The work experience of a primary school teacher of MOAU "Secondary School No. 15 of Orsk" Vinnikova L.A.
Development of mathematical abilities of primary school students in the process of solving text problems.
The work experience of a primary school teacher of MOAU "Secondary School No. 15 of Orsk" Vinnikova L.A.
Compiled by: Grinchenko I. A., methodologist of the Orsk branch of IPKiPPRO OGPU
Theoretical base of experience:
- theories of developmental learning (L.V. Zankov, D.B. Elkonin)
- psychological and pedagogical theories of R. S. Nemov, B. M. Teplov, L. S. Vygotsky, A. A. Leontiev, S. L. Rubinstein, B. G. Ananiev, N. S. Leites, Yu. D. Babaeva, V. S. Yurkevich about the development of mathematical abilities in the process of specially organized educational activities.
- Krutetsky V. A. Psychology of mathematical abilities of schoolchildren. M.: Publishing house. Institute of Practical Psychology; Voronezh: Publishing House of NPO MODEK, 1998. 416 p.
- The development of mathematical abilities of students is consistent and purposeful.
Using the material of the textbook "My Mathematics" by T.E. Demidova, S. A. Kozlova, A. P. Tonkikh allows to identify and develop the mathematical and creative abilities of students, to form a steady interest in mathematics.
Relevance:
In elementary school age there is a rapid development of the intellect. The possibility of developing abilities is very high. The development of mathematical abilities of younger students today remains the least developed methodological problem. Many educators and psychologists are of the opinion that the elementary school is a “high-risk zone”, since it is at the stage of primary education, due to the primary orientation of teachers to the assimilation of knowledge, skills and abilities, that many children block the development of abilities. It is important not to miss this moment and find effective ways to develop the abilities of children. Despite the constant improvement of the forms and methods of work, there are significant gaps in the development of mathematical abilities in the process of solving problems. This can be explained by the following reasons:
Excessive standardization and algorithmization of problem solving methods;
Insufficient inclusion of students in the creative process of solving the problem;
The imperfection of the teacher's work in developing the ability of students to conduct a meaningful analysis of the problem, put forward hypotheses for planning a solution, rationally determining the steps.
The relevance of the study of the problem of developing the mathematical abilities of younger students is explained by:
Society's need for creative thinking people;
Insufficient degree of development in practical methodological terms;
The need to generalize and systematize the experience of the past and present in the development of mathematical abilities in a single direction.
As a result of purposeful work on the development of mathematical abilities in students, the level of academic performance and quality of knowledge increases, interest in the subject develops. .
Fundamental principles of the pedagogical system.
Progress in the study of the material at a rapid pace.
The leading role of theoretical knowledge.
Training at a high level of difficulty.
Work on the development of all students.
Students' awareness of the learning process.
Development of the ability and need to independently find a solution to previously unseen educational and extracurricular tasks.
Conditions for the emergence and formation of experience:
Erudition, high intellectual level of the teacher;
Creative search for methods, forms and techniques that provide an increase in the level of mathematical abilities of students;
The ability to predict the positive progress of students in the process of using a set of exercises to develop mathematical abilities;
The desire of students to learn new things in mathematics, to participate in olympiads, competitions, intellectual games.
Essence experience is the activity of the teacher to create conditions for the active, conscious, creative activity of students; improving the interaction between the teacher and students in the process of solving text problems; the development of mathematical abilities of schoolchildren and the education of their industriousness, efficiency, exactingness to themselves. By identifying the causes of success and failure of students, the teacher can determine what abilities or inability affect the activities of students and, depending on this, purposefully plan further work.
To carry out high-quality work on the development of mathematical abilities, the following innovative pedagogical products of pedagogical activity are used:
Optional course "Non-standard and entertaining tasks";
Use of ICT technologies;
A set of exercises for the development of all components of mathematical abilities that can be formed in primary grades;
A cycle of classes on the development of the ability to reason.
Tasks contributing to the achievement of this goal:
Constant stimulation and development of the student's cognitive interest in the subject;
Activation of the creative activity of children;
Development of the ability and desire for self-education;
Cooperation between the teacher and the student in the learning process.
Extracurricular work creates an additional incentive for the creativity of students, the development of their mathematical abilities.
Novelty of experience thing is:
- the specific conditions of activity that contribute to the intensive development of the mathematical abilities of students have been studied, reserves for increasing the level of mathematical abilities for each student have been found;
- the individual abilities of each child are taken into account in the learning process;
- identified and described in full the most effective forms, methods and techniques aimed at developing the mathematical abilities of students in the process of solving word problems;
- a set of exercises for the development of the components of the mathematical abilities of primary school students is proposed;
- requirements for exercises have been developed that, by their content and form, would stimulate the development of mathematical abilities.
Productivity.
