Opción, Año 33, No. 85 (2018): 605-631
ISSN 1012-1587 / ISSNe: 2477-9385
Organization of educational
and developmental training of school students at mathematics lessons
Rustem Gabdullin Sh.Ualikhanov Kokshetau State University, Kokshetau,
Kazakhstan rustem_GS_79@mail.ru
Kairzhan Kozhabaev Sh.Ualikhanov Kokshetau State University, Kokshetau,
Kazakhstan labdid_2008@mail.ru
Alma Kostangeldinova Sh.Ualikhanov Kokshetau State University, Kokshetau,
Kazakhstan alma_a2006@mail.ru
Abstract
The purpose of this paper is to highlight the importance of teaching
and
teaching mathematics
in the country and will seek to address the
educational weaknesses
in this field. The objectives of math education, which
include:
Educational,
cultural and
emotional goals,
will
be discussed in general. This necessity is stipulated by the innovative processes occurring in the educational
system
of the Republic of
Kazakhstan.
The results of the research show that students getting the
profession
of mathematics teachers
lack in their training
process orientation at the development of schoolchildren’s personal traits.
Keywords: educational
and developmental training, education
(upbringing), personality, personality formation, development.
Recibido: 10-01-2018 ·Aceptado: 09-03-2018
Organización de la formación educativa
y de
desarrollo de los estudiantes de la
escuela en las
lecciones de
matemáticas
Resumen
El propósito de este documento es resaltar
la importancia de la enseñanza y la enseñanza de las matemáticas en el país, y buscará abordar
las debilidades
educativas en este campo. Los objetivos de la educación
matemática incluyen: metas educativas, culturales y emocionales,
serán discutidos en general. Esta necesidad está estipulada por los procesos
innovadores
que se producen en el sistema educativo de la República de
Kazajstán. Los resultados de la investigación muestran que los estudiantes que obtienen la profesión de profesores de matemáticas carecen de orientación en su proceso de capacitación en el desarrollo de los rasgos
personales de los escolares.
Palabras clave: capacitación educativa y de desarrollo, educación
(educación del hogar), personalidad, formación de personalidad,
desarrollo.
1.
INTRODUCTION
In the contemporary
world, governments are aiming to raise their intelligence
and intelligence capabilities, increase their ability
and
capacities,
and advance their
determined
goals. In
the
past, about a hundred years ago, students enrolled as a group without knowledge and "ability" at the school, and the teacher attended the classroom with all kinds
of intelligence. For example, a seven-year-old student with an IQ of
150 followed a seven-year-old student with an IQ of 80, and he repeatedly repeated what the teacher said or wrote on the blackboard. The main task
of improving mathematical education at the present stage of the society
development is to repurpose the methodological system and focus it on
the
developmental function of instruction whose aim is to forms kills of
efficient employment
of
the educational
content. Such system
of education demands shifting of attitudes: school students, who used to be the objects of the educational
activity, should be regarded now as its
subjects.
Activity is the driving force of personality development
and education: a child can develop
and
study only in the conditions of activity.
The outstanding
psychologist L. Vygotsky considered bringing-up processes to be the inherent basis of the child’s development responsible
for the formation
of new patterns in his/her behavior. He believed that bringing-up
processes must be based on the student’s
individual activity, thus reducing the art of an educator only to the direction and regulation of
this activity (VYGOTSKY, 1991). Consequently, without organizing comprehensive activity for students, the school cannot expect any
significant progress in his/her development (self-development) and
training. Establishing a hierarchy of relations between education
(upbringing),
development and training is "the most urgent issue" of the educational and developmental system. Having determined these relations,
the
teacher will be able to let students
into the "zone of proximal
development" (Vygotsky, 1991). In the present paper the author draws on the fundamental principles of the theory of educational and developmental training offered by L.Vygotsky, A.Leontyev, L.Zankov,
P.Galperin,
D.Elkonin,
V.Davydov (Zimnyaya, 1997). The reference to the works of these authors, as well as to a number of dissertational studies on the
educational and developmental
training, speaks of the scale of this
problem, a large variety of approaches to its solution and the ever-growing
interest to educational and developmental training (Vygotsky, 1987). The analysis of publications on the problem of teachers’ training, as well as the
study of its solutions and practical implementations
in universities and secondary
schools, show that there has not yet been elaborated a methodology for preparing future mathematics teachers to give effect to
the
educational and developmental
functions of the school mathematics course.
