1. Introduction.

In recent years, various programs of education in primary grades have been developed and implemented in practice. In some cases, alternative education differs from standard primary education mainly in the content of the curriculum (for example, author’s schools, the developmental education system of D. B. Elkonin — V. V. Davydov). In other cases, the difference from a regular primary school concerns mainly the conditions of education (small class sizes, a special regime of alternating study and play, etc.).

The variety of forms of primary education requires constant monitoring of the mental development of students. Such monitoring should be based on certain psychological ideas about the essence and uniqueness of the development of thinking in primary school age.

According to the concept put forward by V. V. Davydov [1], during education in primary school, a change in the type of thinking occurs. Children move from an empirical (non-generalized) approach to solving problems to a theoretical (generalized) one. In the first case, similar, uniform problems are considered by the child as having no internal relationship. As a result, most or even all of the problems are solved unsuccessfully. In the second case, such a real relationship of problems (the general principle of their construction and solution) is revealed. Therefore, all problems are solved successfully.

2. Materials and methods.

Based on the above-mentioned ideas about the uniqueness of different approaches to solving similar problems, requirements were developed for an experimental situation designed to determine the characteristics of the method for solving problems as a whole, i.e. to clarify how (empirically or theoretically) the proposed problems were solved.

First, the subject must be offered not one, but several problems to solve.

Second, these problems must have a common principle of construction and solution.

Third, their conditions must differ in external, directly perceived features.

These requirements are met, for example, by a technique that includes the following problems:

1. Transform the sequence of letters H, K, P, T into the sequence K, H, T, P in 2 steps.
2. Transform the sequence L, М, F, С into the sequence М, L, С, F in 2 actions.
3. Transform the sequence G, S, P, В into the sequence S, G, В, P in 2 actions.

The transformation of one sequence of letters into another is carried out in each task by mutually exchanging the places of any two letters. For example, the sequence Р, В, К can be transformed into the sequence К, В, Р in 1 action, by exchanging the letters Р and K.

If the child solved 1, 2 or all 3 problems incorrectly, it means that he did not discover their internal relationship and, therefore, acted empirically, i.e., when solving each problem, he deployed the search actions anew. The internal relationship of these problems, the general principle of their construction and solution is that in each problem only adjacent letters are mutually exchanged.

If the child solved all the problems correctly, it was considered that he had found the general principle of their solution and acted theoretically, i.e. deployed search actions only to solve the first problem [2,3],

According to the concept under consideration, the following main components are distinguished within the framework of any method of solving problems: a mental action associated with the analysis of the conditions of the problems (analysis); a mental action associated with the child’s appeal to his own actions to solve the problem (reflection); a mental action associated with the construction of a program of steps to achieve the desired result (planning).

2.1. Characteristics of types of analysis of problem conditions.

In some cases, analysis as a mental action associated with the analysis of the conditions of the problems is characterized by the fact that the data contained in the conditions and their relationships are considered equivalent, equally important for a successful solution. This is a formal, dissecting analysis, characteristic of the empirical method of solving problems.

In other cases, the analysis of the conditions of the problem is connected not only with the consideration of the data contained in the conditions and their relationships, but also, most importantly, with the clarification of their role in a successful solution: what of them is essential and necessary, and what is inessential and accidental. This is a meaningful, comprehending analysis, characteristic of the theoretical method of solving problems.

In accordance with these ideas, a general scheme of a two-part experimental situation was developed, intended to determine the characteristics of analysis in solving problems.

In the first part of this situation, the subject was asked to analyze the solution of a sample problem. In the second, he was required to solve a series of problems similar (of the same type) to the sample problem. It was believed that successful solution of the problems in this series indicates an understanding of the principle on the basis of which these problems are constructed and which is specifically presented in the sample problem.

The selection of problems in the series met the following requirements: 1) the conditions of the problems should differ in external, directly perceived features; 2) the number of these features (as inessential moments in solving problems) should increase towards the last problem in the series; 3) there should be at least two problems with the same number of inessential moments; 4) the first problem in the series should be as easy (in terms of the number of actions required for the solution and the number of inessential moments) as the sample problem.

For example, the following technique meets these requirements. First, the subject is offered a solution to the sample problem: “In the word CAMP, the letters were rearranged so that PAMC was obtained.” Then he is asked to solve a series of problems: “What will happen if the letters in the words FEET, GIRL, CITY, BEST, HELP,SKIN” are rearranged in the same way?”

If the child solved all the problems correctly, then it can be assumed that when analyzing the conditions of the sample problem, he performed a comprehending analysis and discovered the principle of the solution — a mirror correspondence of the places of the same letters in the original and derivative words: the first letter becomes the last, and the last — the first.

If the problems were solved incorrectly, it is considered that when analyzing the conditions of the sample problem, he carried out a dissecting, empirical, formal analysis [2,3].

