CC BY
3
DOI: 10.24412/2413-2527-2022-432-41-45
A Module of Educational Program for Distance Learning
Grand PhD V. N. Fomenko, PhD V. V. Garbaruk Emperor Alexander I St. Petersburg State Transport University Saint Petersburg, Russia vfomenkol943@gmail. com, vvgarbaruk@mail. ru
Abstract. A training program on two sections of linear algebra is proposed for educational purposes. The student, using basic operations, should transform the given matrix into the identity one. Mastering this skill makes it possible to solve systems of linear algebraic equations by the Gauss method and find the inverse matrix. An important feature of the suggested software segment is that it provides the ability to monitor the process of solving the problem by a student, for whom there are always several possible approaches. The program can be used in distance technologies in computer training.
Keywords: methodology of teaching, training programs, linear algebraic equations, inversed matrix.
Introduction
Educational programs at universities are aimed at students who independently acquire skills to solve problems of a particular subject. In such a program, at the pace chosen by the student, the training material is presented in a convenient and understandable form. Monitoring of acquired skills provides feedback in training programs. Not all branches of mathematics have algorithms that allow you to create programs that meet the learning objectives [1-5]. It is difficult to create a training program in which a student, solving a problem, chooses a sequence of actions. A similar program was created, for example, to study the operations of vector algebra [6]. The proposed software product is characterized by the ability to monitor the process of solving the problem by the student, who always has several possible approaches. A large number of programs (Excel, Mathcad, MATLAB and others) can solve problems, but they do not explain how the result is obtained [7, 8]. The program consists of two modules: «solving a system of linear algebraic equations» and «finding the inverse matrix», united by a single methodological approach. The modules of the training program are written in Fortran 95 [9].
Features of program modules
In the learning process, the student can apply for various kinds of information, necessary both for the correct sequence of operations and for answers to the questions offered by the program. These include a quick reference (F1 key), a text with a detailed description of the methods, five presentations with explanations of the sequence of actions when solving examples, a set of tasks consisting of four systems of equations (defined, indefinite, incompatible, homogeneous) and one third-order matrix. This set of tasks makes it possible to use software package to verify learning outcomes. An important element of training is the student's independent choice of the trajectory of achieving the goal. If the student is having difficulty solving the problem, then he can see in step-by-step mode a detailed solution to a similar example.
After solving the problem, a digital code is displayed on the screen. It is calculated by the program according to a certain algorithm known to the teacher based on the parameters of the problem. This code is a hash code of the problem description. The student can get this code either from the program, having solved the problem correctly, or guessing it, which is extremely unlikely.
The program has the following features:
• at any time, you can turn to theoretical material;
• the «Go back», «Go on» keys enable the student to both view all the steps of the solution and cancel unsuccessful operations;
• when solving an example, the mistakes made by student are recorded and their nature is determined. As a result, a list is composed and then communicated to the student before the program is closed.
Calculating the inverse matrix
Let us consider in detail the operation of the module «cal-culating the inverse matrix». After specifying the dimension of the matrix to be unversed, its elements are entered in my order in the table that appears. You can enter both decimal and common fractions. Pressing the «Complete Entry» button takes the student to the next page. If a degenerate matrix is entered, the message «A singular matrix! Enter a valid matrix». Usually, in the classroom, a third-order matrix is selected, for example:
/0 1 1 \
(2 1 0 ).
V3 0 -1/
After specifying the dimension of the matrix, its elements are entered into the table.
Fig. 1. An invertible matrix
On the screen, the student sees an invertible matrix, an identity matrix of the same dimension, and a menu «MATRIX TRANSFORMATION» that consists of three items (Figure 1):
• swap rows;
• single row linear transformation;
• two rows linear transformation.
The proposed operations are performed on two matrices identically. The student, choosing and performing various steps of the menu, should receive an identity matrix instead of the inverse. In this case, the second matrix becomes inverse [10, 11].
Possible initial steps to achieve the desired result are provided below. In the matrix shown in Figure 1, to get a non-zero element in the upper-left corner, you can select «Swap rows» item and highlight with the mouse, for example, two rows.
Fig. 2. Swap rows
In the matrix shown in Figure 1, to get a non-zero element in the upper-left corner, you can select «Swap rows» item and highlight with the mouse, for example, two rows (Figure 2). The result can be seen in Figure 3 after selecting «Continue».