The development of mathematical abilities of students is achieved through consistent and purposeful work by developing methods, forms and techniques aimed at solving text problems. Such forms of work provide an increase in the level of mathematical abilities of most students, increase productivity and creative direction of activity. The majority of students increase the level of mathematical abilities, develop all the components of mathematical abilities that can be formed in the primary grades. Students show a steady interest and a positive attitude towards the subject, a high level of knowledge in mathematics, successfully complete tasks of the Olympiad and creative nature.
Labor intensity.
The complexity of the experience is determined by its rethinking from the standpoint of the creative self-realization of the child's personality in educational and cognitive activity, the selection of optimal methods and techniques, forms, means of organizing the educational process, taking into account the individual creative capabilities of students.
Possibility of implementation.
Experience solves both narrow methodological and general pedagogical problems. The experience is interesting for primary and secondary school teachers, university students, parents and can be used in any activity that requires originality, unconventional thinking.
Teacher work system.
The teacher's work system consists of the following components:
1. Diagnosis of the initial level of development of mathematical abilities of students.
2. Predicting the positive results of students' activities.
3. Implementation of a set of exercises to develop mathematical abilities in the educational process within the framework of the School 2100 program.
4. Creation of conditions for inclusion in the activities of each student.
5. Fulfillment and compilation by students and the teacher of tasks of an Olympiad and creative nature.
The system of work that helps to identify children who are interested in mathematics, teach them to think creatively and deepen their knowledge includes:
Preliminary diagnostics to determine the level of mathematical abilities of students, making long-term and short-term forecasts for the entire course of study;
The system of mathematics lessons;
Diverse forms of extracurricular activities;
Individual work with schoolchildren capable of mathematics;
Independent work of the student himself;
Participation in olympiads, competitions, tournaments.
Work efficiency.
With 100% progress, a consistently high quality of knowledge in mathematics. Positive dynamics of the level of mathematical abilities of students. High educational motivation and motivation for self-realization in the performance of research work in mathematics. Increase in the number of participants in Olympiads and competitions at various levels. Deeper awareness and assimilation of program material at the level of application of knowledge, skills in new conditions; increased interest in the subject. Increasing the cognitive activity of schoolchildren in the classroom and extracurricular activities.
Leading pedagogical idea experience is to improve the process of teaching schoolchildren in the process of classroom and extracurricular work in mathematics for the development of cognitive interest, logical thinking, and the formation of students' creative activity.
Perspective of experience is explained by its practical significance for increasing the creative self-realization of children in educational and cognitive activities, for the development and realization of their potential.
Experience technology.
Mathematical abilities are manifested in the speed with which, how deeply and how firmly people learn mathematical material. These characteristics are most easily detected in the course of solving problems.
The technology includes a combination of group, individual and collective forms of learning activity of students in the process of solving problems and is based on the use of a set of exercises to develop the mathematical abilities of students. Skills develop through activity. The process of their development can go spontaneously, but it is better if they develop in an organized learning process. Conditions are created that are most favorable for the purposeful development of abilities. At the first stage, the development of abilities is characterized to a greater extent by imitation (reproductivity). Gradually, elements of creativity, originality appear, and the more capable a person is, the more pronounced they are.
The formation and development of the components of mathematical abilities takes place already in the primary grades. What characterizes the mental activity of schoolchildren capable of mathematics? Capable students, perceiving a mathematical problem, systematize the given values in the problem, the relationship between them. A clear holistically dissected image of the task is created. In other words, capable students are characterized by a formalized perception of mathematical material (mathematical objects, relations and actions), associated with a quick grasp of their formal structure in a specific task. Pupils with average abilities, when perceiving a task of a new type, determine, as a rule, its individual elements. It is very difficult for some students to comprehend the connections between the components of the task, they hardly grasp the totality of the diverse dependencies that make up the essence of the task. To develop the ability to formalize the perception of mathematical material, students are offered exercises [Appendix 1. Series I]:
1) Tasks with an unformulated question;
2) Tasks with an incomplete composition of the condition;
3) Tasks with redundant composition of the condition;
4) Work on the classification of tasks;
5) Drawing up tasks.
The thinking of capable students in the process of mathematical activity is characterized by fast and broad generalization (each specific problem is solved as a typical one). For the most capable students, such a generalization occurs immediately, by analyzing one individual problem in a series of similar ones. Capable students easily move on to solving problems in literal form.
The development of the ability to generalize is achieved by presenting special exercises [Appendix 1. Series II.]:
1) Solving problems of the same type; 2) Solving problems of various types;
3) Solving problems with a gradual transformation from a concrete to an abstract plan; 4) Drawing up an equation according to the condition of the problem.