2.2. Characteristics of types of reflection.

 Reflection as a mental action associated with the child’s appeal to his own actions to solve a problem can occur in different ways.

Sometimes the noted appeal is characterized only by the identification of specific operations in the actions. This is an external, formal reflection, characteristic of the empirical method of solving problems.

In other cases, the child’s attention to his own actions is connected not only with the selection, but also, most importantly, with the generalization of specific operations, the identification of a common method of their implementation. This is an internal, meaningful reflection, characteristic of the theoretical method of solving problems.

In accordance with these ideas, a general scheme of a two-part experimental situation was developed to determine the characteristics of reflection in solving problems. In the first part of this situation, the subject was asked to solve several problems. Then, if they were successfully solved, these problems had to be grouped. It was believed that by the nature of the grouping, one can objectively judge the implementation of internal reflection, since only in this case are the problems united on the basis of the common method of their solution.

The selection of problems in the first part met the following requirements: 1) the problems should belong not to one, but to two classes. This means that some of the problems are solved on the basis of one principle, and some — using another; 2) the conditions of the problems should differ in external, directly perceived features.

For example, the following technique meets these requirements. First, the subject is asked to solve 3 problems related to the transformation of one sequence of letters into another by mutually swapping any two letters.

1. Transform the sequence P, T, H, V into the sequence T, P, V, H in 2 steps.
2. Transform the sequence M, K, S, D into the sequence D, S, K, M in 2 steps.
3. Transform the sequence F, B, L, J into the sequence B, F, J, L in 2 steps.

After successfully solving all three problems, they are asked to group them. To do this, you need to choose and justify one of the following five opinions about the problems:

1) all problems are different;

2) all problems are similar;

3) the first and second problems are similar, and the third is different from them;

4) the first and third problems are similar, and the second is different from them;

5) the second and third problems are similar, and the first is different from them.

In this method, the problems are selected in such a way that the first and third belong to the same class. Their general principle of construction and solution is that the transformation of one sequence of letters into another is based on the mutual exchange of places of only adjacent letters.

The solution of the second problem is based on another principle: the transformation of one sequence of letters into another is based on the mutual exchange of places of such letters that occupy a mirror position in both sequences — the first letter becomes the last, and the last — the first.

 Thus, if, when grouping successfully solved problems, the second problem was singled out (as having a different solution principle), then it was considered that the child carried out an internal, meaningful reflection associated with the generalization of the solution method.

If the child indicated that all the problems were different (because the letters were different everywhere) or that they were all the same, similar (because the letters had to be rearranged everywhere), then it can be assumed that in all these cases only external, formal reflection was carried out, not related to the generalization of the solution method [2,3]

2.3. Characteristics of types of planning.

Planning as a mental action related to the construction of a program of steps to achieve the desired result is carried out in different ways. In some cases, it is characterized by the fact that each step within a certain sequence of actions is planned and carried out separately. This is formal, partial planning, characteristic of the empirical method of solving problems.

In other cases, the entire sequence of steps is planned at once, before the first step is completed. This is meaningful, holistic planning, characteristic of the theoretical method of solving problems.

In accordance with these ideas, a general scheme of a two-part experimental situation was developed, designed to determine the characteristics of planning.

In the first part of this situation, the subject is asked to master some simple action. In the second part, he is required to solve several problems on constructing a sequence of these actions.

In our studies [2], it was established that the selection of problems in the second part of this situation should meet the following requirements.

First, the sequence of executive actions should gradually increase from the first problem to the last. Second, there should be at least two problems with the same number of executive actions. Third, and most importantly, the problems should not have a common solution principle.

For example, the following technique meets these requirements. First, the subject is asked to solve 3 training problems:

1. Which letters should be swapped so that in the sequence P, B, H the letters are arranged in the same way as in the sequence B, P, H?
2. Which letters should be swapped so that in the sequence K, P, S the letters are arranged in the same way as in the sequence K, S, P?
3. Which letters need to be swapped so that in the sequence L, G, М the letters are arranged in the same way as in the sequence М, G, K?

Then the subject was asked to solve, for example, 6 basic problems related to the transformation of one sequence of letters into another by means of a simple action of mutually swapping any two letters.

1. Transform the sequence К, G, Н, Р, D into the sequence G, К, Н, D, Р in 2 actions.
2. Transform the sequence С, P, К, В, Н into the sequence Н, В, К, P, С in 2 actions.
3. Transform the sequence М, Р, К, L, B into the sequence Р, К, L, М, B in 3 actions.
4. Transform the sequence Т, Х, P, С, В into the sequence С, Х, Т, В, P in 3 actions.
5. Transform the sequence P, N, Z, F, M into the sequence F, P, M, Z, N in 4 actions.
6. Transform the sequence P, G, L, S, B into the sequence L, B, S, P, G in 4 actions.