Fig. 3. Single row linear transformation
To get «1» in the upper left corner, the student selects the item «Single row linear transformation», mark the first line and enter the required coefficient «1/2» (Figure 3). Figure 4 shows the result of this operation.
To get zero in the matrix, the last item «Two row linear transformation» is selected. First, the student selects the row of the matrix that needs to be changed, it becomes highlighted in red. Then indicates the line with which the change will be implemented. To complete the operation, the student enters the
Step No. 3
Go back
Go on
Invcrsed matrix Matrix to be invcrsed is to appear here
1 1/2 0 0 1/2 0 3 0
0 1 L 1 0 0 o
3 0 -1 0 0 1 -1 0
MATRIX TRANSFORMATION
Select two rows, insert multipliers to (lie right of them and press «Continue»
Continue
Transformations are complete
The i cd row of the two selected ones is changed
Fig. 4. Two rows linear transformation
coefficients by which the selected rows will be multiplied (Figure 4). Instead of the first selected line, the sum of the two marked lines multiplied by the given coefficients appears. Figure 5 shows the final view of the matrices after several steps.
Step No. 8
Go back Go on
Inversed matrix
Matrix to be inversed is to appear here
1 0 0
0 1 0
0 0 1
t -1 1
_2 3 _2
3 -3 2
MATRIX TRANSFORMATION
0 Swap rows
0 Single row linear transformation 0 Two rows linear transformation
Select two rows, inserl multipliers to the right of them and press ((Continue»
Continue
Transformations aie complete
The red row of the two selected ones is changed
Fig. 5. Transformations completed
The key «Transformations completed» is used if the student believes that he has completed all the necessary actions (Figure 5). If the left matrix is different from the identity matrix, then the inscription appears: «Transformations are not finished yet», and the unsuccessful action will be included in the list of errors. At the end of the work, the message may appear: «You have not made a single mistake», or a list of errors indicating their type and quantity.
During the learning, one should limit oneself to the dimension of the matrix 6*6 (although the maximum allowable size is 20*20), because in this case, the transformed initial matrix, the obtained inverse matrix, and all menu items (Figure 4) are visible simultaneously on the 15-inch monitor screen.
Solving a system of linear algebraic equations
The module «solving a system of linear algebraic equations» has a more diverse menu (Figure 5). Three operations are added: «Swap columns», «Remove zero row», «Remove proportional row».
r2x1 + 4x3 = 6; x2 + x3 =2; 2x2 + 2x3 = 4; ^3x1 + 6x3 = 9 .
The extended matrix shown in Figure 6 corresponds to this system.
Step No. 1
Go back
Go on
-Ï1 X2 X3 b
2 0 4 6
0 1 1 2
0 2 2 4
3 0 6 9
MAT RL\ TRANSFORMATION
O Swap rows
Single row linear 0 transformation
_ Remove
O Swap columns
q Two rows linear transformation Remove
proportional row
O
Select two rows, insert multipliers to the right of them and press «Continue»
Analysis of solution
O Inconsistent system
O Definite system -
O Indefinite system Continue
The red row of the two selected ones is changed
Fig. 6. The extended matrix
In this module there is another final step of the solution. After the transformation of the system matrix, the student must specify the type of system in the «Analysis of solution» menu:
• definite system;
• indefinite system;
• indefinite system.
If the student makes the right choice, he needs to type the answer. The problem, for example, is considered solved if the first rows and columns in the matrix of the system form the identity matrix [10, 12].
X1 X2 X3 ^ 1002
( 0 1 0 3 1 .
V0 0 1 4/
In this case the student selects the «Definite system» item. Then you need to type the answer received in the last column.
Slep No. 1
Go back
Go on
Xl Xl b
0 4 6 3
0 1 1 2
0 2 2 4
3 0 6 9 -2
MATRIX TRANSFORMATION
O Swap rows
Single row linear ® transformation
[ Remove zero row
O Swap columns
q Two rows linear
transformation 0 Remove proportional row
Select two rows, insert multipliers to the right of them and press «Continue»
Analysis of solution
Olnconsistenl system O Definite system
O Indefinite system The red row 01 the two selected ones is cha nged
Continue
Fig. 7. Two rows linear transformation
The zero line obtained after the operation shown in Figure 7 is obligatorily deleted.