The thinking of capable students is characterized by a tendency to think in folded conclusions. For such students, the curtailment of the reasoning process is observed after solving the first problem, and sometimes after the presentation of the problem, the result is immediately given. The time to solve the problem is determined only by the time spent on the calculations. A folded structure is always based on a well-founded reasoning process. Average students generalize the material after repeated exercises, and therefore the curtailment of the reasoning process is observed in them after solving several tasks of the same type. In students with low ability, curtailment can begin only after a large number of exercises. The thinking of capable students is distinguished by great mobility of thought processes, a variety of aspects in the approach to solving problems, easy and free switching from one mental operation to another, from direct to reverse thought. For the development of flexibility of thinking, exercises are proposed [Appendix 1. Series III.]
1) Tasks that have several ways to solve.
2) Solving and compiling problems that are inverse to this one.
3) Solving problems in reverse.
4) Solving problems with an alternative condition.
5) Solving problems with uncertain data.
It is typical for capable students to strive for clarity, simplicity, rationality, economy (elegance) of the solution.
The mathematical memory of capable students is manifested in the memorization of types of problems, methods for solving them, and specific data. Able students are distinguished by well-developed spatial representations. However, when solving a number of problems, they can do without relying on visual images. In a sense, logicality replaces "figurativeness" for them; they do not experience difficulties in operating with abstract schemes. While completing the learning tasks, students at the same time develop their mental activity. So, when solving mathematical problems, the student learns analysis, synthesis, comparison, abstraction and generalization, which are the main mental operations. Therefore, for the formation of abilities in educational activities, it is necessary to create certain conditions:
A) positive motives for learning;
B) students' interest in the subject;
C) creative activity;
D) a positive microclimate in the team;
D) strong emotions;
E) providing freedom of choice of actions, variability of work.
It is more convenient for the teacher to rely on some purely procedural characteristics of the activity of capable children. Most children with mathematical abilities tend to:
- Increased propensity for mental action and a positive emotional response to any mental load.
- The constant need to renew and complicate the mental load, which leads to a constant increase in the level of achievements.
- The desire for independent choice of affairs and planning of their activities.
- Increased performance. Prolonged intellectual loads do not tire this child, on the contrary, he feels good in a situation where there is a problem.
In the search for ways to more effectively use the structure of lessons for the development of mathematical abilities, the form of organization of educational activities of students in the lesson is of particular importance. In our practice we use frontal, individual and group work.
In the frontal form of work, students perform a common activity for all, compare and summarize its results with the whole class. Due to their real capabilities, students can make generalizations and conclusions at different levels of depth. The frontal form of organization of learning is implemented by us in the form of a problematic, informational and explanatory-illustrative presentation and is accompanied by reproductive and creative tasks. All textual logical tasks, the solution of which must be found using a chain of reasoning, proposed in the 2nd grade textbook, are analyzed frontally in the first half of the year, since their independent solution is not available to all children of this age. Then these tasks are offered for independent solution to students with a high level of mathematical abilities. In the third grade, logical problems are first given to all students for independent solution, and then the proposed options are analyzed.
The application of acquired knowledge in changed situations is best organized using individual work. Each student receives a task for independent completion, specially selected for him in accordance with his training and abilities. There are two types of individual forms of organizing tasks: individual and individualized. The first one is characterized by the fact that the student’s activity in fulfilling tasks common to the whole class is carried out without contact with other students, but at the same pace for all, the second allows using differentiated individual tasks to create optimal conditions for the realization of the abilities of each student. In our work, we use the differentiation of educational tasks according to the level of creativity, difficulty, volume. When differentiated by the level of creativity, the work is organized as follows: students with a low level of mathematical abilities (Group 1) are offered reproductive tasks (work according to the model, performing training exercises), and students with an average (Group 2) and high level (Group 3) are offered creative tasks. tasks.
- (Grade 2. Lesson No. 36. Problem No. 7. 36 yachts participated in the race of sailing ships. How many yachts reached the finish line if 2 yachts returned to the start due to a breakdown, and 11 due to a storm?
Task for the 2nd group. Solve the problem in two ways. Come up with a problem with a different plot so that the solution does not change.
Task for the 3rd group. Solve the problem in three ways. Make a problem inverse to this one and solve it.
It is possible to offer productive tasks to all students, but at the same time, children with low abilities are given tasks with elements of creativity in which they need to apply knowledge in a changed situation, and the rest are given creative tasks to apply knowledge in a new situation.
- (Grade 2. Lesson No. 45. Task No. 5. There are 75 budgerigars in three cages. There are 21 parrots in the first cage, 32 parrots in the second. How many parrots are in the third cage?
Task for the 2nd group. Solve the problem in two ways. Come up with a problem with a different plot, but so that its solution does not change.
Task for the 3rd group. Solve the problem in three ways. Change the question and the condition of the problem so that the data on the total number of parrots become redundant.
