=Paper=
{{Paper
|id=Vol-2393/paper_209
|storemode=property
|title=Computer Simulation as a Method of Learning Research in Computational Mathematics
|pdfUrl=https://ceur-ws.org/Vol-2393/paper_209.pdf
|volume=Vol-2393
|authors=Lyudmila Bilousova,Oleksandr Kolgatin,Larisa Kolgatina
|dblpUrl=https://dblp.org/rec/conf/icteri/BilousovaKK19
}}
==Computer Simulation as a Method of Learning Research in Computational Mathematics==
https://ceur-ws.org/Vol-2393/paper_209.pdf
Computer Simulation as a Method of Learning Research
in Computational Mathematics
Lyudmila Bilousova[0000-0002-2364-1885], Oleksandr Kolgatin[0000-0001-8423-2359]
and Larisa Kolgatina[0000-0003-2650-8921]
H. S. Skovoroda Kharkiv National Pedagogical University,
Alchevskyh Str., 29, Kharkiv, Ukraine
lib215@ukr.net
Abstract. The paper deals with a problem of students’ independent work through
learning research. By solving research problems students gain experiences in a
way closest to the real scientific study. A student does not receive the “ready”
knowledge and decisions, but a teacher puts for him the goal, the cognitive
problem that this student should solve during independent research activity. We
suggest appropriate computer models in Mathcad environment (dynamic support
synopsis) to provide such learning research in the “Methods of Computing”
course. Students research some problems of global polynomial interpolation,
spline interpolation, choose the appropriate function for approximation, verify
the efficiency (accuracy and convergence rate) of some methods of numerical
calculation of integrals and solving of non-linear equations. The results of
approbation of this learning methods in the training process of H. S. Skovoroda
Kharkiv National Pedagogical University are shown and discussed. The
efficiency of our approach was checked with use of such criteria: completeness
of knowledge and depth of knowledge. Statistical analysis of obtained data shows
advisability of implementation of learning research in training process.
Keywords: Computer Simulation, Students’ Independent Work, Computational
Mathematics.
1 Introduction
The existence of the wide software spectrum should be reflected in the technique of the
numerical methods learning by future experts in applied fields of science and
technology. The main questions of teaching are the following: interpretation of the
whole picture of possible approaches to numerical problem solving; realization of the
idea and peculiarities of each method; conditions of the method application; advantages
and disadvantages in comparison with other methods; determination of problems
classes, for which the method can be the most efficient. We should mention that
theoretical analysis of possibilities of using some methods becomes very complicated
in practice in many cases, so the investigator’s intuition is considered to be very
important in process of the proper method choice. All these facts can explain the
necessity of carrying out the special teaching methods, which will give the future
specialists the opportunity to gain their own experience in the use of different methods
and to make proper conclusions.
The realization of these methods is a very difficult task and is determined by the role
of computers in the process of teaching. According to the tradition, computer is
considered as a calculating device in the process of learning of numerical methods, so
practical work in this course is transformed to programming where certain algorithms
of methods are translated into one or another language. The availability of software for
supporting professional mathematical activity directs us to transfer the practical work
to the higher level by the use of computers as a means of scientific research. In this
case, we can avoid the necessity of programming, but the process of calculation is
hiding from a user and it can reduce the laboratory work to mindless copying the
numbers from the computer screen. It is possible to give the laboratory work real
educational and research character only by means of the problem situation that can fill
in the whole work with purport by putting its final aims and determining the leading
questions. We have prepared our guidebook for students – future teachers to put such
problem situation and show the steps of problem solving [1].
The fundamentals of numerical methods have been published in scientific and
education books of well-known authors [2], [3], [4], [5], [6], [7], [8], [9] etc. Modern
applications of numerical methods are undoubtedly connected with using computers.
So computer-based environment became a significant element in teaching numerical
methods. There are enough scientific works in the field of mathematics education that
use a computer-based environment to show the calculation procedure at using
numerical methods, for example, the authors [10] use MS Excel “... to perform
numerical integration, specifically trapezoidal rule and Simson’s rule ...” and the
procedure of generation of Lagrange’s interpolation polynomial [10]. The other
direction of studies in the field of education in numerical methods is to use special
teaching software [11]. Graphical visualization is another interesting approach to this
difficult educational problem. Thus, the authors [12] take attention for visualization of
complicated iterative and recursive process for solving non-linear equations and use
GeoGebra software that “... makes possible to deal with these methods by means of
their geometrical interpretation and to visualize their behavior and procedure” [12].