Hence, it is very important to work out a methodology that, alongside with professional
knowledge, would provide future specialists
with the theory and methodology
of students’ development and upbringing
through mathematics teaching, and would help to work out an adequate system of didactic aids for its implementation in the conditions of the university
training practices. What can educational and
developmental mathematics training give to students? To answer this
question, it is necessary to reconsider the very notion of training, to get an insight into the problem by giving due consideration to the content of the
school course of mathematics
and
carry out a thorough critical analysis (Vygotsky,
1987). For the time being there has not been carried out yet a
special
study that would take into account modern regalia
and
development
trends in
higher
and
secondary schools
and offer a systematic approach to the creation of theoretical foundations of educative
and developmental
mathematics training
as
well as
a
technology for exercising the
educational and developmental functions of the
school
course of mathematics. Finding solutions to the above problems is the basic objective in this study.
1.1. The expedience
of the research results from
- New social demands for personality upbringing and development and for the development of the ability to actively and creatively
acquire knowledge;
- The existence of factors (educational, ideological, psychological, pedagogical, methodological) encouraging the formation of scientific and theoretical foundations of educational and developmental mathematics training;
- Psychological readiness of teachers to accept the idea of
educational and developmental training.
1.2. The relevance of the research
Mathematical science is also a requirement of human life, and
those who know numbers
and figures and law are well-known simple and logical calculations
and calculations,
their minds are ready to accept a complex series of difficulties and difficulties
in life, and in the context of
the
problems and inadequacies
of society and personal and social affairs
Preparedness
and
ability to complete as much as their mental capacity. Is stipulated by a number of contradictions, which include:
- contradictions
between the existing opportunities of educational and developmental
aspects of the content of the school course of
mathematics,
the insufficient professional
level of teachers
required for their application, and the lack of theoretical bases and practical technologies of training essential to ensure intensification
of
upbringing and development outcomes of students in the process of exercising activities connected with mathematical content;
- The necessity to organize mathematical education of
schoolchildren on the basis of developmental
technologies and the
existing knowledge-orientated training of teachers;
- The requirements of the school curricula to the level of mathematical development
of students and the real results of their mathematical development observed so far.
1.3. The aim of the research
It is to identify potential opportunities
of the school course of mathematics for the implementation
of
educational and developmental functions,
to elaborate the structure of teachers’ training
activity to ensure
the
organization of the corresponding educational process.
1.4. Hypothesis of the research
If we determine and implement educational
and developmental
opportunities of the school course of mathematics in the light of the new
educational paradigm, this will considerably
contribute to the students'
conscious learning of the program material, enhance schoolchildren’s
creative activity, promote their intellectual development, increase
motivation to study the subject and turn the student into a full-fledged subject of the learning activity. Proceeding
from the aforesaid, there were
set
the following research objectives:
1) To determine the role and place of the educational
and developmental training in the modern educational paradigm and define its methodological bases in the process
of teaching
mathematics;
2) To identify and demonstrate
the
basic trends of implementing
educational and developmental functions in the process of teaching mathematics at school;
3) To work out a theoretical model of the system of educational
and
developmental training for teaching mathematics at school;
4) To develop a system of requirements to the knowledge, skills and competences of the future mathematics teacher to promote the
development of methodological culture and proficiency;
5) To make an experimental evaluation
of
the elaborated
methodology of educational and developmental teaching.