X1 X2 X3 b
2 0 4 6
0 1 1 2
0 2 2 4
0 0 0 0
X1 X2 X3
1 0 4
0 1 1
0 2 2
This action can be accelerated by selecting the menu item «Remove proportional row». After selecting two proportional rows, the indicated row is deleted first:
X1 X2 X3 "
a o 4 6^
0 112
V0 2 2 4V
X1 X2 X3 "
/2 0 4 6) f0 1 1 2/ '
Select the item «Single row linear transformation», the student marks the first line and enters the required coefficient «1/2».
Xi X2 X3 b
X1 X2 X3
/2 0 4 ^ /1 0 2 3^ V0 1 1 2^ V0 1 1 2/ .
In this case, in the «Analysis of solution» menu, select the «Indefinite system» item. Then the student must type the answer:
fxi = 3 - 2x3; 1*2 = 2 —
specifying first the additional basis variables (for instance, x\ and X2) in the given example.
In the case of obtaining a matrix of the form
X1 X2 X3
f1 f0
002 0 0 13
) .
It is necessary to select the «Swap rows» item in the «Matrix Transformation» menu (Figure 6). By selecting the second and third columns, the student receives a matrix
X1 X3 X2
/1 0 0 2) f0 1 0 3) '
In the «Analysis of solution» menu, student selects the «Indefinite system» item. Then he must type the answer
fxi = 2;
1x3 = 3,
specifying basis variables X1 and X3.
For an incompatible system
X1 X2 X3 ^
/1 0 5 2\ (0 1 6 3 1 , Vo 0 0 4/
it is necessary to specify the decisive row, for example, the third row in the example above.
Conclusions
The sudden transition of universities around the world from traditional forms of education to individual self-training has required a change in the teaching methods of all disciplines, including mathematics. The success of technologies for remote learning depends on the correct methodological approach within software. A student cannot effectively comprehend new information if the educational material comes only in the form of texts [13, 14]. In the proposed learning program, the student has a starting position — the initial matrix, and the finishing position — the identity matrix. In each module the student is invited to think through a sequence of actions and implement
them, having access to texts and presentations in case of difficulties. When the pandemic ends and universities return to traditional in-person education, such distance learning will continue to be in demand [15, 16]. This program can be used on a local computer, in a classroom, or via the Internet in online mode.
REFERENCES
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Модуль образовательной программы для дистанционного обучения
д.ф.-м.н. В. Н. Фоменко, к.т.н. В. В. Гарбарук Петербургский государственный университет путей сообщения Императора Александра I
Санкт-Петербург, Россия vfomenko 1943@gmail. com, vvgarbaruk@mail. ru
Аннотация. Предложена обучающая программа по двум разделам линейной алгебры. Студент, используя элементарные операции, должен преобразовать заданную матрицу в единичную. Освоение этого навыка дает возможность решать системы линейных алгебраических уравнений методом Гаусса и находить обратную матрицу. Важной особенностью предлагаемого программного обеспечения является то, что в нем отслеживается процесс решения задачи студентом, у которого всегда есть несколько возможных способов получения ответа. Программа может быть использована в дистанционных технологиях при компьютерном обучении.
Ключевые слова: методология обучения, обучающие программы, системы линейных алгебраических уравнений, обратная матрица.
Литература
1. Ochkov, V. F. Teaching Mathematics with Mathematical Software / V. F. Ochkov, E. P. Bogomolova // Journal of Humanistic Mathematics. 2015. Vol. 5, Is. 1. Pp. 265-285. DOI: 10.5642/jhummath.201501.15.
2. Tchounikine, P. Computer Science and Educational Software Design: A Resource for Multidisciplinary Work in Technology Enhanced Learning. — Heidelberg: Springer-Verlag, 2011. — 193 p. DOI: 10.1007/978-3-642-20003-8.
3. Жулёва, Л. Д. Использование обучающих программ по высшей математике в учебном процессе иностранных студентов // Научный вестник Московского государственного технического университета гражданской авиации. Серия «Международная деятельность вузов». 2007. № 116. С. 102-107.