Differentiation of educational tasks according to the level of difficulty (the difficulty of a task is a combination of many subjective factors depending on personality characteristics, for example, such as intellectual capabilities, mathematical abilities, degree of novelty, etc.) involves three types of tasks:
1. Tasks, the solution of which consists in the stereotypical reproduction of learned actions. The degree of difficulty of the tasks is related to how complex the skill of reproducing actions is and how firmly it is mastered.
2. Tasks, the solution of which requires some modification of the learned actions in changing conditions. The degree of difficulty is related to the number and heterogeneity of elements that must be coordinated along with the features of the data described above.
3. Tasks, the solution of which requires the search for new, still unknown methods of action. Tasks require creative activity, a heuristic search for new, unknown patterns of action or an unusual combination of known ones.
Differentiation in terms of the volume of educational material assumes that all students are given a certain number of tasks of the same type. At the same time, the required volume is determined, and for each additionally completed task, for example, points are awarded. Creative tasks can be offered for compiling objects of the same type and it is required to compose the maximum number of them for a certain period of time.
- Who will make more tasks with different content, the solution of each of which will be a numerical expression: (54 + 18): 2
When solving problems independently, individual work is also effective. The degree of independence of such work is different. First, students perform tasks with a preliminary and frontal analysis, imitating a model, or according to detailed instruction cards. [Annex 2]. As learning skills are mastered, the degree of independence increases: students (especially with an average and high level of mathematical abilities) work on general, non-detailed tasks, without the direct intervention of a teacher. For individual work, we offer worksheets developed by us on topics, the deadlines for which are determined in accordance with the desires and capabilities of the student [Appendix 3]. For students with a low level of mathematical abilities, a system of tasks is compiled, which contains: samples of solutions and tasks to be solved on the basis of the studied sample, various algorithmic prescriptions; theoretical information, as well as all kinds of requirements to compare, compare, classify, generalize. [Appendix 4, fragment of lesson No. 1] Such an organization of educational work enables each student, by virtue of his abilities, to deepen and consolidate the knowledge gained. The individual form of work somewhat limits the communication of students, the desire to transfer knowledge to others, participation in collective achievements, so we use a group form of organizing educational activities. [Appendix 4. Fragment of lesson No. 2]. Tasks in the group are carried out in a way that takes into account and evaluates the individual contribution of each child. The size of the groups is from 2 to 4 people. The composition of the group is not permanent. It varies depending on the content and nature of the work. The group consists of students with different levels of mathematical abilities. Often we are preparing students with a low level of mathematical ability in extracurricular activities for the role of consultants in the lesson. The fulfillment of this role is sufficient for the child to feel himself the best, his significance. The group form of work makes clear the abilities of each student. In combination with other forms of education - frontal and individual - the group form of organizing the work of students brings positive results.
Computer technologies are widely used in mathematics lessons and optional courses. They can be included at any stage of the lesson - during individual work, with the introduction of new knowledge, their generalization, consolidation, for the control of ZUNs. For example, when solving problems for obtaining a certain amount of liquid from a large or infinite volume of a vessel, reservoir or source using two empty vessels, setting different volumes of vessels, various required amounts of liquid, you can get a large set of tasks of different levels of complexity for their hero " Overflows". The volume of liquid in the conditional vessel A will correspond to the volume of the drained liquid, the volumes B and C will correspond to the given volumes according to the condition of the problem. An action denoted by a single letter, for example, B, means filling a vessel from a source.
Task. Breeding instant mashed potatoes "Green Giant" requires 1 liter of water. How, having two vessels with a capacity of 5 and 9 liters, pour 1 liter of water from a tap?
Children look for a solution to a problem in different ways. They come to the conclusion that the problem is solved in 4 moves.
| № | Action | A | B (9l) | B (5l) |
| 0 | 0 | 0 |
||
| 1 | IN | 0 | 0 | 5 |
| 2 | V-B | 0 | 5 | 0 |
| 3 | IN | 0 | 5 | 5 |
| 4 | V-B | 0 | 9 | 1 |
For the development of mathematical abilities, we use the wide possibilities of auxiliary forms of organization of educational work. These are optional classes on the course "Non-standard and entertaining tasks", home independent work, individual lessons on the development of mathematical abilities with students of low and high levels of their development. In optional classes, part of the time was devoted to learning how to solve logical problems according to the method of A. Z. Zak. Classes were held once a week, the duration of the lesson was 20 minutes and contributed to an increase in the level of such a component of mathematical abilities as the ability to correct logical reasoning.
In the classroom of the optional course "Non-standard and entertaining tasks", a collective discussion is held on solving a problem of a new type. Thanks to this method, children develop such an important quality of activity as awareness of their own actions, self-control, the ability to report on the steps taken in solving problems. Most of the time in the classroom is occupied by students independently solving problems, followed by a collective verification of the solution. In the classroom, students solve non-standard tasks, which are divided into series.