2. METHODOLOGY
The present study is part of a broader research
whose aim is to
examine the components that influence the learning
of
teachers in
mathematics. The theoretical basis
for this study
was the "theory based on
data". For a deeper description of the experiences
and beliefs of mathematical teachers about professional learning, the "phenomenology" method was chosen because it allows them to understand the experiences
and
how to conceptualize
the
participants in the research. This research
method usually requires interviewing
people to understand why and how
they are confronted with the problem
that the researcher poses, and for accessing rich data, "interviewing"
is the strongest tool. Over a period of years we have been developing, specifying
and testing in schools the conception of educational and developmental teaching mathematics. Its
main idea consisted in recognizing the child as a full-fledged subject of the educational activity; in developing the need for knowledge; fostering
responsible attitude toward learning and discipline; ensuring
the unity of education and upbringing. In the course of the research there were conducted several surveys
and polls; the respondents
included 100
students from Sh. Ualikhanov
Kokshetau State University, 120 students from Kostanay
and North Kazakhstan Universities, as well as about 200 teachers and 560 secondary
school students of the region.100% of the
teachers participating in the survey have a university degree, i.e. they attended some courses related to methods of teaching mathematics when
they used to be university students. Mathematics teachers’ accumulated
period of work can be seen in the following diagram (Figure 1).
Figure 1.Mathematics teachers’ accumulated
period of employment
In the process of the experiment we tested our research hypothesis,
determined the key factors affecting
the content of the school course of mathematics, studied teachers’ and students’ expectations from
educational and developmental
training, analyzed the influence of the
school
mathematical
education on the formation of the student's
personality and specified the peculiarities of forming reasoning capacity at
mathematics lessons. We proceeded from the proposition that cognitive interest should be the groundwork of the educational and developmental
training: the upbringing and educational processes will never be complete and thoroughgoing if the student is indifferent to the subject and has no
motivation to get new knowledge. The main objectives of the conducted
survey were as
follows:
- To determine the state of the
educational and developmental
training in the school practice of teaching mathematics;
- To itemize the educational and developmental
content of the
learning material;
- To find methodological instruments and techniques for
implementing the model of educational and developmental
training; to give logically-psychological and pedagogical definitions of the teacher's activity connected with the
implementation of educational and developmental learning content;
- To identify the influence of the educational
and developmental content on the development of students’ cognitive interest in the course of studying mathematics,
and
to find out the opinion of teachers on the development of students’ cognitive interest.
The conducted survey with the participation of teachers, students and students showed the following:
31.7% of all the interviewed teachers understand the importance of
personality formation in the process of teaching mathematics and the need
to solve the
problem, and 60% of them
give their preference to the
classical forms of lessons. In their opinion, the main way
to promote
students’ development is to solve as many tasks as possible.42% of young
teachers
find it very difficult
to actualize the educative (upbringing) function of mathematics, 23% experience difficulties in organizing extracurricular activities
on the subject. The level of mathematical training does not satisfy 31% of teachers and 58% are not satisfied with their level
of
methodological competence. In the questionnaire section ‘notes’ the
respondents noted that they had never, or very rarely, applied educational and developmental
materials in their classroom, therefore they could not evaluate their influence on the development of their students' cognitive
interest. The question: "Do your students realize the connection
between
real life and the tasks they solve?"
was answered by the school
teachers in
the
following ways:
31% of teachers chose the answer "Yes, since they solve contextual (practice-oriented) tasks with interest ":14% out of them are teachers with more than 20 years of work experience,
5% of teachers have been in the
profession
for
10-15 years, 5% of teachers have 15-20 years of work
experience, 3% are teachers whose length of service is 5-10 years and 4%
are
recent university graduates:
45% of teachers chose the following answer:
"Only those students
who have already come across practice-oriented
tasks in their practice realize the connection";
16.3% believe that schoolchildren mostly do not see
the
connection because they have rarely dealt with the tasks having some life context;
7.7% of teachers noted that students do not see the connection because of the small number of such tasks.
According to the results of the questionnaire survey, only 61.2% of the third-year students gave the affirmative answer to the question "Will
you be able to use the content of mathematics
for educational and
developmental purposes?" In the questionnaires, school students also expressed their personal attitude to studying mathematics at school. To the question "What, in your opinion, is the role of mathematics
in the educational process? The following answers were given: 60% of the 5th-
11th graders think that mathematics is one of the
most important and
interesting subjects; 39.1% of the surveyed pupils note its role in the formation
of their life stance and moral position; 0.9% find it difficult to answer the question. The answers to the question "What motivates you to
solve tasks" showed the following
motives of students: the most frequent motive (42%) is the "interest in the decision process";
25.4% are "interested in obtaining
the result"; 19.2% indicated as their motive the "desire to overcome difficulties"; 10.3% solve tasks on account of the
"habit of meeting the requirements
of teachers and parents;3.1%
of the respondents
answered that they have some other motives not specified in
the
questionnaire.