Last time, great attention is paid to particularities of using the numerical methods in
some non-standard conditions [1], [9]. Therefore, teaching methods, which oriented on
using computer-based environment for learning research of these particularities, for
deeper mastering in computational mathematics, need in further development.
The goal of this work is to verify the methods of learning research organizing with
suggested computer-based models for numerical methods investigation in Mathcad
environment (dynamic support synopsis).
2 Curricula and objectives
The purpose of teaching the discipline “Methods of Computing” at Physics and
Mathematics department of the H. S. Skovoroda Kharkiv National Pedagogical
University is to familiarize students with the general characteristics of the basic
methods of numerical solution of applied tasks based on the construction of
mathematical models; to form students skills in mathematical-oriented computer
environments and the ability to conduct computational experiments in these
environments; to equip students with research skills and analysis skills; to provide
students with the skills of competent use of numerical methods in practice.
The main objectives of studying the discipline “Methods of computing” are the such:
to reveal the place and significance of knowledge on numerical methods in general
and vocational education; to show psychological and pedagogical aspects of
mastering the subject; to establish relationship of the discipline “Methods of
computing” with other educational subjects; to show the practical significance of
numerical methods, mathematical modeling, computational experiment, their
applicability to solving of humanitarian, technical and scientific problems; to present
examples of realization of great opportunities that open the efficient use of
computers;
to provide future teachers with grounded study of the concepts and methods of
computing that can be used by them in the process of teaching some topics of the
school courses of mathematics and computer science, familiarizing pupils with
elements of numerical methods, conducting optional classes in secondary schools;
to form students with sufficient knowledge, skills and abilities, necessary for the
practical conduct of educational work in the secondary schools with the wide use of
modern information technologies.
According to the requirements of the educational-professional program, students must
know the main stages of solving the problems on a computer, the basic numerical
methods for solving mathematical problems and the peculiarities of its computer-based
implementation. Students should be able substantiate the choice of a numerical method
for solving a mathematical problem, realize the method algorithm using the
programming languages or some specialized computer-based environment (such as
Excel, Mathсad, etc.), perform the necessary calculations and analyze the obtained
results.
According to curricula, students study such chapters of numerical methods:
Elements of the Error Theory, Functions Approximation (Polynomial Interpolation,
Spline Interpolation, Least Squares Method), Numerical Integration, Solution of One-
Variable Equations, Solution of Systems of Linear Equation, First-Order Differential
Equation and its Systems. In this paper we focus on approximation and integration
problems.
3 Theoretical background
There are two main tasks at the statement of the approximation problem:
Choosing of approximation function;
Choosing the criterion of proximity of the function.
Different kinds of approximation may emerge depending on this choice. One case of
approximation problem is the polynomial interpolation problem, where it is required
that interpolating polynomial should take the same values as the interpolated function
at the given points x0, x1, ..., xn of the interval [A; B]. A system of linear equations should
be solved to obtain the coefficients of polynomial and some useful formulas are well
known (Lagrange’s and Newton’s polynomial interpolation formulas). Students study
these formulas and should use it in practice, this part of educational material is
reproductive. The interpolation polynomial of degree not above n = N–1 always can be
construct, if N interpolation nodes are given. But problem is in error estimation of
approximation the given function by polynomial. To solve this problem a student
should work on high levels of cognitive activity.
We may get the error estimation of the function approximation by interpolation
polynomial:
| | − || − |…| − |, where ( )( ).
( )| ≤ = max
( + 1)! ∈[ ; ]
This formula for polynomial interpolation error estimation can be simplified for
equidistant nodes:
| ( )| ≤ ℎ | ( − 1) … ( − )|.
( + 1)!
Thus, | ( )| = 0(ℎ ), that is interpolation error should decrease as ℎ with
decreasing of the interpolation step. But many factors affect at this error:
the form of the function f(x), in particular, the value of its derivative of the (n+1)
order;
the interpolation nodes arrangement on the interval [A; B] and the size of the interval.
