Formation of research skills of students in solving non-standard problems in mathematics in basic school



This scientific article is aimed at considering the problem of developing students' research skills in solving non-standard problems in mathematics in primary school. The aim of the work is to identify the conditions for the formation of the research personality of a schoolchild in the process of using non-standard tasks in mathematics in basic school. The paper provides an analysis of the text tasks of many modern methodologists and psychologists. The results obtained will be of interest to teaching staff of schools, with the aim of rational planning of mathematics lessons in the main school.

Keywords: research skills, non-standard tasks.

The scientific article is devoted to the problems of the formation of research skills in primary school, when solving non-standard problems. The relevance of the topic is that modern society sets the bar high for today's high school students.

Most of the specialists whose activities are aimed at modernizing the problem material of school mathematics presented in modern textbooks, as a rule, represent an algorithmic way of solving problems, thus reducing the operational and information field of students' activities.

The tasks of the main course of mathematics can be divided into two types: standard (closed) and non-standard (open).

The main part of the problem book program is occupied by standard tasks. The solution of such tasks requires the student to be able to work «according to the existing model», i.e. to solve such problems, there is an algorithm, following which, you can get the right solution. Difficulties that arise in solving standard problems can be overcome due to the technical nature, that is, to train the ability to solve the same type of exercises.

The second type of tasks is non-standard, such tasks, as a rule, are tasks the solution of which is unknown to the student, or solutions to such problems are not considered in this mathematics course, which is the consequence of the algorithm for solving such tasks is not familiar to the student. Non-standard tasks include tasks in which a tense situation is created for the student, requiring ingenuity, criticality and flexibility of thinking, distribution of attention, and development of new methods of action for its resolution.

The concept of «non-standard task» is used by many methodologists. Yu. M. Kolyagin reveals this concept as follows: «A non-standard task is understood as a task, upon presentation of which students do not know in advance either the method of solving it or what educational material the solution is based on» [10, p. 36].

We present the classification of mathematical and non-standard problems, thereby trying to generalize the material on the problem under study. Classification of mathematical problems according to B. A. Kordemsky [3, p. 7]:

• Tasks related to the school mathematics course, but of increased difficulty — such as tasks of mathematical Olympiads. They are intended mainly for schoolchildren with a definite interest in mathematics; thematically, these tasks are usually associated with one or another specific section of the school curriculum. The exercises related to this deepen the educational material, supplement and generalize the individual provisions of the school course, expand the mathematical horizons, and develop skills in solving difficult problems.

• Tasks like mathematical entertainment. They are not directly related to the school curriculum and, as a rule, do not require much mathematical preparation. This does not mean, however, that the second category of tasks includes only easy exercises. Here there are problems with a very difficult solution and such problems, the solution of which has not yet been obtained. «Non-standard tasks, presented in a fun way, bring an emotional moment to mental activities. Not connected with the need to apply memorized rules and techniques to solve them every time, they require the mobilization of all accumulated knowledge, teach them to search for original, non-standard ways of solving, enrich the art of solving with beautiful examples, make them admire the power of the mind» [3, p. 7].

Now, consider the classification of mathematical problems according to V. A. Dalinger [2]:

– By the number of unknowns in the problem structure.

– By the nature of the task objects.

– In relation to theory.

– In relation to theory.

– By mathematical content.

– By correlating tasks with each component of educational and cognitive activity.

– By the predominance of one or another type of thinking in the process of solving problems.

– Tasks of the «object» type; tasks of the «procedure» type.

– By types and types of tasks.

– By the nature of the requirements.

From the above classifications, we can conclude that there are no generally accepted classifications of non-standard tasks.

The basis for solving non-standard tasks is that the types of such tasks are a mathematical problem for which there is no clear algorithm. However, the solution of non-standard problems is based on the students' knowledge of basic mathematical concepts and facts, algorithms and laws, which is one of the indicators of their mathematical training.

It should be noted that when solving these problems, students show curiosity, and also stimulate cognitive interest through research detail.

The next concept that we will consider will be «research activities of students».

The research activity of students is aimed at developing research abilities through the use of certain forms and methods of work. [5, p. fourteen].

The concept of «ability» B. M. Teplov considers from three sides. [5, p. 101]:

1) abilities are individual psychological characteristics that distinguish one person from another;

2) abilities are not any individual characteristics, but only those that are related to the success of the implementation of a particular activity;

3) abilities are not the knowledge, skills or abilities that a particular person has.

We can also draw a relationship between knowledge, abilities and skills. Knowledge, skills and abilities are acquired faster than abilities, however, it is abilities that facilitate the assimilation of knowledge and skills. In the future, the formed knowledge and skills contribute to the development of abilities. Also, like non-standard tasks, abilities do not have a generally accepted classification.