The question "What tasks are the most interesting for
you?" was answered in the following
ways: 39% of the respondents
consider "tasks with practical content reflecting real life situations, i.e. contextual tasks"
to be the most interesting
ones; 29.3% chose "puzzle
tasks that require an original solution, i.e. creative tasks";6.3% consider
"difficult tasks" to be interesting;25.4% of the respondents think no tasks are interesting. In their comments to the survey, the students noted that classes are interesting and exciting, if in the process of explaining new material and when solving tasks, there is used some material related to
real life and life situations.
Both university and secondary school students
think that the most interesting tasks are the following:
- Tasks with explicit contradictions (tasks - problems, tasks - paradoxes);
- Optimization tasks (whose objective is to select an optimal
solution);
- Tasks for reviewing (whose objective is to find errors or check the result);
- Concretization tasks (whose objective is to make more precise the
aim
of the task, its conditions and requirements);
- Logical
tasks (whose
objectives
are to specify
cause-effect
relations, to find proofs, to define concepts, etc.).
The questionnaire survey
and interviews
helped to learn
the attitude of teachers to the organization of teaching mathematics and to the students’ active cognitive activity. They also helped to know the teachers'
opinion on the school curricula and textbooks
and their compliance with the concept of educational and developmental training.
3.
RESULTS
In the present article the author speaks of the necessity
to train a teacher of a new format: a modern teacher must be competent
enough to work in the market conditions using all the progressive experience of the
developed countries. This necessity
is stipulated by the innovative
processes occurring
in the educational system of the Republic of Kazakhstan. Modern teachers must know how to promote
students’
creativity and develop their skills of independent
acquisition of
knowledge. From
this perspective, the school must be viewed as an institution aimed at facilitating personality formation and creating
conditions for self-education and self-development, timely recognizing the individual characteristics of its students and taking them into account in
the course of education. The idea of teaching mathematics
through
educational and
developmental training
is the directing vector
of the above-mentioned changes. Having analyzed the existing school practices, we may conclude that mathematics teachers still tend to focus primarily
on
transferring
to students the knowledge of their subject. The results of
the
research show that students getting the profession of mathematics teachers lack in their training process orientation at the development of
schoolchildren’s
personal traits. The research study of the problem of
mathematics
teachers’ training exposed a need to review and re-evaluate
the
methodological component of training. The present research is aimed
at clarifying the essence of the concepts "education", "development" and
"training",
as well as at determining their correlations and influence on the formation
of the student's personality. The given research dwells on structural characteristics of the concepts "personality",
"personality
formation" and "education", defines correlations and connections between
them and determines conditions and driving forces
that contribute to
student’s personality formation in the process of studying mathematics. It also formulates a definition of educational and developmental training by
resorting to a minimal system of components
and
eliciting their significance in stimulating the formation and development
of a child’s personality in the process of
studying. The analysis of
philosophical, psychological, pedagogical and methodological
studies, as well as conversations with teachers and methodologists,
allowed us to accept the following definition of the educational and developmental training: It is training organized with due regard for the individuality and singularity of the child’s personality, ensuring comprehensive
mastery of knowledge,
forming active learning activity
and
directly affecting personal
development and growth of the child.
The teacher must remember that knowledge
affects only the
intellect and human thinking, but not consciousness
at
large. Knowledge appeals to the human brain, but not to the whole person. Even
the
mindset
of
a person is not always adequate to his/her knowledge. Therefore, in our opinion, education wherein training prevails, responds primarily to the question "What to do?", whereas the student studying
at school should receive the answer to the question "How to live?". The question
"How to live?" belongs to the domain of educational upbringing. Thus, the mission
of education
is to teach students to live and, what is more important, to
teach students not just to live, but to live with dignity, in conformity
with
the
highest social criteria. When studying at school, students should not only gain knowledge, but also find the meaning of life, fill it with a specific value system.It was found out that educational and developmental
training is driven by the following system of principles facilitating
the formation of the child's personality:
- The process of upbringing plays the leading
part in the formation of the child's personality;
- The source of education
and development lies in the student, his/her subjective experience, personal needs;
- The content of training
should be adequate to the phases of the
child's development and be conducted at such a level, by such methods and means that correspond to these phases and stimulate mental development;
- formation of the personality is carried out through practical
activity, guided by a system of personally-meaningful motives,
interests, beliefs and ideals that are important to the individual;
- Identification of "zones of proximal development"
and special
sensitive periods should be a method of development diagnostics
and
a way of expanding the scope of activity.