For students with a low level of development of mathematical abilities, individual work is carried out after school hours. The work is carried out in the form of a dialogue, instruction cards. With this form, students are required to speak out loud all the ways of solving, searching for the right answer.
For students with a high level of ability, after-hours consultations are provided to meet the needs for in-depth study of the issues of the mathematics course. Classes in their form of organization are in the nature of an interview, consultation or independent performance of tasks by students under the guidance of a teacher.
For the development of mathematical abilities, the following forms of extracurricular work are used: olympiads, competitions, intellectual games, thematic months in mathematics. Thus, during the thematic month "Young Mathematician", held in elementary school in November 2008, the students of the class participated in the following activities: the release of mathematical newspapers; competition "Entertaining tasks"; exhibition of creative works on mathematical topics; meeting with the associate professor of the department of SP and PPNO, defense of projects; Olympiad in mathematics.
Mathematical Olympiads play a special role in the development of children. This is a competition that allows capable students to feel like real mathematicians. It was during this period that the first independent discoveries of the child take place.
Extracurricular activities are held on mathematical topics: "KVN 2 + 3", the Intellectual game "Choosing an heir", the Intellectual marathon, "Mathematical traffic light", "Pathfinders" [Appendix 5], the game "Funny Train" and others.
Mathematical ability can be identified and assessed based on how a child solves certain problems. The very solution of these problems depends not only on abilities, but also on motivation, on existing knowledge, skills and abilities. Making a forecast of the results of development requires knowledge of precisely the abilities. The results of observations allow us to conclude that the prospects for the development of abilities are available for all children. The main thing that should be paid attention to when improving the abilities of children is the creation of optimal conditions for their development.
Tracking the results of research activities:
For the purpose of practical substantiation of the conclusions obtained during the theoretical study of the problem: what are the most effective forms and methods aimed at developing the mathematical abilities of schoolchildren in the process of solving mathematical problems, a study was conducted. Two classes took part in the experiment: experimental 2 (4) "B", control - 2 (4) "C" of secondary school No. 15. The work was carried out from September 2006 to January 2009 and included 4 stages.
Stages of experimental activity
I - Preparatory (September 2006). Purpose: determination of the level of mathematical abilities based on the results of observations.
II - Ascertaining series of experiment (October 2006) Purpose: to determine the level of formation of mathematical abilities.
III - Formative experiment (November 2006 - December 2008) Purpose: to create the necessary conditions for the development of mathematical abilities.
IV - Control experiment (January 2009) Purpose: to determine the effectiveness of forms and methods that contribute to the development of mathematical abilities.
At the preparatory stage, students of the control - 2 "B" and experimental 2 "C" classes were observed. Observations were carried out both in the process of studying new material and in solving problems. For observations, those signs of mathematical abilities that are most clearly manifested in younger students were identified:
1) relatively fast and successful mastery of mathematical knowledge, skills and abilities;
2) the ability to consistently correct logical reasoning;
3) resourcefulness and ingenuity in the study of mathematics;
4) flexibility of thinking;
5) the ability to operate with numerical and symbolic symbols;
6) reduced fatigue during mathematics;
7) the ability to shorten the process of reasoning, to think in collapsed structures;
8) the ability to switch from direct to reverse course of thought;
9) the development of figurative-geometric thinking and spatial representations.
In October, teachers filled out a table of schoolchildren's mathematical abilities, in which they rated each of the listed qualities in points (0-low level, 1-average level, 2-high level).
At the second stage, diagnostics of the development of mathematical abilities was carried out in the experimental and control classes.
For this, the "Problem Solving" test was used:
1. Compose compound problems from these simple problems. Solve one compound problem in different ways, underline the rational one.
2. Read the problem. Read the questions and expressions. Match each question with the correct expression.
IN
a + 18
class 18 boys and a girls.
3. Solve the problem.
In his letter to his parents, Uncle Fyodor wrote that his house, the house of the postman Pechkin and the well were on the same side of the street. From the house of Uncle Fyodor to the house of the postman Pechkin 90 meters, and from the well to the house of Uncle Fyodor 20 meters. What is the distance from the well to the house of the postman Pechkin?
With the help of the test, the same components of the structure of mathematical abilities were checked as during observation.
Purpose: to establish the level of mathematical abilities.
Equipment: student card (sheet).
table 2
The test tests skills and mathematical abilities:
| № Tasks | The skills required to solve the problem. | Abilities manifested in mathematical activity. |
| № 1 | The ability to distinguish the task from other texts. | Ability to formalize mathematical material. |
| № 1, 2, 3, 4 | Ability to write down the solution of the problem, to make calculations. | The ability to operate with numerical and symbolic symbols. |
| № 2, 3 | The ability to write the solution of a problem in an expression. Ability to solve problems in different ways. | Flexibility of thinking, the ability to shorten the process of reasoning. |
| № 4 | Ability to perform the construction of geometric figures. | The development of figurative-geometric thinking and spatial representations. |
At this stage, mathematical abilities have been studied and the following levels have been determined:
Low level: Mathematical ability manifests itself in a general, inherent need.
