CC BY
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PEDAGOGICAL SCIENCES / «<g©LL©(MUM~JOy©MaL» #16»3)), 2©2fl
PEDAGOGICAL SCIENCES
UDC 510.6:378.147
Hulivata Inna,
ORCID: https://orcid.org/0000-0003-4752-535X Ph.D. in Pedagogical Sciences, Associate Professor
Husak Liudmyla, ORCID: http://orcid. org/0000-0002-0022-9644 Ph.D. in Pedagogical Sciences, Associate Professor
Kopniak Kateryna ORCID: http://orcid.org/0000-0003-0618-0359
Senior Lecturer
Department of Economic Cybernetics and Information Systems Vinnytsia Institute of Trade and Economics of Kyiv National University of Trade and Economics, Vinnytsia, Ukraine DOI: 10.24412/2520-6990-2021-16103-54-58 FORMATION OF LOGICAL COMPETENCE OF A SPECIALIST IN THE PROCESS OF
MATHEMATICAL TRAINING
Abstract.
Developed logical reasoning is a necessary backgroundfor successful work as a lawyer, economist, manager, scientist; this is reflected in the standards of higher education in Ukraine. This is also a sign of universal and professional well-educated specialist, who is able to solve complex specialized issues and practical problems in his own professional activity. Such activity involves possession of a certain logical arsenal: methods of analysis and synthesis, abstraction and generalization, ability to prove and disprove, to draw the right conclusions, to make reasonable and rational decisions. In connection with all before mentioned, there arises a problem of finding ways to developing logical thinking ofspecialists in various fields, taking into account the historical connection of Logic with Mathematics.
The theoretical and empirical research methods were applied to achieve the goal and solve the tasks of the research: scientific study of scholarly papers, analysis of educational and methodological literature, pedagogical observations of the learning process of students.
This paper clarifies the role and application of Logic as a science in Mathematics. It is proved on the example of binary relations, that predicate calculus gives an explicit insight into a general and abstract ideas and relations in Mathematics. This paper explains the connection between mathematical logic and other mathematical disciplines; establishes the ways of implementation of logical laws in other sciences through teaching mathematical disciplines.
Mathematical logic is essentially a formal logic that uses mathematics-based methods, in regard to this the so-called «mathematization» of knowledge is essentially «logical» knowledge. Therefore, the formation of the logical culture of a competitive specialist can achieve by studying various disciplines of Mathematics, or using a system of non-standard logical problems.
Keywords: Logic, Mathematics, logical competence, competitive specialist, logical thinking, logical reasoning, mathematical training.
Problem statement. According to Piaget, human actions and deeds, as well as thought process, have a logical structure, and logic itself is created at some level by spontaneous actions (Piaget, 1969). Indeed, the ability to think logically is the basis for professionals in various fields, which is reflected in the standards of Ukrainian higher education. This is a sign of universal and professional education of specialists who are able to solve complex specialized problems and practical problems in their own professional field. Such activity involves possession of some logical range of methods of analysis and synthesis, abstraction and generalization, the ability to prove and disprove, to draw the right conclusions, to make informed, rational decisions.
Analysis of recent researches and publications. The structure of the professional competence of a specialist is quite complex and ramified. In accordance
with the requirements of the labor market, it is increasingly important to develop the logical competence of the future specialist, which he needs for further successful professional development and career growth [1]. Developing logical thinking is a necessary precondition for the successful work of a lawyer, economist, manager and scientist. Logical relations lie in the cornerstone of many branches of science. It is believed that mathematicians have a high level of logical culture. It is found a causal relationship between logical thinking and mathematical training of students in modern scientific publications [2].
Scientists claim that most part of students' mathematical knowledge is focused on logical thinking. There are no studies on the influence of logic on the formation of a modern specialist in any field, but the fact that civil servants and lawyers are required to know the basic elements of logic - it is an indisputable fact
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[3]. It should be mentioned also that the question of importance of logic in the study of mathematics was also important in Piaget studies [4]. Historical background and genesis of relations between formal logic and mathematics are covered in [5]. In this way there are next follow-up questions: of application of logic in mathematics and the application of logical laws and structures in other sciences.