The analysis
of academic literature
and
the results of the
experimental work have made it possible to determine the model of the learning process essential for the implementation of the educational and developmental
functions of the school course of mathematics.
The model consists of the following
structural components: didactic conditions, actualizing the educational and developmental functions of teaching
mathematics; objectives of training, whose priority is educational and
developmental functions of the educational process; content of learning
adequate to the leading types of mathematical
activity, describing the essence of the subject of mathematics, its leading ideas and methods of cognition of reality, revealing the structure of mathematical
objects;
methods and forms
of
training that provide incentives for the motivational and cognitive spheres of students'
activity, which form the basic mental
processes and reflexive activity of students; educational tools, that complement
the
structure of the educational process and promote
exploration
and
creative activity of students. A particular attention in the study is given to the methodological literacy of the teacher
which includes presentation
of the material and explanation. It has great developmental
and
educational
opportunities that help to ensure vigorous mental activity of students. The ability of the teacher to describe an object and reasonably
explain the material is an important
indicator of his research style and
pedagogical activity. The implementation
of the above is shown in two examples below:
a) Solve the equation х - 2 +
х + 3 = 5 .
The standard solution presupposes splitting of the number
axis into
intervals with the points
х = 2, х = -3 with the respective
zeroing of the
submodular expressions, and further expanding of the modules at each of the obtained intervals. The teacher draws attention to the fact that the
module is the distance, hence the problem of the task is to find such х ,
whose distances to the points
х = -3,
х = 2 , when summed up, amount
to 5 (such х make up the interval -3£ х £ 2).
b) Find the values of the parameter a, for which the system of equations has more than two solutions.
ì7ах + 4 у = -8,
í
îх + 7ау = 49а2
The standard approach to solving the equation presupposes
the
expression of one
of the equation unknowns and
substituting it into another equation. The teacher offers to students to form a geometric image
for each equation (each
equation is a straight line, and straight lines on the
plane can be either parallel, or coincident, or intersecting);
since it is
required to
find the values of the parameter a, for which the system has more than two solutions, coincident straight lines are the most appropriate
variant. Thence we have:
ì 7а
ï у = - 4
í
х - 2,
ì 7а
ï- 4
Þ í
= - 1 ,
7а
ï у = - 1 х + 7а,
ïî 7а
ï7а =
-2,
а = - 2.
7
In the process of description there occurs not only statement
of facts, but also growth of knowledge,
its
development
and enrichment. To
describe an object the researcher resorts to such logical methods as: collation, comparison, analysis, synthesis, generalization, etc. In the
process of solving
tasks, the teacher, having asked a question, gave the
students an opportunity
to think first and only then demanded the right
answer. It is very important for the formation of thinking
that the student should be able to explain how he came to this or that solution, what
actions he made, why he gave up some of the ideas, etc. If training goes in
this way, the teacher
gets an overview of the student's thinking and his/her level of development.
It
is proved that teaching mathematics at school demands achieving a balance between the empirical and theoretical levels
of cognition; it is undesirable
to overuse excessive specifics and visual images. However, a hasty
transition to a theoretical
level without reliance
on
the empirical level leads to a restraint in the development
of children.
The
results of our experimental work also proved that school researches in the
field
of mathematics
are very effective for
the
realization of the
educational and developmental functions of teaching mathematics. Below
is given one of the variants of the control work for the eleventh form, the
completion of
which
attests of
the
students’
possession of
heuristic
techniques:
1. Solve the inequality: 8
3 х-1
3 <
х+3
4 2
2. Solve the equation: log 3 ( x + 1) + log 3 ( x + 3) = 1
3. Solve the equation: sin x + sin 2x + sin 3x = 1
4. Solve the equation:
x 2 - 3x + 5 + x 2 = 3x + 7
5. For
what
values of
the parameter
a does
the equation (а -1) х 2 + 2(2а + 1) х + (4а + 3) = 0 have only solution
satisfying
the
condition -2<x<3?
