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FEATURES OF SOLVING LOGIC PROBLEMS BY YOUNGER SCHOOLCHILDREN
Zak A.
Leading Researcher, Psychological Institute RAE, Moscow, Russia
Abstract
In the article, the characteristics of solving logical problems by younger schoolchildren are considered. The content of two cycles of individual experiments with the fourth-graders is described: in total, 103 students participated in the study. It was shown that the change in the form of the action in the solution of the logical tasks of the proposed species, from visual to figurative to the substantive, facilitates the transition of a number of children from the implementation of unsuccessful solutions to successful ones.
Keywords: fourth graders, logical tasks, form of action, comparison and collation, ways of solving problems.
1. Introduction
The new federal Standard for primary general education contains requirements for metasubject results of mastering the basic educational program of primary school. According to these requirements, during the period of study in primary grades, children must master logical actions associated with solving various logical problems [7].
The purpose of this study was to study the conditions for the successful solution of logical problems by younger schoolchildren, in particular, by fourth-grade students.
Two cycles of experiments were carried out. In the first cycle, children solved problems where it was required to find the missing elements of classes (sets), in the second cycle, they were asked to solve problems related to the definition of an unknown class (set) by indirect signs.
2. Materials and methods
2.1. Solving problems "to find missing elements of classes"
The purpose of the experiments in this cycle was to study the characteristics of solving problems "to find the missing elements of the classes." In this cycle, two series of experiments were carried out.
The purpose of the first series was to identify the features of successful and unsuccessful solution of problems of the noted type, the purpose of the second series was to clarify the nature of the relationship between successful and unsuccessful methods of solving problems of the noted type with the form of action when the required result is achieved.
This cycle of individual experiments involved 47 fourth-graders in the middle of the school year.
First series
The following problem was used as a technique for the first series of experiments:
Fig. 1. Problem condition for the first series of experiments
The student was told: "The artist drew nine large triangles and in each small figures. But I forgot to draw the figures in the ninth triangle. Guess what figures should have been drawn in the ninth triangle? "
The structure of this problem can be represented in the sign model as follows:
A 1 Q A 2 W A3 Z
(1) (2) (3)
B 1 Z B 2 Q B 3 w
(4) (5) (6)
C 1 W C 2Z ? ? ?
(7) (8) (9)
In this model, the letters A, B, C denote small figures located in the left corner of each large triangle, numbers 1, 2, 3 - figures located in its upper corner, letters Q, W, Z - the figures located in the right corner.
In the first series, children were asked to solve the above problem. The purpose of the experiments in this series was, as noted above, to find out the features of successful and unsuccessful methods of solving problems "to find the missing elements of classes." As a result of the experiments, it turned out that 12 people (25.5% of the sample) solved the problem successfully (group A), the remaining 35 people (74.5% of the sample) solved this problem unsuccessfully (group B).
The subjects of group A were characterized by the following actions when solving the problem. Having skimmed all the triangles and the small figures located in them, they immediately assumed that there was not enough circle in the upper corner of the triangle.
However, after that, they carefully examined the eight drawings proposed in the problem statement so that, as they said, "... find out what will happen in the lower corners ...". On examination, it was possible to notice that these subjects revealed the principle of placing small figures in the left corner of the triangle: "... in each row the same ..." and indicated that there should be a rhombus in the ninth triangle.
Then they examined all the triangles for a relatively long time, so that - as could be understood by their actions (some children even held two hexagons with their fingers in images 2 and 6, or two small triangles in images 4 and 8) - to find out how the small figures are distributed in the right corner of the triangles. As a result, some children came to the conclusion that "... in each row there are three different ..." and therefore there should be a rectangle in the ninth triangle. Other children (just those who "held" the images with their fingers) found the "diagonal" principle of placing small figures in the right corner of the triangles "... one figure in different places in rows ..." and also said that "... a rectangle is needed here ...".
Subjects in group B solved the problem differently. Most of the children in this group (subgroup B.1, - 23 people, 65.7%), as a rule, did not carefully examine all nine, more precisely, eight triangles with small figures inside, limiting themselves to examining the images, or in one horizontal row (in in particular, images 7 and 8), or in one vertical row (in particular, images 3 and 6) with image 9.