The purpose of the article. The purpose of an article is to determine the role of logic in mathematics and other sciences for the effective formation of a competitive specialist. Theoretical and empirical research methods were used to achieve the goal and solve tasks set in the research: the study of scientific publications, analysis of educational and methodological literature, pedagogical observations of the learning process of students.
Results. It is known that all mathematical statements, except for axioms, are proved logically. This principle is necessary for proving mathematical theories. For example, the statement: "The diagonal of the square is incommensurable with the side, the length of which is equal to one", cannot be proved by anything but logical reasoning. Even the most accurate measuring tools are not capable of determining its accuracy.
Famous German mathematician Karl Theodor Wilhelm Weierstrass argued that the essence of mathematical knowledge lies in the absolute completeness of its proving. The first complete system of mathematical logic on the basis of strict logical-mathematical language - the algebra of logic, was proposed by J. Bull (1815-1864). The logical system was first used to analyze the foundations of mathematics and substantiate mathematical theories. The purpose of a logical system is to exclude intuitive concepts from mathematical theories and to provide logical proofs and logical structures. However, it's not easy to carry out a complete formalization of mathematics and thus clear mathematical knowledge of any intuitive ideas or concepts.
Logical methods have been used in mathematics since ancient times by Thales, Parmenides and Pythagoras, who greatly contributed to the turning mathematics into a correct and accurate science. G. Frege and B. Russell tried to reduce mathematics to logic. Deductive and axiomatic reasoning, the essence of which is logic, played a significant role in this situation. They contributed not only to the mathematical rigor of proofs in mathematics, but also to the building of mathematical
theories. The founders of inductive and deductive methods were Francis Bacon and Rene Descartes.
The essential difference between mathematics as a formal system and logic is that logic requires a stricter formalization, which is basic for proving theorems and building of mathematical theories. The influence of logic on mathematics is huge and indisputable. This comes mainly from propositional and predicate logic, which are considered as two subsystems of mathematical logic. In this respect, predicate calculus plays more important role, which is more flexible and, therefore, more accurately describes the logical dependencies in mathematics. This is an axiomatic-deductive system based on statements that are accepted without proof -axioms, and from which by deduction derive all other contents.
Predicate calculus gives a particularly clear idea of more general and abstract ideas and relations in mathematics [6]. Here are examples of such relations (Table 1). Let's consider the equivalence relation, which plays an extremely important role in algebra. This is the most elementary type of binary relation.
For example, the statements: "a is equal to b", "a is less than b", "a is greater than b" are a relation between two elements of a certain nature, and in the language of logic has the form P (a, b). It is reflexive, symmetrical and transitive:
- reflexivity relation: each element of some set is in the relation R with itself, i.e. aRa. For example, in geometry, each line is parallel to itself (a || a), or each figure is similar to itself (S ~ S). In the language of predicates, reflexivity is presented in the form: (VaeX) (P (a, a));
- symmetric relation: for each pair of elements of some set, if the relation aRb is applied, then the relation bRa is also applied. For example, if line a is parallel to line b, then line b is parallel to line a (if a || b, then b || a). Or, if figure A is similar to figure B, then figure B is similar to figure A (if A ~ B, then B ~ A). In the language of predicates, this statement is written: (Va, beX) (P (a, b) ^P (b, a));
- transitivity relation: for any a, b, c, if a is related to b, and b is related to c, then a is related to c. For example, if a || b and b || c, then a || c. In the language of predicates, this relation is denoted as: (Va, b, ceX) (P (a, b) a p (b, c) ^ P (a, c)).
