate that the most relevant actions take place only in some moments of the class and that, although the problems proposed are worked on, the verification of the results, argumentative skills and collaborative work are not encouraged. We also concluded that the most frequent aids have little impact on mediated learning and that the interactions between teacher and student do not point towards feedback.

Numerikus számítások hatékonyságának vizsgálata Python, Matlab és Octave programcsomagokkal

A numerikus analízis a matematikai problémák közelítő, számítógéppel is hatékonyan elvégezhető megoldásával foglalkozik. A cikkben bemutatott kutatás célja, hogy összehasonlítja a Python NumPy és SciPy, a Matlab és GNU Octave programcsomagok alapvető numerikus algoritmusainak sebességét. A bemutatásra kerülő sebességtesztek olyan alapvető numerikus műveletekre koncentrálnak, mint a mátrix műveletek, interpolációk, lineáris egyenletrendszer megoldása, valamint a numerikus integrálás és deriválás. Különböző méretű tesztfeladatok esetén, az egyes műveletek elvégzéséhez szükséges időket fogjuk összevetni, valamint a mérés során figyelembe vesszük az egyes implementációk közötti különbségeket is.

Kemampuan Representasi Matematis Siswa Kelas VIII SMP Ditinjau dari Gaya Kognitif Visualizer dan Verbalizer

Mathematical representation skill is students' ability to express mathematical ideas (such as problems, statements, and definitions) in various ways to solve problems through multiple representations, such as images, words, tables, and symbols mathematics. Students are struggling in representing mathematical ideas. It hampers them in determining the solution of mathematical problems. They are careless in reading the word problems, lacking problem analysis, less thorough, and struggling to connect concepts. The subjects of this study were in two students from one of the junior high school in Jambi. The instruments used for this research were VVQ, Mathematical Representation Ability Test and interviews. This study used a descriptive qualitative method. The results showed that the representation abilities of students with visualizer and verbalizer style were quite good. However, each subject had a different way of solving problems. Visualizers were more interested in questions with image information in solving the problem. Verbalizer tended to prefer information with detailed wording.

Optimal Maintenance of DiscreteObjects on the Plane, in Space and in Given Boundaries of the Sphere

The authors of the article simulate a mathematical solution of problems associated with optimal distribution and maintenance of discrete objects belongng, particularly, to transport environment. Options for location from one to ten objects, which are intended to be in space, on a straight line or on the plane, within the boundaries of a sphere of a certain radius, are considered. The results of the study can be applied for planning residential areas, optimal location of public transport stop points on roads and of transport rescue services. Following the suggested model, similar calculations can be used for any aspect of human life.