In the first case, the children suggested drawing a pentagon and a triangle in the upper and right corners, respectively, and a rhombus in the left corner. They explained this solution by the fact that in the lower corner the small figures should be repeated, because in images 7 and 8 the rhombus is repeated.
In the second case, the children, also relying on the idea of the repetition of figures, suggested drawing a circle in the upper corner, a trapezoid in the left corner, and a hexagon in the right corner. This means that they were, in fact, proposing to fill triangle 9 with the same shapes that were in triangle 6.
A smaller part of the children in this group (subgroup B.2, - 12 people, 34.3%) examined all the triangles and, as a rule, came to the conclusion that a circle should be drawn in the upper corner of triangle 9, thereby marking the repeatability of the figures along the vertical ... Further, some children suggested drawing figures in the lower corners, relying on the repetition of the rhombus in triangles 7 and 8, and others - to draw a rhombus from triangles 7 and 8 in the left corner of triangle 9, and a hexagon from triangle 6 in the right corner.
Analysis of the actions of the subjects of both subgroups of group B (B.1 and B.2) shows that they could not reveal the principle of placing small figures in the right corner of triangles. Comparing their actions with the actions of the subjects of group A in solving the problem, it can be noted that the analysis of the conditions of the problem by children who successfully solved the problem (group A) was based on the orientation towards the location of a small figure in triangles and clarification of the principle of placing figures in each of the three corners along all eight triangles. And only after the discovery of such a principle, a conclusion was made about which figure should be in the corresponding corner of triangle 9.
The approach to the analysis of the conditions of the problem by children who unsuccessfully solved the problem (group B) was based on an orientation towards the features of the figures, that is, these children tried to directly answer the question of what figures should be in triangle 9 in the upper, left and right corners. Therefore, noticing the repetition of some figures, they directly used this to infer which figures and where should be in triangle 9.
On the whole, it can be said that the children of group A acted indirectly, analyzing the relations between the elements of the conditions of the problem and thereby revealing the essential relations of these elements. This means that they solved the problem in a general, theoretical way. The children of group B did not carry out such an analysis and relied in their search for a solution only on a simple comparison of elements of conditions. This indicates that they solved the problem in a particular, empirical way (for more details about the characteristics of the general, theoretical and particular, empirical methods, see, for example, in our works: [2], [3], [4], [5]).
Second series
The purpose of the second series, as noted, was to determine the nature of the connection between the method for solving logical problems of the type under
consideration and the form of action in which it is proposed to solve the problem. 35 subjects who did not successfully solve the problem "with triangles" (ie, by a particular, empirical method) were asked to solve a
0
In the second problem, the same three classes of 9 elements of three types in each were used: small figures located, firstly, at the top of the circle, secondly, at the bottom of the circle on the left, and thirdly, in the middle of the circle on the right.
The noted similarity of both problems makes it possible to use in the analysis of the second problem the same model representation of elements of the three classes as in the first problem.
The second series of experiments was carried out as follows. 35 people who unsuccessfully solved the first problem (group B) were divided in this series into three almost equal groups: B.1 (11 people, - 31.6% of group B), B.2 and B.3 (12 people - 34.3% each from group B).
The subjects of group B.1 were asked to solve the second problem in the same way as they solved the first, that is, in a visual-figurative plan.
Subjects group B.2 again dealt with the first problem. They were asked to first copy the content of eight images by sequentially tracing large triangles and small figures in them - first they had to circle triangle 1 and three small figures in it, then triangle 2 (and three small figures in it), then triangles 3 - 8 in succession (with small figures in them). After that, it was proposed to
second problem of the same type, which, with an external difference, structurally repeated the problem from the first series.
determine which three small figures should be in triangle 9.
Subjects of group B.3 also dealt with the first problem for the second time. They were asked to first copy the content of eight images by successively copying large triangles (and small figures in them) onto another sheet, and then find a solution to the problem.
The division of the subjects into three marked groups pursued the goal of correlating the success of the repeated solution of the problem in a visual-figurative form (for this purpose, the children solved the second problem, similar to the first) with the success of its solution (in this case, the first problem was solved) in a materialized form of various types: circle and drawing images that make up a particular class of objects.