Intermediate level: abilities appear in similar conditions (according to the model).
High level: creative manifestation of mathematical abilities in new, unexpected situations.
Qualitative analysis of the test showed the main reasons for the difficulty in performing the test. Among them: a) the lack of specific knowledge in solving problems (they cannot determine how many actions the problem is solved, they cannot write down the solution of the problem by the expression (in 2 "B" (experimental) class 4 people - 15%, in 2 "C" class - 3 people - 12%) b) insufficient formation of computational skills (in 2 "B" class 7 people - 27%, in 2 "C" class 8 people - 31%.
The development of mathematical abilities of students is ensured, first of all, by the development of the mathematical style of thinking. To determine the differences in the development of the ability to reason in children, a group lesson was conducted on the material of the diagnostic task “different-same” according to the method of A.Z. Zach. The following levels of reasoning ability have been identified:
High level - tasks #1-10 solved (contain 3-5 characters)
Intermediate level - solved tasks #1-8 (contain 3-4 characters)
Low level - tasks #1 - 4 solved (contain 3 characters)
The following methods of work were used in the experiment: explanatory-illustrative, reproductive, heuristic, problem presentation, research method. In real scientific creativity, the formulation of the problem goes through the problem situation. We strived to ensure that the student independently learned to see the problem, formulate it, explore the possibilities and ways to solve it. The research method is characterized by the highest level of cognitive independence of students. At the lessons, we organized independent work of students, giving them problematic cognitive tasks and assignments of a practical nature.
FRAGMENT OF THE LESSON.
Theme "Dividing the amount by a number" (Grade 3, lesson No. 17)
Purpose: To form ideas about the possibility of using the distribution property of division with respect to addition to rationalize calculations when solving problems.
I. Actualization of knowledge.
II. "Discovery of new knowledge". It is done on the basis of an inciting dialogue, while hypothesizing at the same time.
Students read the text and look at the pictures. The teacher asks questions:
What interesting things have you noticed?
What surprised you?
Children are aware of and formulate the problem, offer opportunities and ways to solve it.
| Teacher (uses prompting dialogue) | Students (formulate the topic of the lesson) |
| Now you will be divided into groups and will solve problem number 1. Write down the solution. Suitable for each group: What other hypotheses are there? Where to start? (Incitement to put forward hypotheses). | Break into groups and start working. After completing the work, the groups hang out on the board and voice hypotheses: 4 + 6: 2 = 5 (c.) - erroneous hypothesis (4 + 6): 3 \u003d 5 (c.) - decisive 4: 2 + 6: 2= 5 (c.) hypotheses |
Based on the analysis of figures and text, the discovery of an algorithm for dividing a sum by a number occurs. The students explain their solutions and compare them with those of the boys. Obviously, Denis' solution came down to the fact that he first gathered all the chickens together (found the sum of the given values), and then seated them in two boxes (divided equally). Kostya's solution boiled down to the fact that
He divided the chickens in such a way that each box got an equal number.
Black and yellow chickens (divided chickens by color).
Working with signed text?
Purpose of the work: primary reflection on the discovered property of actions on numbers; the initial formulation of this property.
Compare your output with the rule in the textbook.
Students suggest replacing numbers with letters and using formulas to solve similar problems.
Confirmation of their hypotheses and the final formulation of the algorithm for dividing the sum by a number.
III. Primary fastening.
Front work. 1. Task number 2, p. 44 2. Task number 3, p. 45.
We consider 3 solutions: 12: 3 + 9: 3; 9:3 + 12:3; (12 + 9) : 3
IV. Independent work in pairs. Task number 4, p. 45. After checking the solution, all solutions are necessarily considered and compared.
During the experiment, we identified the most effective forms of work aimed at developing mathematical abilities:
- frontal, individual and group work
- differentiation of educational tasks according to the level of creativity, difficulty, volume
New forms of educational work:
- optional classes on the course "Non-standard and entertaining tasks"
- home independent work
- individual sessions
- olympiads
- contests
- Mind games
- math themed months
- issue of mathematical newspapers
- project protection
- meetings with famous mathematicians
- open championship in problem solving
- Correspondence Family Olympiad
Expediency such activities is that they contribute to the development of all components of mathematical abilities that can be formed in the primary grades.