4. Жуковская, Т. В. Повышение эффективности процесса обучения математике на основе специального программного обучения / Т. В. Жуковская, Е. А. Молоканова // Вестник Тамбовского университета. Серия: Гуманитарные науки. 2010. № 12 (92). С. 53-61.
5. Дайняк, И. В. Подход к построению интерактивных мультимедийных страниц для автоматизированной обучающей системы / И. В. Дайняк, К. Дзержек, Т. Хустё // Дистанционное обучение — образовательная среда XXI века: Материалы III Международной научно-методической конференции (Минск, Беларусь, 13-15 ноября 2003 г.). — Минск: Белорус. гос. ун-т информатики и радиоэлектроники, 2003. — С. 201-203.
6. Карпович, С. Е. Компьютерная программа для изучения операций векторной алгебры с возможностью применения в дистанционном обучении / С. Е. Карпович, В. С. Баев // Информатизация обучения математике и информатике: педагогические аспекты: Материалы международной научной конференции, посвященной 85-летию Белорусского государственного университета = Informatization of Teaching Mathematics and Informatics: Pedagogical Aspects: Proceedings of the
International Scientific Conference in Honour of 85 Years Jubilee of Belarusian State University (Минск, Беларусь, 25-28 октября 2006 г.). — Минск: Белорус. гос. ун-т, 2006. — С. 176-180.
7. Walkenbach, J. Excel 2016 Bible. — Indianapolis (IN): John Wiley & Sons, 2015. — 1152 p. — (Bible).
8. Gilat, A. Numerical Methods for Engineers and Scientists: An Introduction with Applications Using MATLAB. Third Edition / A. Gilat, V. Subramaniam. — John Wiley & Sons, 2013. — 576 p.
9. Chapman, S. J. Fortran 90/95 for Scientists and Engineers. — McGraw-Hill Companies, 1998. — 888 p.
10. Квасов, Б. И. Численные методы анализа и линейной алгебры. Использование Matlab и Scilab: Учебное пособие. — Санкт-Петербург: Лань, 2016. — 328 с. — (Учебники для вузов. Специальная литература).
11. Anton, H. Elementary Linear Algebra: Applications Version. Twelfth Edition / H. Anton, C. Rorres, A. Kaul. — John Wiley & Sons, 2019. — 800 p.
12. Воеводин, В. В. Энциклопедия линейной алгебры. Электронная система ЛИНЕАЛ / Вал. В. Воеводин, Вл. В. Воеводин. — Санкт-Петербург: БХВ-Петербург, 2006. — 544 с.
13. Bolshakov, M. A. Comparative Analysis of Machine Learning Methods to Assess the Quality of IT Services / M. A. Bolshakov, I. A. Molodkin, S. V. Pugachev // Proceedings of the Workshop «Models and Methods for Researching Information Systems in Transport 2020» on the Basis of the Departments «Information and Computer Systems» and «Higher Mathematics» (MMRIST 2020) (St. Petersburg, Russia, 11-12 December 2020). CEUR Workshop Proceedings. 2021. Vol. 2803. Pp. 142-149. DOI: 10.24412/1613-0073-2803-142-149.
14. Integration of the MATLAB System and the Object-Oriented Programming System C# Based on the Microsoft COM Interface for Solving Computational and Graphic Tasks / S. E. Ada-durov, Y. S. Fomenko, A. D. Khomonenko, A. V. Krasnovidov // Intelligent Algorithms in Software Engineering: Proceedings of the 9th Computer Science On-line Conference (CSOC 2020), (Zlin, Czech Republic, 15 July 2020). Volume 1 / R. Silhavy (ed.). — Cham: Springer Nature, 2020. — Pp. 581-589. — (Advances in Intelligent Systems and Computing. Vol. 1224).
DOI: 10.1007/978-3-030-51965-0_51.
15. Adnan, M. Online Learning Amid the COVID-19 Pandemic: Student's Perspectives / M. Adnan, K. Anwar // Journal of Pedagogical Sociology and Psychology. 2020. Vol. 2, Is. 1. Pp. 45-51. DOI: 10.33902/JPSP.2020261309.
16. Toquero, C. M. Challenges and Opportunities for Higher Education amid the COVID-19 Pandemic: The Philippine Context // Pedagogical Research. 2020. Vol. 5, Is. 4. Art. No. em0063. 5 p. DOI: 10.29333/pr/7947.