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Table 1
Binary relation: Logic and Mathematics
Binary relations Properties of binary relations of logic Mathematical relation
Reflexive Vx e X, xRx = equal to ^ less than or equal to ^ greater than or equal to
Symmetric Vx, y e X, xRy ^ yRx - equality of real numbers - similar figures
Transitive Vx, y, z e X, (xRy a yRz) ^ xRz - Divisibility of numbers: if a is divisible by b and b is divisible by c, then a is divisible by c - Equality of numbers: if a=b and b=c, then a=c
Equivalence Vx, y, z e X, (xRx) a (xRy ^ yRx) a ((xRy a yRz) ^ xRz) - Equality in set of real numbers
Irreflexive Vx e X, —( xRx) ^ not equal to < less than > greater than
Antisymmetric Vx, y e X, xRy a (yRx) ^ x = y - Not strict inequality of an ordered set: if a < b and b < a, then a=b - Divisibility of integers: if a is divisible by b and b is divisible by a, then a=b
Antitransitive Vx e X, (xRy ^ yRz) ^ -(xRz) - for every three real numbers a, b, c no transitivity
Regarding the latter, it is important to cite the German philosopher Hegel that any science uses logic. It should be mentioned that logic studies propositions and their relations in terms of form, ignoring the content. But in reality we do not have propositions, instead we have real objects and processes for which to study we use verbal expressions. In this regard, it is necessary to implicitly present logical laws in other sciences. Mathematics is most suitable for this, because it applies laws of logic directly. A striking example of application is integer theory, group theory, abstract algebra, set theory, and so on. These mathematical theories are so intertwined with modern logic that it is hardly possible to draw a line between them. Mathematical logic has just as much in common with set theory. For example, consider the logical connections in statements and operations on sets that have the same meaning (Table 2).
Table 2
Analogy between ^ logical formulas and operations on sets
Logical formulas Operations on sets
Conjunction p a q A n B Intersection
Disjunction P v q A U B Inclusion
Implication p ^ q A œ B Conditional (subset)
Negation -P A Addition
Taking into account the above-mentioned relations, we can assume that predicate logic, as one of the subsystems of mathematical logic, is based on assumptions and arbitrary statements (propositions), those which are devoid of any specific meaning. Considering these examples, we ignore both the nature of the elements a, b, c, and the nature of proposition, which are written in mathematical symbols, as we are interested only in the logical consequences that follow from the accepted statements (axioms) and the relevant rules. This procedure is quite of a formal nature. And this, of course, does not make it lack scientificity. On the contrary, it is what provides the necessary accuracy, consistency, and, most importantly, aggregation - a condition important for the introduction of logic to Mathematics, and through Mathematics into other sciences.
Thereby, though set theory has direct application of laws of logic, other mathematical sciences apply them indirectly, that means, either through set theory or directly through related theories as a theory of integer, group theory, and other important branches of Abstract algebra. Mathematics cannot do without the aid of logic, no matter how it is applied - either with common sense, or with logical structuring. In general, logic is a necessary condition for theory building in Mathematics. There is an opinion that any further development of mathematical logic will largely depend on the introduction of topology methods.
Nowadays, logic has its place in various fields of knowledge and technology, where it plays a crucial role. Such "takeover" occurs through the application of logical laws and methods, which are mainly provided by mathematics. In this case, Mathematics plays the role of a link between logic and other sciences. In this way, the formation of a logical culture of a specialist in any field can be effectively used during the study of mathematical disciplines. However, field of study in the Humanities do not have such an advantage, moreover, have biases towards the study of Mathematics. One way out of this problem is solving non-standard logical
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problems, which are not themed, age-oriented, and do not require professional knowledge [7].
The professor can suggest different ways of presenting the following tasks: handouts, group demonstrations with a choice of other types of information visualization (presentations, open knowledge maps, etc.), means of modern digital learning technologies (computer games, online tests, etc.). Here are some types of such tasks.
1. Building logical chains: - establish a principle on which the sequence of numbers is constructed:
+4
+4
-1
-1
- select combination which will complete the series of given figures?