If the child solved all the problems correctly, then it can be considered that he carried out holistic planning, since the successful solution of problems in 3 and especially in 4 actions presupposes (as observations of subjects in individual experiments have shown) a preliminary plan for the entire sequence of actions.

 If the child coped with the problems in only 2 actions, then it can be assumed that when solving the problems of this series he carried out only partial planning [2,3].

2.4. General characteristics of the diagnostics of theoretical thinking.

The diagnostics of theoretical thinking in primary school students includes two main approaches. The first approach is associated with determining the characteristics of the method of solving problems as a whole, i.e. with finding out how (empirically or theoretically) the child acted when solving these problems. The second approach is associated with determining the characteristics of individual components of the problem-solving method, i.e. with finding out what kind of analysis (dissecting or comprehending), what kind of reflection (external or internal), what kind of planning (partial or holistic) took place in the actions of the subject.

The first approach can be characterized as an incomplete diagnosis of theoretical thinking, and the second — as its complete diagnosis. In this case, the degree of completeness is determined by the number of diagnosed components of the problem-solving method. The minimum degree of incomplete diagnosis is associated with finding out the characteristics of one component (analysis, reflection or planning), the average degree of incomplete diagnosis takes into account the characteristics of two components, and complete diagnosis concerns all three components.

In a number of our studies (see, for example, [4]), it was also shown that when solving problems in a theoretical way, the ability to carry out comprehending analysis, internal reflection and holistic planning depends on the form in which the problems are proposed to be solved.

When studying this pattern, it was established that the more specific the form of action in solving problems (object-active in relation to visual-figurative and especially to verbal-symbolic), the more often the theoretical method and its components are used.

Based on these data, we can talk about the possibility of an expanded and collapsed diagnosis of the theoretical method of solving problems and its individual components. In the first case, the subject is asked to solve problems in all forms of action (object, figurative and verbal), and in the second — only in one or two forms of action.

Distinguishing between complete and incomplete diagnostics of the method of solving problems is fundamentally important for improving the control of the development of theoretical thinking in primary school students. Thus, the successful solution of problems in a visual-figurative form of action by two groups of children or two students does not provide sufficient grounds to assert that they have the same level of development of theoretical thinking, since the results may change when solving problems in a verbal-symbolic form.

A generalization of the results of our research allows us to conclude that the condensed and expanded diagnostics of theoretical thinking can be complete and incomplete.

Complete expanded diagnostics has one implementation option, since it is necessary to determine the characteristics of each component of the method of solving problems in all forms of action.

Incomplete expanded diagnostics and incomplete condensed diagnostics include many options, since these types of diagnostics are based on a combination of a certain number of components of theoretical thinking with a certain number of forms of action when solving problems. The possibilities of such a combination are presented in Table 1.

Table 1

Composition of complete and expanded diagnostics of theoretical thinking.

Using this table, you can assess how fully and extensively the diagnostics of theoretical thinking in primary school students was carried out in each specific case.

2.5. Diagnostics of types of analysis. Based on the table presented above, an incomplete diagnostics of theoretical thinking was carried out, in particular, a group diagnostics of types of analysis in solving problems in verbal-symbolic form in two contingents of primary school students. The first contingent consisted of second-graders (29 people), the second — third-graders (31 people).

At the beginning of the diagnostic lesson, each student received a Form with the conditions of 10 plot-logical problems of the author’s method «Presence — Absence».

                                                  FORM

1. Andrey and Vova collected stamps: someone — English, someone — German. Andrey had English stamps. Which of the children collected German stamps?
2. Masha and Ira were reading: someone was reading a book, someone was reading a magazine. Masha was reading a book. Who didn’t read a magazine?
3. Yulia and Nadya bought clothes in a store: someone was buying a jacket, someone was buying a coat. Nadya didn’t buy a coat. What did Yulia buy?
4. Dasha and Inna were knitting: someone was knitting a scarf, someone was knitting a hat. Inna was knitting a hat. Who didn’t knit a scarf?
5. Dmitry and Evgeny were guessing riddles: someone was guessing easy riddles, someone was guessing difficult riddles. Evgeny got easy riddles. Who got difficult riddles?
6. Lyova and Vadim were playing checkers. One of them won twice, someone won three times. Misha lost four times, Petya lost six times. Kolya won once, Vadim won three times. How many times did Lyova win?
7. Petya and Sasha competed in running. One of them covered the distance in a minute, another in two minutes. Yura ran in three minutes, Kolya in five, Vitya in four, and Sasha in one. How many minutes did Petya run in?
8. Misha and Vova had dogs: one had a greyhound, another had a bulldog. Pasha had a shepherd, Dima had a Great Dane, Misha had a poodle, Vitya had a husky, and Vova had a greyhound. What kind of dog did Misha have?
9. Sasha and Kolya were sailing down the river: one in a boat, another in a kayak. Vitya was sailing on a motorboat, Petya was sailing on a steamboat, Igor was sailing on a raft, Kolya was sailing on a boat. Who was sailing on a kayak?
10. Kostya and Gena left the house: one of them put on a cap, the other — a baseball cap. Igor put on a baseball cap, Vova — a cap, Oleg — a beret, Gena — a baseball cap. What did Kostya put on?