Analysis of indicators of the development of mathematical abilities of students in the control and experimental classes:
Table 3
 Stages of experiment-Ment level Mathematical kih abilities | Ascertaining experiment | Control experiment |
||
| 2 "B" | 2 "B" | 4 "B" | 4 "B" |
|
| High | 4 hours (15%) | 3 hours (12%) | 11 hours (43%) | 6 hours (22%) |
| Average | 14 hours (54%) | 14 hours (54%) | 10 hours (38%) | 13 hours (48%) |
| Short | 8 hours (31%) | 9 hours (34%) | 5 hours (19%) | 8 hours (30%) |
As can be seen from the table, in the class where the experimental classes were held, there was a significant increase in the indicators of mathematical abilities compared to the control class. Eight students improved their mathematical abilities. The number of students with a high level of mathematical abilities increased by 2.7 times, with one person from low to high. In the control class during the same period, the shift in the development of mathematical abilities was less significant. It increased in six students. The number of students with a high level of mathematical abilities has doubled. The number of students with a high level of mathematical abilities in the experimental class at the end of the experiment was 43%, with a low level - 19%, in the control class - 22% and 30%, respectively. The number of students with excellent marks in mathematics at 4 "B" during the experiment period increased by 2 times and amounted to 12 people (46%) at the final stage, in the control class the number of students with excellent marks in mathematics was 6 people (23%) .
The results of the ascertaining and control stages of the experiment are given in Appendix No. 6.
Comparison of the results of tests, the quality of teaching in mathematics allow us to conclude that with an increase in the level of mathematical abilities, success in mastering mathematics increases. The results of the Olympiads show that students with a high level of mathematical abilities confirm their level.
Table 4
Olympiad results:
| class place | 2 "B" | 2 "B" | 3 "B" | 3 "B" | 4 "B" | 4 "B" |
| I | 1 hour | 1 hour | 2h | 1 hour | 2 hours | - |
| II | - | - | 1 hour | - | 1 hour | - |
| III | 1 hour | 1 hour | 1 hour | 1 hour | 3 o'clock | 1 hour |
The number of students who won prizes in the Olympiad increased by 3 times.
At the end of the experiment (December 2007), the indicator of the quality of knowledge in mathematics was 84.6% in the experimental class, and 77% in the control class (experimental class - 4 "B" (2 "B"), control - 4 "C" ( 2 "B").
Analyzing the work done, a number of conclusions can be drawn:
1. Classes on the development of mathematical abilities in the process of solving text problems in mathematics lessons in the experimental class were quite productive. We managed to achieve the main goal of this study - on the basis of theoretical and experimental research, to determine the most effective forms and methods of work that contribute to the development of mathematical abilities of younger students in solving word problems.
2. The analysis of the educational material by T. E. Demidova, S. A. Kozlova, A. P. Tonkikh according to the program "School 2100", preceding the practical part of the work, made it possible to structure the selected material in the most logical and acceptable way, in accordance with the objectives of the study.
The result of the work carried out is several methodological recommendations for the development of mathematical abilities:
1. The formation of skills in solving problems must begin on the basis of taking into account the mathematical abilities of students.
2. Take into account the individual characteristics of the student, the differentiation of mathematical abilities in each of them, using effective forms, methods and techniques.
3. In order to improve mathematical abilities, it is advisable to further develop effective forms, methods and techniques in the process of solving mathematical problems.
3. Systematically use tasks in the lessons that contribute to the formation and development of the components of mathematical abilities.
4. By purposefully teaching schoolchildren to solve problems with the help of specially selected exercises, techniques, teach them to observe, use analogy, induction, comparisons and draw conclusions.
5. It is advisable to use tasks for ingenuity, joke tasks, mathematical puzzles in the lessons.
6. Provide targeted assistance to students with different levels of mathematical abilities.
7. When working with groups of students, it is necessary to ensure the mobility of these groups.
Thus, our study allows us to assert that the work on the development of mathematical abilities in the process of solving word problems is an important and necessary matter. The search for new ways to develop mathematical abilities is one of the urgent tasks of modern psychology and pedagogy.
Our research has a certain practical significance.
In the course of experimental work, based on the results of observations and analysis of the data obtained, it can be concluded that the speed and success of the development of mathematical abilities does not depend on the speed and quality of assimilation of program knowledge, skills and abilities. We managed to achieve the main goal of this study - to determine the most effective forms and methods that contribute to the development of students' mathematical abilities in the process of solving word problems.
As the analysis of research activity shows, the development of children's mathematical abilities develops more intensively, since:
A) appropriate methodological support has been created (tables, instructional cards and worksheets for students with different levels of mathematical abilities, a software package, a series of tasks and exercises for the development of certain components of mathematical abilities;
B) the program of the optional course "Non-standard and entertaining tasks" was created, which provides for the implementation of the development of mathematical abilities of students;
C) diagnostic material has been developed that allows you to timely determine the level of development of mathematical abilities and correct the organization of educational activities;
D) a system for the development of mathematical abilities has been developed (according to the plan of the formative experiment).