Difficulty level depends on the quantity of figures, colors, forms of objects in one figure.
4. Insert numbers to solve the equation using the available choices.
? ? = 13
Answer choices:
51 35 4 38 52
1. 3, 5, 2, 9, 1, ...?
- continuing numerical series: 24, 21, 19, 18, 15,
13, 12, ...?
Complexity level of such problems can be adjusted by selecting sequences in the chain, taking into account the accuracy and speed of solving.
2. Word formation from a given array of letters.
- connect the letters from columns to make words;
- find the maximum number of words with a certain number of letters, using specific letters.
Depending on the audience, you can adjust the complexity level of problems due to the total number of letters, words, word length, hiding some letters during the task, time constraints.
3. Elimination grids and selecting objects out of group of objects.
Which of the following combinations is not correct?
All types of mathematical operations have application in problems of this type. Inverse problems are also effective, in which you need to choose an action, not a number.
5. Logical reasoning problems.
Brown, Jones and Smith have been charged with a bank robbery. The robbers fled the bank and got into a car drove them away from the area. During the investigation, Brown testified that the robbers were driving a blue Buick; Jones said it was a black Chrysler; Smith testified that it was Ford and by no means blue. It became known that the accused had made false statements while testifying, each of them correctly indicated either only the brand of the car or its color. What color was the car and what was the car brand?
Answer choices: Blue Chrysler; Blue Buick; Blue Ford; Black Chrysler; Black Buick; Black Ford.
Conclusions and further research prospects. The influence of logic on mathematics is huge and indisputable. Mathematical logic is a formal logic itself that applies mathematical methods. The interdependence of Law and methods of Modern mathematics and their introduction into almost all fields of knowledge means both the introduction of law and methods of modern logic into the sciences.
The so-called "mathematization" of knowledge is the cornerstone of "logical" knowledge. Therefore, the formation of logical thinking of a competitive specialist can be achieved through mathematical disciplines, or the usage of a system of non-standard approach to the logical problems in the educational process. For further research work, it would be of interest to further substantiate the methodological system of forming students' logical thinking in higher education.
References
1. Kopniak, K. (2020). Structural and functional model of formation of managerial competence of future economists in the professional training process. Scientific Bulletin of Uzhhorod University. Series: «Pedagogy. Social Work», No. 1(46), pp. 55-58. DOI: https://doi.org/10.24144/2524-0609.2020.46.54-58. [in Ukrainian].
2. Nunes, T. & Bryant, P. & Evans, D. & Bell, D. & Gardner, S. & Gardner, A. & Carraher, J. (2007). The contribution of logical reasoning to the learning of mathematics in primary school. British Journal of Developmental Psychology, Vol. 25, pp. 147-166. DOI: http://dx.doi.org/10.1348/026151006X153127. [in English].
3. Test for analytical and operational skills and general erudition (test example in the selection to the Foreign Intelligence Service of Ukraine). URL: https://testderz.com/2018/03/23/general-skills/. [in Ukrainian].
4. Piazhe, Zh. (1969). Izbrannyye psikholog-icheskiye trudy [Selected psychological works]. Mos-kow : Prosveshcheniye. [in Russian].
5. Sikic, Z. (1996). Mathematical logic: mathematics of logic or logic of mathematics. DOI:
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http://dx.doi.org/10.13140/RG.2.1.1251.9441. [in English].
6. Khomenko, I. V. (1998). Lohika - yurystam [Logic - to lawyers]. Kyiv : Chetverta khvylia. [in Ukrainian].
7. Hulivata, I. O. & Nikolina, I. I. (2020). The role
of logic in mathematics and the formation of a competitive specialist. Modern informational technologies and innovative methods in professional training: methodology, theory, experience, problems, No. 57, pp. 86-92. [in Ukrainian].