The factor of the function form is very essential, as an example, we can remember the
classical Runge’s function, which have growth of derivatives with its order so high that
the interpolation of this function with polynomial of high degree leads to unacceptable
error growth. It is not easy to analyze the function derivative on all interval [A; B] in
many practical cases. In addition, the total error of calculation of the function value
according to an interpolation formula doesn’t only consist in interpolation error
(method error) but includes irremovable error, which is connected with inaccuracy of
ascertainment of the function values in nodes and calculation error. So we should
provide the student with some competence, may be intuition, for an expedient choice
between global interpolation, piecewise interpolation and spline interpolation as well
as choice of nodes arrangement. Such competence can be formed only by own
experience in study of interpolation of different functions in different conditions, that
is, practical solving problems or learning research.
A similar situation takes place with the numerical integration, because its error also
depends on the form of the function at the interval [A; B]. We should provide our
students with competences to make an expedient choice between global or piecewise
quadrature formulas, to recommend the order of the quadrature formula, to use Newton-
Cotes or Gaussian formulas, to estimate the real accuracy of the integration.
4 Methods of students’ learning research management
The laboratory works on the discipline “Methods of Computing” is constructed as a
series of learning research of gradually increasing complexity. Students do not obtain
ready-made knowledge, ready decisions, but the problem is posed that becomes a basis
of students’ independent research activity. Gradual complication of the content of the
cognitive tasks, which are proposed for experimental study, is an important element in
organizing such laboratory practice. The possibility of using the research method in the
laboratory practice is conditioned by the advent and improvement of professional-
oriented environments that provide the students with comfortable tools for the
computational experiments. As a basis for the learning research in numerical methods
we have taken the Mathcad environment, which is widely used to solve applied
mathematics problems, and, at the same time, has a number of certain attractive
qualities that makes it convenient for use in learning: intuitive interface, developed
system menu and help, ability to input textual, symbolic and graphic information in any
place of the workspace, fairly flexible embedded programming language, etc. [13].
To provide students’ learning activity we have designed some models that realize
certain algorithms of numerical methods in Mathcad environments. The interface of
these models is designed as a set of special electronic pages that play the role of
dynamic support synopsis (DSS) that is convenient for entering the input data of the
task and structured displaying the result according to algorithm. Thus, the student
actually receives a virtual laboratory for a computational experiment.
Management of the learning research was carried out according to the following
scheme [13]:
setting objectives;
providing observation and accumulation of experimental data;
doing evaluation and comparative analysis of the obtained data;
predicting characteristics of the investigated method;
formulating hypothesizes;
studying the features of the method being studied;
analyzing the received data;
correcting the hypothesis;
summarizing the results of work and formulating conclusions.
We have developed didactical support for the laboratory works in the form of plans-
reports to provide meaningful and targeted student’s activities and to ensure the
achievement of the predicted learning effect. Plans-reports are built according to a
single scheme and consist of two parts – informative and instructional. The informative
part contains the subject of the work, its purpose, description of the software,
characteristic of entered and displayed numeric and graphic data. The instructional part
contains the order of execution of the work, where its key points are marked and fixed.
To motivate the student to a study, he is initially offered a chain of appropriately
selected questions. Then the work is carried out according to the proposed plan, which
defines the stages of the study, tasks that are solved at each stage, experimental material
to be obtained, the form of its submission, etc. As students gain experience, the
instructions to them become less detailed. Some experiments the student must plan, put
and implement on their own. Individual variants of sets of tasks are suggested to each
student to perform the laboratory work. These tasks are selected to show features of
every algorithm that are studied and correspond to the purpose of the work [13]. The
results of the work are proposed to be done in the form of conclusions, outlines of which
with a greater or lesser detailing are fixed in the plan-report. It helps the student to
record the results of work, to structure them, to pay his attention to those moments of
research that can remain unnoticed.
5 Learning research in polynomial interpolation
We suggest students to investigate the problems of polynomial interpolation during
three laboratory works:
Lagrange’s interpolation formula – the students study the influence of interpolation
polynomial degree on the accuracy of global interpolation for different functions at
the given interval [A; B]. Each student works with the individual set of functions,
including functions with great growth of its derivative of high orders, polynomials,
even and odd functions etc. An example of such student’s investigation are shown
in Table 1. Part 2 of this laboratory work is devoted to influence of the interpolation
nodes localization on the interpolation accuracy and using Chebyshev’s polynomial
roots as nodes of interpolation;
Newton’s interpolation formula – the students are introduced with technics of
building the polynomial across the table of function values as well as the methods of
accuracy control and its limitation in “bad” situations;
Piecewise interpolation, splines – the students operate with the various kind of
interpolation models and get an experience in choice the appropriate method in
different cases.