The level of complexity of the first three tasks
corresponds to most tasks
of the textbook.
The fourth and the fifth tasks are heuristic:
they test the level of development
of
students and require creative application
of knowledge, analysis of non-standard situations, independence. The results of the test work
are
shown in Table 1.As we can see in the table, although
the
results of the first three tasks in both classes are almost identical, the
ratio of excellent and good grades in the experimental
class is about 18% higher. Consequently, significant changes in the mental development of
schoolchildren of the experimental class are experimentally
confirmed through the use of the developmental method of teaching.
Table 1.
Results of the test
|
Number of
the solved tasks
|
Completed the test successfully
|
|
Experimental
class EC(27
pupils)
|
Control class CC(26
pupils)
|
|
Number of
pupils
|
%
|
Number of
pupils
|
%
|
|
5
|
2
|
7,4
|
-
|
-
|
|
4
|
5
|
18,5
|
2
|
7,6
|
|
3
|
11
|
40,7
|
11
|
42,3
|
|
2
|
7
|
25,9
|
10
|
38,4
|
|
1
|
2
|
7,4
|
3
|
11,5
|
|
0
|
-
|
-
|
-
|
-
|
The results that we achieved due to the use of our methodology are
shown in the following histograms (histograms on the left show the results
obtained before the experiment,
and
histograms
on the right show the results after the experiment).
Having completed the experiment, we came to the conclusion that teachers have accumulated a solid experience of using productive
methods of teaching mathematics; schoolchildren and university students are interested in acquiring learning skills that contribute to revealing the educational and developmental potential of the content of mathematics. However, both teachers and students need to understand the essence of the educational and developmental teaching
mathematics and broaden their knowledge of innovative technologies.
In
cooperation with students and teachers, we identified a
range of new problems that
may serve as the basis for further research:
- The formation of professional qualities
of the future mathematics teacher, ensuring the
effective implementation of the
educational and
developmental
functions of
the school course of
mathematics in the
context of the core, baseline and special competencies;
- The study of various aspects of educational
and
development
mathematics training at school in the context of globalization
and
integration into the world educational space.
A teacher, putting forth a question, should give students
an opportunity
to think it over first and only then ask for the right answer, since for the formation of thinking it is very important to get students to explain how they came to the solution
of
this task, what steps they made, why they had to give up this or that idea, etc. By organizing training in
such a way, the teacher will get an idea of the student's thinking
and the
level of his/her mental maturity. To check the level of students’ logical thinking maturity, it is useful to give students problems and exercises on
the
operations: comparison; classification; generalizations;
analogical
inference; analysis and synthesis.
4.
DISCUSSIONS
The problem of educational and developmental content, as well as
education and upbringing, has long
concerned scientists around the world. There have been conducted numerous studies
in
this sphere by Kazakhstani and foreign scientists.
There exist different points of view on this
issue.
Thus,
according to one of
them,
training is development (James, 1985, Morgan and Sørensen, 1999, Schmidt, et al., 2007): in its
supporters ‘opinion, training merges fully with development,
and
every step in learning corresponds to a
step in the development. Another point of view, proposed by the French psychologist J. Piaget and his school, denies
the connection between training and development. J. Piaget believes that
training is just an external condition
of maturation (Horn, 2005). Most
researchers adhere to the point of view offered by L. Vygotsky. According
to him, training leads to development and must go ahead of it, playing a leading role in the child's mental development ( Vygotsky, 1987). Thus, according to the fundamental
thesis of L. Vygotsky, training and development represent a unity, wherein training comes ahead of
development and stimulates it, basing itself, at the same time, on the
actual development. The idea of the developmental
training offered by
Vygotsky was later supported and developed in the works and educational practices of (Sechenov, 1947, Vaillant, 1971, Leontiev, 1983), etc. The
theory of developmental training does not have a common conceptual
foundation today and is presented in various distinctive directions. In
modern Russian pedagogy the concept of person-centered developmental
training was studied by I. Yakimanskaya
from the stand point of
psychology. According to I. Yakimanskaya, the aim of training consists in the development of the student’s individuality (Yakimanskaya, ET AL.,
2017). In the academic and pedagogical
literature of Russia and Kazakhstan, references to the researches on the problems of education, upbringing
and
development can be found in the works of I. Depman
(1946, On the Educational Significance of Mathematics), G. Ganeeva (1997, Theoretical Foundations of Developmental
Teaching Mathematics in Secondary
Schools), S. Slepkan (1987, Methodological
system for the realization of the developing function of teaching mathematics),
K. Kozhabaev (1988, Teaching mathematics at school and its role in the bringing-up process), etc. The problems of development and training were also
considered by the outstanding mathematicians (Smith, 2000, Chen,
2012, Tagunova, et al., 2016) and others.