As a result of the experiments of the second series, it turned out that only 3 people (27.3% of group B.1) solved the second problem successfully, acting in a theoretical way, i.e. based on the analysis of the essential relationships of the elements of the conditions of this problem The remaining 5 (respectively, 72.7% of group B.1)) in solving the second problem also acted unsuccessfully, using, as in the first problem, an empirical approach to the solution associated with the direct search for suitable figures by direct comparison of condition elements.
These children were asked to circle the images in the condition of this problem. As a result, it turned out that after tracing the lines, all the children solved the problem correctly: already during copying, they found that some figures are repeated in vertical columns (images 1, 4, 7; 2, 5, 8; 3, 6, 9), others - in horizontal rows (images 1, 2, 3; 4, 5, 6; 7, 8, 9), and the third are in three rows in different places (images 1, 6, 8; 2, 4, 9; 3, 5, 7).
All subjects of groups B.2 and B.3 correctly determined the composition of the small figures in triangle 9 after copying the images. Regarding the actions of the subjects of group B.2, it should be said that some children of this group (subgroup B.2.1, - 4 people, 33.3% of this subgroup) revealed the recurrence of small figures horizontally after tracing the triangles of the first row, - 1, 2 and 3, and the rest of the children (subgroup B.2.2, - 8 people, 67.3% of this subgroup) - after circling the triangles of the second row, - 4, 5 and 6. However, the repetition of small figures vertically, all children of this group found after outlining the triangles of the second row, and the peculiarities of the distribution of elements in the right corner of the large triangles, they could establish only after outlining all the triangles.
Children of group B.3 also acted differently when sketching images. Most of the children (subgroup B.3.1, - 9 people, 75.0% of this subgroup) found a "horizontal" repeatability of figures after copying the images of the first row, and a smaller part (subgroup B.3.2, - 3 people, 25.0% of this subgroups) - after copying the images of the second row. Children of subgroup B.3.1 found the "vertical" repetition and, even, the nature of the distribution of figures in the right corner of the large triangles after copying the images of the second row, and children of subgroup B.3.2 after copying all the images.
Thus, as a result of the experiments of the second series, it turned out that the change in the form of action when solving problems where it was required to find the missing elements of the class, contributed to the transition of 24 subjects (51.1% of the sample of this cycle of experiments) from an unsuccessful (private, empirical) method of solving problems to successful (general, theoretical).
2.2. Solving logical problems "to identify classes"
The purpose of the experiments in this cycle was to study the characteristics of solving problems "to determine classes". In this cycle of experiments, two series were carried out. The first series was aimed at identifying the features of successful and unsuccessful solution of problems "to determine classes". The second series was aimed at clarifying the nature of the relationship between successful and unsuccessful methods of solving problems of this type with the form of action when the required result is achieved. This cycle of individual experiments involved 56 fourth-graders at the end of the school year.
First series
The following problem was used as a technique for the first series of experiments:
M2 A3 N 2 E 5
(1) (2)
M 2 E 5 N2 A3
(3) (4)
M 2 A 5 N2E3
(5) (6)
M 2 E 3 N2 A5
(7) (8)
Each child was given a sheet on which the condition of this problem was depicted: eight combinations of signs, four each. In each combination, the signs took places different in function: the left extreme - the first place, the left middle - the second place, the right middle - the third place, the right extreme - the fourth place. On each of the named places there was a sign of a certain category: in the first place was a consonant letter, in the second - an even number, in the third - a vowel letter, in the fourth - an odd number. This material allows you to guess any, - from one to three, - the number of signs.
At first, the experimenter drew the child's attention to the presence of eight combinations of signs, four in each, and to the fact that different combinations contain the same signs.
Then the experimenter said that he could point to any of the eight combinations on the playing field and guess some two or three signs in it. The subject's goal was to identify the hidden signs. For this, it was possible to name any combination in which the desired signs could be found. If this combination contained all the hidden signs, then the experimenter answered "Yes", and if there were not all the desired signs (for example, two signs out of three conceived, or only one of two) and, even more so, if there was none of required signs, the experimenter answered: "No".