The last laboratory work is integrative. So let analyze it more detail. The aim of the
work is to investigate interpolation of the function using piecewise interpolation
polynomial and cubic spline. The software (DSS) contains models of spline, global and
piecewise polynomial interpolation and gives the students possibilities to choose
parameters of this model easy (Fig. 1, 2). Students entered the input data: A, B are the
ends of the interpolation interval; f(x) is the interpolated function.
At the beginning of the work students discuss some leading questions to actualize
theoretical knowledge and formulate problems for experimental study:
Are there cases, when increasing of the global interpolation polynomial degree leads
to increasing, but not decreasing of the interpolation error?
What conditions does the global interpolation polynomial become useless for
function approximation under?
What factors limit the interpolation polynomial degree in practice?
How does the number of interpolation nodes, which have been taken into account,
influence on the computational complexity with global, piecewise polynomials and
spline interpolation?
How does the number of interpolation nodes, which have been taken into account,
influence on the volume of data, which must be kept for calculation of the value of
global or piecewise interpolation polynomials and spline?
What are the differences between the function approximation by piecewise
interpolation with cubic polynomials, global polynomial interpolation and cubic
spline interpolation? Consider cases, when there are 3m+1 (where m = 1, 2, ...) nodes
on the interpolation interval.
Can one forecast such situations, when it is necessary to prefer the global
interpolation polynomial, piecewise interpolation polynomials or spline?
Table 1. Student’s investigation of influence of the global polynomial degree n on interpolation
accuracy
Error of interpolation in cases:
1
n 1⁄ 2 − +3 − +1 sin cos
[0.5; 1.5] [0.5; 1.5] [–1; 1] 1 + 25 [–2; 2] [–2; 2]
[–1; 1]
1 0.35 1.4 5.1 0.96 0.39 1.5
2 0.024 0.48 0.87 0.64 0.39 0.14
3 1.6.10-3 0.18 0.40 0.71 3.5.10-2 0.11
4 8.7.10-5 0.065 1.0.10-15 0.44 2.5.10-2 7.4.10-3
5 4.4.10-6 0.025 3.6.10-15 0.30 1.4.10-3 5.3.10-3
6 2.0.10-7 9.2.10-3 1.8.10-15 0.61 9.7.10-4 2.2.10-4
7 7.9.10-9 3.6.10-3 3.6.10-15 0.25 3.4.10-5 1.6.10-4
8 2.9.10-10 1.4.10-3 4.5.10-15 1.0 2.4.10-5 4.5.10-6
9 9.4.10-12 5.1.10-4 6.2.10-15 0.28 5.4.10-7 3.2.10-6
10 2.8.10-13 2.0.10-4 8.9.10-15 1.5 3.8.10-7 6.2.10-8
According to recommendations each student investigate global and piecewise
polynomial interpolation and spline interpolation for given individual set of function.
Students construct the global interpolation polynomial, piecewise interpolation
polynomials of different degree and spline, using a fixed set of interpolation nodes N
for the given functions. They find the error of each function approximation.
Then the students use the necessary number of interpolation nodes to provide given
accuracy (10-4) of function approximation with spline, global and piecewise
interpolation polynomial of the degree n = 2, 3, 4, 5. The example of such investigation
is shown in the Table 2. Students also investigate dependence of approximation
accuracy at a fixed point from the number of considered nodes from a fixed set of nodes
for spline interpolation.
Fig. 1. Interface of the spline interpolation model
Fig. 2. Interface of the piecewise and global polynomial interpolation model
Table 2. Student’s investigation of the number of nodes, which provide the given accuracy
of interpolation
Approximation polynomial Number of
Number of Interpolation
considered
type degree subintervals error
nodes
( )= , interpolation interval [0.5; 4.5]
Global interpolation
9 1 10 5.5.10-5
polynomial
2 76 153 9.9.10-5
Piecewise
3 19 58 8.4.10-5
interpolation polynomial
4 8 33 6.7.10-5
Spline 3 28 29 9.6.10-5
1
( )= , interpolation interval [−1; 1]
1 + 25
Global interpolation Necessary accuracy was not reached; interpolation error
polynomial increases with increasing the interpolation polynomial degree
2 71 143 1.0.10-4
Piecewise 3 32 97 8.7.10-5
interpolation polynomial 4 17 69 5.0.10-5
5 14 71 6.0.10-5
Spline 3 27 28 9.3.10-5
( ) = sin , interpolation interval [−3; 3]
Global interpolation Necessary degree of interpolation polynomial exceeds 20,
polynomial calculation was stopped because the time is beyond
2 140 281 1.0.10-4
Piecewise 3 53 160 9.6.10-5
interpolation polynomial 4 26 105 9.4.10-5
5 17 86 9.6.10-5
Spline 3 81 82 9.9.10-5
As a combined result of the investigations, students formulate the conclusions.