Now, consider the classification of abilities, which can be extended:

We give the following classification of abilities, which is not exhaustive and can be expanded.

Table 1

№	Basis of classification	Ability type
1.	origin	– Natural abilities — abilities that have a biological structure. – Social abilities — abilities that provide life in a social environment, these abilities appear in the process of training and education.
2.	direction	– General abilities. – Special abilities — these are the abilities necessary for a special kind of inclinations, and their subsequent development, these abilities determine the success of a person in specific activities and communication.
3.	development conditions	– Potential abilities are abilities that appear due to changes in social conditions. – Actual abilities are the abilities that are needed at the moment and are being implemented in a particular type of activity.
4.	state of the art	– Giftedness. – Talent. – Genius.
5.	composition and structure	– Elementary (eye, feeling, ear for music). – Complex (communicative, labor).

Research abilities are manifested in individual psychological characteristics, as they provide a successful and high-quality search process, which manifests itself in comprehending and acquiring new information, so we can conclude that research abilities relate to special abilities, presented in Table 1 above.

From the theoretical model of Savenkov A. V., it follows that research abilities should be considered as a complex of 3 components [6, p. 154–157]:

1) search activity, which is characterized by the motivational component of research abilities;

2) divergent thinking, which is characterized by the ability to find several solutions to a creative problem, that is, multivariate thinking;

3) convergent thinking, characterized by a strategy of precise use of algorithms for solving certain problems, already learned earlier.

Thus, research abilities are a complex dynamic formation, and the relative development of individual parameters is not always the key to successful research. Thus, the use of non-standard tasks is necessary in the process of teaching mathematics.

Now consider the line of equations with a parameter. Among the tasks of the school course, this line occupies a special place. This is due to the fact that only the topic is considered in the school course: «Quadricular Equations», which is studied in the 8th grade. However, when studying the 8th grade textbooks, we came across the fact that the topic «Quadric Equations» does not cover equations with a parameter. Only in the student of Mordkovich in paragraph 25 are the concepts given: a parameter, an equation with a parameter, and 2 examples are given.

An analysis of the practical material of scientific manuals for the 8th grade showed that they do not contain non-standard tasks that could form students' research abilities on such a topic as quadratic equations with a parameter. [7].

As mentioned earlier, the 8th grade textbooks do not contain enough tasks for quadratic equations with a parameter, but we analyzed the small number of exercises on this topic and were able to identify the following types of tasks:

1) quadratic equations of standard form;

2) incomplete quadratic equations;

3) finding the value of the parameter, under the conditions;

4) quadratic equations solved using the direct and inverse Vieta theorem;

5) finding the number of roots of a quadratic equation.

I would also like to note that in the basic school, depending on the textbook, the study of the topic: «Quadricular Equations» is given from 21 to 26 hours. Also, numerous conditions for the location of the roots of the trinomial, for maintaining the sign of the trinomial at a certain interval, and others that are necessary when preparing for the OGE in grade 9 are not considered. The reason is the lack of a base, existing curricula do not provide for the study of such a topic as: «Solving quadratic equations with parameters».

For a more in-depth study of this topic, you can introduce an elective course in mathematics grades 7–9, on the topic «Solution of non-standard problems in mathematics», since the variety of non-standard tasks covers the entire course of school mathematics.

Based on the foregoing, we can conclude that the solution of quadratic equations with a parameter will contribute not only to the development of research abilities, but also to improve the quality of mathematical training of schoolchildren.

References:

1. Introduction to psychology. Ed. Petrovsky A. V. — M.: Academy, 1996.
2. Dalinger V. A. Improving the process of teaching mathematics based on the implementation of intra-subject communications. / Omsk.state. ped. in-t im. A. M. Gorky. — Omsk, 1993.
3. Kordemsky B. A. Extracurricular tasks for ingenuity as one of the forms of development of mathematical initiative in adolescents and adults: Abstract of the thesis. dis. for the competition scientist step..cand. ped. Sciences. — M., 1956.
4. Episheva O. B. General methods of teaching mathematics in secondary school: Course of lectures: Textbook for students. physical — mat. specialist. ped. in-comrade. — Tobolsk: Ed. TSPI them. D. I. Mendeleev, 1997.
5. Teplov BM Psychology and psychophysiology of individual differences. — M.: Institute of Practical Psychology, Voronezh: NPO «MODEK», 2008.
6. Potapova M. V. Continuity in the development of research abilities in the process of performing experimental tasks by secondary school students // World of Science, Culture, Education. No. 5, 2012, p. 154–157.
7. Federal list of textbooks for the 2019–2020 academic year. [Electronic resource]. — Access mode: https://4ege.ru/materials_podgotovka/58282-federalnyy-perechen-uchebnikov-na-2019–2020-uchebnyy-god.html — Last updated 12/05/2019