For example, at first the experimenter thought of the signs M, 2, 3 in combination (1), - M 2 A 3. Then he informed the child that in combination (1) some three signs were envisioned. After that, the child could point, for example, to the combination (3), - M 2 E 5 and ask: "Are there three hidden signs in this combination?" The experimenter had to answer: "No." Then the subject could indicate the combination (1), - M 2 A 3. The experimenter had to answer: "Yes." As a result, the subject could say that the signs M, 2 and 3. The experimenter in this case had to report that the problem was solved correctly.
The first series involved 56 fourth-grade students. Based on the above eight combinations of signs, each child was asked to solve the problem of finding two hidden signs (these were signs E-5): "In combination 3, two signs are conceived. Guess what are these signs? "
In this series, as noted, the specifics of solving these problems by empirical and theoretical methods were clarified. As a result of the experiments, it turned out that 12 people (21.4% of the entire sample, group A) solved the proposed problem successfully (i.e., in the smallest number of choices, usually in four or five
choices), and 44 people (respectively, 78.6% of the entire sample - group B) either guessed the desired signs with the help of a large number of extra choices (more than 10 choices), or could not guess them at all.
Subjects of group A acted as follows. So, not yet indicating any combination (that is, without making any choice), these subjects carefully examined the playing field, figuring out where the combinations containing the signs of the sample combination (M 2 E 5) are located, that is, they found (as could be judged by their remarks and the movements of the pencil between the combinations of signs) that the combinations with the letter M are in the left column, and with the number 2, the letter E and the number 5 in both columns. At the same time, they also noted that the number 2 does not change, but the letters A and E, numbers 3 and 5 alternate.
After that, the subjects of group A consistently tested their assumptions about the pairs of hidden signs. In doing so, different strategies were used.
8 pupils (66.7% of group A, subgroup A1) acted in general as follows. First, they pointed to combination 1, asking: "Are these signs in combination 1?" After the experimenter gave a negative answer, these children said, for example: "... that means not M-2 ...".
Secondly, they asked: "Is there 7 in the combination?" After the experimenter answered "No", they said, for example: "... then, not M-E ... and not 2-E ...".
Third, they asked: "Is there a combination of 5?" After the experimenter answered "No", they said, for example: "... that means not M-5 and not 2-5 ...".
After this choice, two children immediately gave the answer: "The signs of E-5 are hidden," and the other two made another choice: "Do you have any in group 2?" And after an affirmative answer, the experimenter was offered a solution: "E-5 is conceived."
Two students (16.65% of group A, subgroup A2) acted differently. First of all, they pointed to the combination of 5 and after the experimenter gave a negative answer, they noted: "... then this is not M-2, not M-5, not 2-5 ...". Secondly, they pointed to the combination of 7 and after the experimenter answered "No", they said: "... so, not M-E and not 2-E..." and then proposed a solution: "... E-5.".
Two students (16.65% of group A, - subgroup A3) made their choices, compared with the previous subjects, in the reverse order. They first pointed to the combination of 7, and after the experimenter gave a negative answer, they concluded: "... then this is not M-2, not M-E, not 2-E ...". Then they pointed to the combination of 5 and after the experimenter answered "No", they said: "... so, not M-5 and not 2-5,... but. E-5".
Thus, observations of the actions of the subjects of group A indicate that they perform two types of analysis: analysis of the structure of the playing field, which is expressed in elucidating the features of the placement of combinations with the same signs, and analysis of the result of testing hypotheses about possible pairs of signs in a given combination. -sample.
Subjects in group B solved the problem differently. In contrast to the subjects of group A, they did
not analyze the structure of the field, nor did they analyze the combinations of signs by two — they acted in a random way.
For example, at first they pointed to combination 1, and after the experimenter had a negative answer, they did not draw any conclusions, but simply pointed to another combination - combination (5). After a negative answer, they again did not comprehend the current situation and pointed to the combination 7. After a negative answer, most of the children (28 people, 63.6% of group B, subgroup B.1) refused to further solve the problem, stating: "You cannot guess ... "Or" I don't know how to proceed ... ".
Another part of the subjects (16 people, 36.4% of group B, - subgroup B.2) continued the solution. They pointed further to the combination of 2 and, after the experimenter answered in the affirmative, randomly named one of three possible pairs of signs: 2-5, 2-E or E-5.
When asked why this or that pair of signs is the answer, they usually answered that "... these signs are there and there ..." or "... they are in combinations 3 and 2 ...". The fact that you still need to prove your answer and that not any pair of signs from combination 2 is suitable for the answer, they did not take into account.