In our study a particular importance was attached to the works of
(Smith, 2000), which offer certain solutions to the theoretical and
methodological problems of implementing developmental function of
teaching
mathematics in secondary schools
as
well
as
their
ways of
encouraging unconventional
and critical thinking. Reference to the works of the mentioned authors, as well as to a number of dissertational theses
on
educational and developmental training, indicates the inexhaustibility of this problem, diversity of approaches to its solution and the ever-
growing interest in educational and developmental training. Having
analyzed academic literature and the results of the experimental
work, we
can
conclude that any work is carried out in two directions: external, or
visible, and internal, not visible for an outsider and whose fulfillment the doers themselves are not conscious of. Observation over the course of the
student's "inner thought" will allow the teacher to learn the strong and
weak sides of the
student's thinking so
that the
teacher
might,
when
necessary,
direct the student's thinking on the right track. Therefore, in
order to teach students
to
solve mathematical problems, it is
very
important to understand the students’ reasoning when they are solving the task. How did
the
students understand the statement of the problem? What
are
they going to do? What is the sequence of their thoughts in search of
the
solution
to the problem?
The teacher can receive
this information asking such questions as "Why?", "On what grounds?",etc. Mathematics
training material can mostly conform to the objectives of the educational and developmental training when it gets in line with the following principles:
-school course of mathematics
must be worked out on the basis of
the
contemporary science, pedagogically
adapted to the age of the child and the objectives of the instruction;
- Theoretical material must, whenever possible, accord with its state and interpretation
in science, and be presented to students within a
certain didactic system;
- Theoretical
material must be presented
as an object for finding a
solution to the problem, actualizing the new knowledge;
- Presentation of the learning material should focus not on
memorization, but on understanding;
presentation of the learning
material should base on the cross-cutting
(key) issues of
mathematics;
- Introduction of the theoretical material and its studying must be
motivated and justified;
- Humanization and humanitarization must be prioritized
in the content of training; the applied and practical orientation of the school course of mathematics should be enhanced;
- Interdisciplinary and interdisciplinary connections must be
realized; worldview (educational) orientation of the knowledge obtained in school should be increased;
- The content of the learning
material must be adequate to the development of the students' logic thinking; the level of complexity
of
the learning material should be taken into account
in order to
require minimal assistance from the teachers;
- The interest of pupils to mathematics should be encouraged
and developed;
- The content of the learning material
should be enriched by using additional material that contributes to the upbringing
and
development of students;
- The presented material must be given in the form of a harmonious
didactic system with delimited levels of complexity;
- Training tasks should be widely used.
The process of activating students’
learning and cognitive activity
can
be presented as two stages: 1) the awakening and development
of interest as a means of motivating to activity; 2) directing this awakened activity of students.
5.
CONCLUSION
Since the early
eighties, in response to many protests that ignore
teacher as a job and not a profession, new approaches to teachers were
introduced as
"professional interns." One
of the most
significant and influential events of this decade was the use of the "teacher-researcher" approach, rooted
in
critical theories, and
in particular, the "action
research" or "action
research" view. The great goal of mathematical
education is to strengthen the power of thought. Students need to learn knowledge and skills in mathematics, science and reading, in order to
deal with the challenges of life
so
that they can solve complex life problems by processing information from the world around them. Taking into account special characteristics of mathematics as a science and a discipline,
proceeding
from the psychological
and pedagogical approaches to the creation of pedagogical systems and methodology of the system approach, basing
on the
outcomes achieved in
the course of
the research,
we obtained the following results:
A theory of educational and developmental
teaching mathematics was worked out, and there was also created a model of using the content
of the school course of mathematics
to promote upbringing and
development of
the student’s personality. The theoretical and
experimental research proved that when the conditions formulated in the hypothesis are fulfilled, the quality of future teachers’ preparedness
for
implementing
educational and developmental training at mathematics
lessons are considerably improved. Only specially organized training for
students will help them to acquire efficient learning skills essential for the organization of educational and developmental training by means of the
school course of mathematics.