УДК 378:371.134:331.45
Абтьтарова ЕльвЬа Нуривна кандидат педагоггчних наук, доцент кафедра охорони працг в машинобудуванш та соцгальнт сфер1 Кримський 1нженерно-педагог1чний унгверситет ORCID: 0000-0001-9747-3303 DOI: 10.24412/2520-6990-2021-16103-58-60 СОЩАЛЬНО-ЕКОНОМШШ ЧИННИКИ ФОРМУВАННЯ КУЛЬТУРИ БЕЗПЕКИ ПРОФЕС1ЙНО1 ДШЛЬНОСТ! У МАЙБУТН1Х 1НЖЕНЕР1В З ОХОРОНИ ПРАЦ1
Abiltarova Elviza Nurievna
Candidate of Pedagogical Sciences Associate professor of the Department of Labor Protection in Engineering and Social Sphere Crimean Engineering and Pedagogical University ORCID: 0000-0001-9747-3303
SOCIO-ECONOMIC FACTORS OF FORMATION OF THE CULTURE OF SAFETY OF PROFESSIONAL ACTIVITY IN FUTURE OCCUPATIONAL SAFETY AND HEALTH ENGINEERS
Аннотация.
Статья посвящена поиску путей эффективного формирования культуры безопасности профессиональной деятельности у будущих инженеров по охране труда. Автором сосредоточено внимание на социально-экономических факторах, которые влияют на эффективность формирования культуры безопасности профессиональной деятельности у будущих инженеров по охране труда. Определено, что важными социально-экономическими факторами являются состояние производственного травматизма, социальные выплаты и компенсации вследствие несчастного случая и профессиональных заболеваний на производстве, социальный статус и престижность профессии.
Abstract.
The article is devoted to the search for ways to effectively form the culture of safety ofprofessional activity in future occupational safety and health engineers. The author focuses on the socio-economic factors that affect the effectiveness of the formation of the culture of occupational safety among future safety engineers. It was determined that important socio-economic factors are the state of industrial injuries, social payments and compensation for accidents and occupational diseases at the workplace, social status and prestige of the profession.
Ключевые слова: культура безопасности профессиональной деятельности, профессиональная подготовка, инженер по охране труда, безопасность, факторы.
Keywords: safety culture of professional activity, vocational training, labor protection engineer, safety.
Вступ. Актуальнють формування культури безпеки професшно! дiяльностi у майбутшх шже-HepiB з охорони пращ обумовлена потребою суст-льства у падготовщ висококвалiфiкованих кадр!в, здатних ввдповвдати та оргашзовувати систему уп-равлшня охороною пращ на щдприемсга, яка спря-мована на забезпечення безпечних умов пращ та збереження здоров'я робгтнишв у процес професшно! дiяльностi. Прийняп Резолющя Генерально! асамбле! ООН ввд 25 вересня 2015 р. «Перетво-рення нашого свгту: Порядок денний в област ста-лого розвитку на перюд до 2030 року» [2], Страте-пя сталого розвитку Укра!ни до 2030 року [1, 3] ви-магають пошуку нових пiдходiв та шляхiв щдвищення якосп тдготовки фахiвцiв до рiвня, до-сягнутого в кра!нах ЕС. Для розв'язання означено!
проблеми важливо врахувати чинники, як! сприяти-муть ефективному формуванню культури безпеки професшно! д!яльносп у майбутшх !нженер!в з охорони пращ У нашому дослвдженш було визна-чено наступну групу чиннишв: макрор!вень (р!вень держави) - полгтичш, нормативш, юридичщ соща-льно-економ!чш; мезор!вень (р!вень вищого навча-льного закладу та оргашзацп) - технолопчщ орга-шзацшш, медико-бюлопчш, педагопчщ психоло-пчш; мшрор!вень (суб'ективш) - !ндивщуальш властивосп особистосп, професшно-особиспсш якосп, мотиващя до безпечно! поведшки. У данш робот! розглянемо бшьш детально сощально-еко-ном!чш чинники формування культури безпеки професшно! д!яльносп у майбутшх !нженер!в з охорони пращ Мета статп - обгрунтування та роз-