Comparing the actions of the subjects of groups A and B, we can say that the first of them solved the problem using a theoretical method, since they controlled their search, acting indirectly: they analyzed the conditions of the problem (in particular, they considered the features of the distribution of signs in combinations), put forward hypotheses before the next choice and analyzed its result.
In contrast, there is reason to believe that the subjects of group B acted empirically: they did not analyze the conditions of the problem, did not put forward hypotheses during the solution (without thereby controlling their actions), did not analyze the results of the performed moves, and in situations of coincidence of the found combinations of signs with the original combination relied only on a direct comparison of these combinations.
As it turned out in a conversation after the experiment, many subjects of subgroup B.1 chose the following strategy: first, name the combinations one after the other until they found one about which they would say that it had hidden signs, and then choose which ones. -some two signs.
Second series
The second series of experiments in this cycle was organized in the same way as in the previous cycle: 44 subjects who solved the problems empirically (group B) were asked to solve the second problem "to determine the classes", where group 4 (N 2 A 3), in which two signs were guessed, - A - 3. This problem could be optimally solved in two choices, for example, the first choice is a combination of 8 (checking such pairs of signs as N-2, N-A and 2-A), the second choice is a combination of 6 (checking such pairs of characters as N-3, 2-3). After that, you can already name the solution -signs A-3.
In the second series, the subjects of subgroup B.2 (16 people, 36.4% of group B) solved the proposed
problem in the same way as in the first series - using the signs of the presented combinations in a visual-figurative plan. The subjects of subgroup B.1 (28 people, 63.6% of group B) were asked to act differently: they had to first copy (copy) the presented combinations of characters onto another sheet (in this case, it was necessary to write off combinations in a row in accordance with the numbers: 1, 2, ..., 7, 8), and then they were asked to solve the second problem (that is, the one where the initial combination was 4).
As a result of the experiments of the second series, it turned out that none of the children of subgroup B.2 acted in the optimal way when solving the problem, that is, did not put forward and did not test hypotheses about the pairs of hidden signs.
Among the subjects of subgroup B.1, a large part of children stood out (16 people, 57.1% of subgroup B.1, subgroup B.1.1), who were able to move to the optimal solution of the problem of the type under discussion, and a smaller part of children (12 people, 42, 9% of subgroup B.1, subgroup B.1.2) who did not change their way of solving (in comparison with the previous series).
Observations of the actions of children of the marked subgroups when they were sketching different combinations of signs showed that children of subgroup B.1.1 paid attention to the repetition of signs in the combination being copied with respect to the previous one. So, they noted that in combination 2 there is the same number with combination 1, in combination 3 - the same three characters with combination 2, etc. And after completing their copying work, they considered all eight combinations, noting the repetition of characters in the columns.
Starting to solve the problem of finding the signs conceived in combination 4, these children began to control their behavior: they called the next combination not just to see what happened, but in order to check for the presence of certain signs in the new combination.
In contrast to the actions of subgroup B.1.1, children of subgroup B.1.2 did not pay attention to the ratio of signs in combinations when copying, simply trying to copy without errors. Then, when solving the problem of finding signs from combination 4, they also acted at random, as in the first series.
Thus, as a result of the experiments of the second series, it was shown that the change in the form of action when solving problems "to determine the classes" contributed to the transition of 16 subjects (28.6% of the sample of this cycle of experiments) from the empirical method of solving problems to the theoretical one. At the same time, it is necessary to specially clarify that the noted change is associated with a change in the form of action not so much when searching for a solution (in this part of the solution to the problem there were no changes), but when contacting the conditions of the problem (in the first series, the named contact was carried out in a visual-figurative form, and in the second series - in a substantively effective form).
3. Conclusions
So, in the above cycles of experiments, techniques were used based on the material of logical problems associated with the search for missing elements of classes
(sets) and with the definition of an unknown class (set) by indirect signs.
The general goal of the experiments of both cycles, as noted, was to establish a relationship between a change in the way these problems are solved with a change in the form of action in which they are solved.