It has been proved that teaching mathematics
will be educational
and
developmental only if there is interrelatedness of three constituent principles of the educational process:
a) Change of the student’s status: transition from being the object
of the learning activity to the position
of the subject;
b) Training is conducted in the zone of proximal development;
c) Training promotes upbringing and development of mathematical abilities.
Findings
of the research showed that
teaching by
providing
visualization-based activities
was significantly influenced by students'
perceptions of the concept of marginalizing and the difference between
the
two groups in this regard. But presentation of visualization activities in the teaching of the limit does not affect the students' spatial ability. In this
regard, the difference between the two groups is not meaningful.
REFERENCES
CHEN, Jason A, 2012. "Implicit theories, epistemic beliefs, and science motivation: A person-centered approach". Learning and Individual Differences, Vol. 22, No.6: 724-735.
HORN, Ilana Seidel, 2005. "Learning
on
the job: A situated account of teacher learning in high school mathematics departments".
Cognition and Instruction, Vol. 23, No.2: 207-236.
JAMES, William, 1985. The varieties of religious experience, Harvard
University Press,
LEONTIEV, AN, 1983. "Selected psychological works". Education, Moscow, USSR, Vol. 43, No.1: 52-87.
MORGAN,
Stephen L and Sørensen, Aage B, 1999.
"Parental networks, social closure, and mathematics learning:
A test of Coleman's social capital explanation
of school effects". American
Sociological Review, Vol. 5, No.7: 661-681.
SCHMIDT, William
H, Tatto, Maria Teresa, Bankov, Kiril, Blömeke, Sigrid, Cedillo, Tenoch, Cogan, Leland,
Han, Shin Il, Houang,
Richard, Hsieh, Feng Jui and Paine, L, 2007. "The preparation gap:
Teacher education for middle school
mathematics in six countries". MT21 Report. East Lansing: Michigan State University, Vol.
32, No.12: 53-85.
SECHENOV, IM, 1947. "Selected philosophical and psychological works". Gospolitizdat, Moscow, Vol. 3,
No.4: 442-520.
SMITH, Roger O, 2000. "Measuring
assistive technology
outcomes in education".
Diagnostique, Vol. 25, No.4: 273-290.
TAGUNOVA, Irina A, Selivanova, Nataliya L and Valeeva, Roza A,
2016. "The category of upbringing in Russian and western studies".
Mathematics Education, Vol. 11, No.1: 3-11.
VAILLANT, George E, 1971. "Theoretical hierarchy of
adaptive ego mechanisms:
A 30-year follow-up of 30 men selected for
psychological health". Archives of general psychiatry,
Vol.
24, No.2: 107-118.
VYGOTSKY, Lev Semenovich, 1987. The collected works of LS Vygotsky:
Volume 1: Problems of general psychology, including the volume Thinking
and
Speech, Springer Science &
Business Media, Berlin (Germany).
VYGOTSKY, Lev Semynovitch, 1991. "Pedagogical psychology".
M.
Pedagogics, Vol. 23, No.3: 34-76.
VYGOTSKY, LS, 1987. "Thinking and speech. The collected
works of
LS Vygotsky, vol. 1". New York: Plenum Press, Vol. 114, No.3:
113-114.
YAKIMANSKAYA, Irina Sergeyevna, Molokostova, Anna Mikhaylovna and Ibragimova, Milyausha Yakubovna, 2017. Metacognition
of Organization Members as the Basis of Learning Strategy in
Higher School. Metacognition
and
Successful Learning
Strategies in Higher Education, IGI Global, Hershey (USA).
ZIMNYAYA, IA, 1997. "Pedagogical psychology". Rostov-on-don:
Phoenix, Vol. 52, No.3: 224-253.