                                             *     *     *

The proposed problems are divided into two groups. The first group includes problems 1–5. These are simple problems, since they include two informative judgments on the basis of which a conclusion should be drawn. The second group includes problems 6 –10. These are complex problems, since they include four informative judgments for drawing a conclusion.

3. Results.

Based on the results of solving the proposed 10 problems, five groups of children were identified. Group A included children who correctly solved all 10 problems. Group B included children who correctly solved problems 1-5 and incorrectly solved one or two complex problems. Group C included children who correctly solved only simple problems and incorrectly solved all complex problems. Group D included children who correctly solved only one or two simple problems and incorrectly solved all complex problems. Group E included children who incorrectly solved all 10 problems.

The number of children who made up groups A, B, C, D and E in each cohort is presented in Table 2.

Table 2

The number of groups A, B, C, D and E for second-graders and third-graders (in %).

The data presented in Table 2 allow us to note the following.

Firstly, third-graders completed the proposed tasks more successfully than second-graders. This statement follows from the fact that all the problems were solved correctly by 19.4% of third-graders and 13.8% of second-graders, and all the problems were solved incorrectly by 12.9% of third-graders and 13.8% of second-graders.

It is important to note that the correct solution of all the problems, as shown by preliminary individual experiments, means that in this case a meaningful, comprehending analysis of the conditions of the problems was carried out. And, accordingly, for the children who made up groups B, C, D and D, a meaningful analysis was not carried out, since they were not able to solve all the complex problems correctly.

Secondly, the ratio of the number of children who solved all the problems correctly and children who solved all the problems incorrectly also indicates greater success in solving the proposed problems of third-graders, compared to second-graders: in the third grade, the noted ratio is 19.4% and 12.9%, and in the second grade — 13.8% and 13.8%. Thus, in the third grade, there are more children who solved all the problems correctly than children who solved all the problems incorrectly, and in the second grade, the number of noted children is the same.

At the same time, it is necessary to note the general trend in the change in the results of the five groups of students in the third and second grades. This trend is manifested in the fact that there are more children in group B than in group A, more in group C than in group B, fewer in group D than in group C and group B, and fewer in group E than in group D, group C and group B.

4. Conclusion.

The conducted research was aimed at studying the possibilities of diagnosing the development of thinking in primary school students. The basis for such diagnostics was the concept of two types of thinking – theoretical and empirical – proposed by S.L. Rubinstein [ 5 ] and developed by V.V. Davydov        [ 1 ].

Within the framework of this concept, the components of theoretical thinking were considered: comprehending (content) analysis associated with the identification of essential relationships in the conditions of problems, internal (content) reflection associated with a person’s understanding of the methods of his own actions when solving a problem, holistic (content) planning. At the same time, experimental situations for diagnosing the noted components of theoretical thinking were characterized.

Based on the consideration of the content of the components of theoretical thinking, ideas were developed, firstly, about complete and incomplete diagnostics of theoretical thinking when solving problems and, secondly, about its expanded and collapsed diagnostics.

In the first case, this refers to determining the formation of either all three components of theoretical thinking — analysis, reflection and planning (complete diagnostics), or any two or one component out of three (incomplete diagnostics).

In the second case, we are talking about the forms of action when solving the proposed problems. Either diagnostics is aimed at determining the possibilities of solving problems in a theoretical way in all three forms of action — subject-active, visual-figurative and verbal-symbolic (expanded diagnostics), or in any two or one form of action out of three (condensed diagnostics).

Based on the characteristics of the types of diagnostics of theoretical thinking, the «Presence — Absence» technique was developed, intended for incomplete and condensed diagnostics of types of analysis in solving plot-logical problems.

As a result of group experiments with second- and third-grade students who solved problems using the «Presence — Absence» technique, children were identified who solved problems using comprehending, content analysis. In the second grade, there were 13.8% of such children, in the third grade — 19.4%. The remaining students in each class were not able to solve all 10 proposed problems correctly. This fact indicates that such students did not use content analysis in solving problems.

In general, the study allowed us to characterize the content and forms of diagnostics of theoretical thinking in primary school students, develop a technique for determining the formation of content analysis in children when solving plot-logical problems and determine the number of second- and third-grade students using content analysis.

In the future, it is planned to conduct research on the formation of content analysis using methods that include tasks of different content: spatial-combinatorial, route and «comparison».