6 Learning research in numerical integration
Numerical integration is built on the polynomial interpolation of the function. So, all
competences that students obtained, when they studied the polynomial interpolation,
are the basic and should be actualize. The aim of the students’ learning research is to
investigate effectiveness of Gauss quadrature formulas in comparison with Newton-
Cotes formulas. At the first, students investigate the application of composite Newton-
Cotes quadrature formulas and formulate conclusions about influence of the integration
step and the degree of the interpolation polynomial that is used for quadrature formula
on the integration error. Also, they use Runge’s rule for integration error estimation and
investigate the limits of its applicability. Then students make a comparative analysis of
composite formulas of not high order of accuracy (trapezoid formula and Gauss formula
with one node) and composite or single formulas of high order of accuracy (Newton-
Cotes formula using interpolation polynomial of 8-th order and Gauss formula with 8
nodes). Leading questions on this second stage of the work are the such:
Will the choice of integration nodes influence on the error of integral calculation?
What is a level?
How can we find out nodes location at integration by Gauss formulas?
Will the error be sensible to slight error of node position from optimal?
When is Gauss formula required to be used? Are there any occasions, when it is not
possible to use it?
What occasions need to use composite Gauss formulas?
Appropriate DSS (dynamic support synopsis) are designed on the base of models of
numerical integrating with Gauss and Newton-Cotes quadrature formulas. One page of
DSS (the last) is shown in Fig. 3.
Fig. 3. Interface of the model for numerical integration
Results of computational experiments are collected in tables, forms of which are
suggested to the students in the corseware. One of such table with student’s results are
shown (Table 3).
The final students’ conclusions are connected with such numerical integration
features:
choice of the quadrature formula at limited number of nodes of integration;
difference between Gauss formula and Newton-Cotes formula by construction;
comparison of accuracy of Gauss and Newton-Cotes quadrature formulas;
sensitiveness of integration error by Gauss formulas to accuracy of determining the
nodes;
functions, for which high order quadrature formulas are recommended;
choice of the quadrature formula for numerical integrating of given function at given
conditions;
situations, where the error of applied numerical integration formula turned out to be
comparable with calculation error.
Table 3. Student’s investigation of the accuracy of various methods of numerical integration
Error of integral calculation
. .
Quadrature N m
= 1.6
formula . . 1 + 25
≈ 10.53377 ≈ 0.549360
IN1 9 8 0.055 0.079 0.0075
IG1 8 8 0.027 0.038 0.0078
IN8 9 1 5.5.10-9 0.0022 0.25
IG8 8 1 8.9.10-15 7.2.10-6 0.041
IN1 17 16 0.014 0.020 1.3.10-4
IG1 16 16 0.0069 0.010 4.2.10-5
IN8 17 2 6.0.10-12 5.8.10-5 0.001
IG8 16 2 7.1.10-15 2.2.10-8 3.0.10-5
7 Results of the pedagogical experiment and discussion
In order to confirm the idea of using learning research with DSS in the study of
numerical methods, a pedagogical experiment was carried out on the basis of
H. S. Skovoroda Kharkiv National Pedagogical University in the process of teaching
the discipline “Methods of Computing”. The sample population was created by students
of the III-IV year of the Physics and Mathematics Department. As a result of this
experiment, there was fixed in 1999 that educational discipline “Methods of
Computing” became more significant, more informative and interesting, it became
possible to expand the set of methods that are considered. Students’ activity and
motivation increased, it was increased completeness of knowledge, depth of
knowledge, research competency. Statistical analysis of obtained data showed
advisability of implementation of learning research in training process [14].
To obtain new data we design the test of 57 items for yes/no answers for checking
the completeness of knowledge in function approximation. The problems of this test
items were reproductive, so we estimate the student’s completeness of knowledge as
the ratio of the correct answers number to the number of item in the test. All students
(11 future teachers of informatics) show the completeness of knowledge above 60%.