As a result of the experiments, it was found that a change in the method of solving logical problems of the indicated types can really be associated with a change in the form of actions when solving them. It was shown, in particular, that in the case when children were asked to act not in a visual-figurative, but in a substantive-effective form when solving the named problems (that is, when it was proposed to operate with elements of the conditions of the problems, - with schematic abstract images, - forming and changing their image by actually tracing and sketching them), then the unsuccessful (particular, empirical) method, based only on a direct comparison of the marked elements of the conditions, changed for them to a successful (general, theoretical) method, which was associated with the analysis of the relationship of these elements.
Thus, the data obtained in these experiments indicate that, in general, when diagnosing the development of logical thinking in children, and in particular when assessing the formation of the skills to solve logical problems of the species under study, it is necessary to take into account the form of action in which their successful or unsuccessful solution occurs.
This means that the unsuccessful solution of problems where it is required in a visual-figurative plan to compare and contrast, group and classify (see, for example, studies of domestic [6] and foreign [8], [9], [12] psychologists) does not give sufficient grounds for asserting that the child being assessed is not at all able to solve such problems.
Convincing grounds can appear only when the child being assessed will be asked to solve such problems not only in a visual-figurative form, but also in a materialized, objective-effective form, that is, in a form where he will have the opportunity to form images of elements of the task conditions more fully and more accurately.
The ability of children to solve logical problems of the indicated types with different success in different forms of action, shown in the experiments under discussion, indicates the presence of different levels of formation of methods for solving such problems.
The fact of the influence of the form of action on the result of solving logical problems, established in the study, is of fundamental importance for diagnosing the thinking of children. This fact serves as the basis for the development of a more specific interpretation of the results of solving logical problems that require comparison and comparison of the proposed graphic images for their solution: such problems are found in tests developed, in particular, by D. Raven [13], G. Eysenck [1; 11] and R. Cattell [10].
Therefore, when making judgments about the level of formation of the assessed child's ability to classify schematically represented images of objects, one should specially take into account the form in which the task was performed. In this case, the expressed value
judgments will be more complete, accurate, specific and reasonable.
With further study of the problem of this study, it is planned to identify the features of solving logical problems of the noted species by other age groups of primary schoolchildren: first graders, second graders and third graders. Based on the results of these studies, it will be possible to determine the main characteristics of the age dynamics of methods for solving logical problems and the features of the conditions that change these methods, in particular, the role of the form of action in solving problems in different age groups of primary school students.
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ТВОРЧЕСКАЯ РЕАЛЬНОСТЬ ДУШЕВНОГО МИРА: ПСИХОЛОГИЧЕСКОЕ УЧЕНИЕ Л.М.
ЛОПАТИНА
Серова О.Е.
ФГБНУ «Психологический институт РАО», ведущий научный сотрудник, история отечественной психологии
THE CREATIVE REALITY OF THE SPIRITUAL WORLD: THE PSYCHOLOGICAL TEACHING
OF L. M. LOPATIN
Serova O.
Psychological Institute of RAE, leading researcher, group history of Russian Psychology
Аннотация
В качестве альтернативы современным тенденциям на аннигиляцию души и отмену ценностного контекста науки, в статье рассмотрены особые черты научно-нравственного подхода к пониманию содержания целей, задач и методов психологии в философском наследии профессора Л.М. Лопатина - выдающегося русского мыслителя и морального лидера эпохи коренного перелома русской жизни конца XIX -начала XX вв.: спиритуалистическая точка зрения как потенциал теоретико-методологического решения основных проблем психологии, метафизическое значение психологии, понимание духа как глубинная сущность души, виды психической причинности, концепция духовной причинности, созидательно-творческий характер душевной жизни, тождество психологии и самопознания, центральная роль внутреннего опыта личности в психологическом познании, самонаблюдение как психическая призма сознания и основной метод психологии, особенности познавательного процесса объективных и психологических компонентов во внутреннем мире, дифференциация предмета психологического, психофизиологического и метафизического познания, влияние самонаблюдения как фактора активности на содержание и процессуаль-ность явлений психической жизни, несостоятельность идеи эффективного произвольного вмешательства во внутренний мир личности, основные правила интроспекции.
Abstract
In the modern context of the growing trends towards the annihilation of the soul and the abolition of the moral values of science, the author considers the special features of the scientific and moral approach to understanding