But some test items where correctly answered by all students, these items cannot be
used for pedagogical measurements. We also extract the items with not satisfactory
discrimination and obtain the test of 18 items for measurement of learning
achievements in the approximation (Table 4). According to the results, its reliability
was estimated with Cronbach alpha as α=0.91. To check the depth of knowledge, the
students were suggested with two problems for full answer: 1) to characterize the
influence of global interpolation polynomial order on the interpolation error; 2) to
characterize the conditions of using the global polynomial interpolation, piecewise
polynomial interpolation, spline interpolation and least square approximation. Depth of
knowledge was estimated as a part of essential conditions of expedient application of
interpolation formulas that the student shows in the answer. The results are shown in
Fig. 4.
Table 4. Test Items Specification
Correct Index of Correlation with
Test Item
answer difficulty the test score
Newton interpolation formula is more convenient Yes 0,91 0,28
for calculating the values of a function given by a
table
Spline gives a piecewise-continuous polynomial of No 0,91 0,21
the second degree
Spline is a cubic polynomial between each pore of Yes 0,91 0,21
neighboring nodes
The replacement of the function by an interpolation Yes 0,82 0,37
polynomial is used for numerical differentiation
The approximating function for the least squares No 0,82 0,66
approximation is chosen so that the sum of the
deviations of the function from the given points
should be less than the specified EPS
The approximating function for the least squares Yes 0,82 0,52
approximation is chosen so that the sum of the
squares of the deviations of the function from the
given points is minimal
In interpolation nodes, the value of the interpolation No 0,73 0,75
polynomial and the value of the interpolated
function differ in absolute value no more than a
given EPS
Newton interpolation formula provides less No 0,73 0,75
interpolation error than the Lagrange formula
The approximating function for the least squares No 0,73 0,71
approximation is chosen so that it passes through all
given points
The approximating function for the least squares No 0,73 0,42
approximation is chosen so that the largest
deviation of the function from the given points is
less than the given EPS
Correct Index of Correlation with
Test Item
answer difficulty the test score
In the case of approximation by the method of least No 0,73 0,58
squares, the number of given experimental values
should be equal to the number of parameters of the
approximation function
Interpolation formulas of Lagrange and Newton No 0,64 0,67
define different interpolation polynomials
Spline uses global interpolation No 0,64 0,59
In the case of approximation by the least squares Yes 0,64 0,79
method, the more values are given, the better the
approximating function describes the experimental
dependence
In the interpolation nodes, the values of the Yes 0,55 0,92
polynomial must coincide with the values of the
function
The formulas of Lagrange and Newton give the Yes 0,36 0,86
same polynomial
The mechanical analog of spline is a flexible rod Yes 0,36 0,86
Coincidence of the values of the source and the Yes 0,27 0,73
approximating functions at some points is not
obligatory in the case of approximation of the
function
50%
45%
40%
35%
30%
25%
20%
15%
10%
5%
0%
High (>70%) Average Low (<30%)
Fig. 4. Percent of students with high, average and low depth of knowledge as a result of testing
Based on many years of experience in the implementation of learning research using a
professional computer-based environments, pedagogical conditions of the effectiveness
of such method of teaching are formulated: student’s formation of the necessary
knowledge and skills at the reproductive level; the formation of basic research skills;
possession of modern ICT tools for research in a particular subject area; presence of
motivation, emotional and volitional attitude to such activity; existence of a multilevel,
individualized system of management of students’ independent work.
8 Conclusions
Method of learning research is used in special laboratory practice with DSS (dynamic
support synopsis – some models of numerical methods) for studying numerical methods
of polynomial interpolation and numerical integration.
Pedagogical observations show that suggested methods work towards forming
students’ motivation and research competency. Students’ completeness and depth of
knowledge are on enough level according to results of testing and observations during
the practical work.
Positive influence of suggested methods on educational results is possible only with
developed methodology of investigations, appropriate didactic materials,
individualized management of students’ independent work and enough level of
students’ readiness to research activity: satisfactory knowledge and skills at the
reproductive level; presence of basic research skills; ability to use modern ICT tools for
research in a particular subject area; presence of motivation, emotional and volitional
attitude to research activity.
We see prospects for further work in designing the methods of detailed diagnostic
of the readiness of the student to learning research and approbation of new methods of
individualized management of students’ independent work.
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