The need to use a set of exercises for the development of mathematical abilities is determined on the basis of the identified contradictions:
Between the need to use tasks of different levels of complexity in mathematics lessons and their absence in teaching; - between the need to develop mathematical abilities in children and the real conditions for their development; - between the high requirements for the tasks of forming the creative personality of students and the weak development of the mathematical abilities of schoolchildren; - between the recognition of the priority of introducing a system of forms and methods of work for the development of mathematical abilities and an insufficient level of development of ways to implement this approach.
The basis for the study is the choice, study, implementation of the most effective forms, methods of work in the development of mathematical abilities.
The general structure of mathematical abilities (according to V.A. Krutetsky)
This paragraph presents the general structure of mathematical abilities at school age according to V.A. Krutetsky. It is considered on the basis of the main stages of problem solving: I. obtaining mathematical information; II. processing of mathematical information; III. storage of mathematical information. Each of the stages I - III corresponds to one or more mathematical abilities. We will give a description of each mathematical ability, highlighting the actions that are inherent in each ability and a description of the protocols for solving problems by capable and incapable students, described by Vadim Andreevich Krutetsky in the book.
Skills Required to Gain Mathematical Information
The ability to formalize the perception of mathematical material, grasping the formal structure of the problem
Ability characteristic. This mathematical ability is manifested in the desire for a kind of formalization of the structure of mathematical material in the process of its perception. Formalization is understood as a quick “grasping” in a specific task, in a mathematical expression of their formal structure, when everything meaningful (numerical data, specific content) seems to fall out and pure relationships between indicators remain, characterizing the assignment of a task or mathematical expression to a certain type. Formalized perception is a kind of generalized perception of functional connections, separate from the objective and numerical form, when its general structure is perceived in the concrete.
highlight various elements in the mathematical material of the problem;
give the elements of the mathematical material of the problem a different assessment;
to systematize the elements of the mathematical material of the problem;
combine elements of the mathematical material of the problem into complexes;
look for relationships and functional dependencies of the elements of the mathematical material of the problem.
The first three actions are aimed at the analytical perception of the mathematical material of the problem, while the others are aimed at the synthetic perception of the mathematical material of the problem.
Features of the implementation of the first stage of solving problems by students with this ability. To clarify the features of the perception of mathematical material, V.A. Krutetsky used a series of "Systems of the same type of tasks". This series is intended for students who are still unfamiliar with abbreviated multiplication formulas. It was studied how students can highlight the main, main, essential in terms of the type of task, abstract from the insignificant, secondary, from the details. With the help of this series, the process of generalization is also explored - the summing up of objects under the concept that has just formed in its basis.
Let us consider the solution of one of the tests of the series "Systems of the same type of tasks" aimed at determining the mastery of this ability by students capable of mathematics and incapable of mathematics. A series is a kind of "task ladder" of the same type, from the simplest to the most complex. It turns out how the subject will be able to prove that the given problem, despite its external difference, belongs to the same type, and how, taking into account the specific features of the problem, he is going to solve it according to the general scheme for solving problems of the type he has established.
Let us give a clear example of how students capable of mathematics and those incapable coped with one of the tasks in this series.
Capable students in solving the problem of applying the abbreviated multiplication formula (a+b)2. They easily highlight moments that are essential for a given type (the sum of two algebraic expressions squared), as well as those that are not essential for a given type (the specific value and nature of algebraic expressions that make up the number a and b). In other words, there was a kind of formalization of the structure of the task when it was perceived, when the task (for example, 6ax+1/2by)2 was grasped in the following form: (+)2=.
Incapable students, on the other hand, had a narrowly limited idea of the “first” and “second” numbers in this formula, it was difficult for them to understand that a and b denote any value and any algebraic expression. Therefore, they did not independently capture the structural "backbone" of the problem.
Abilities Required to Process Mathematical Information
The ability for logical reasoning in the field of quantitative and spatial relations, numerical and symbolic symbolism
Ability characteristic. One of the features of mathematics is the algorithmic nature of solving many problems. An algorithm, as you know, is a specific indication of what operations and in what sequence must be performed in order to solve any problem of a certain type. The algorithm is a generalization, since it is applicable to all problems of the corresponding type. Of course, a very large number of problems are not algorithmized and are solved with the help of special, special techniques. Therefore, the ability to find solutions that do not fit the standard rule is one of the essential features of mathematical thinking.
The actions represented by this ability. With this mathematical ability, students perform the following actions:
logically argue (prove, substantiate);
operate with special mathematical signs, conditional symbolic designations of quantitative quantities and relations and spatial properties;
translated into symbolic language.
Features of the implementation of the second stage of solving problems by students with this ability. To clarify this ability, a series of "Problem Problems" is used. A series is a system of the same type of problems, more and more complicated proofs.
For example, let's take the solution of a problem by a capable and incapable student.
Here is how a capable student solved the problem: “Prove that the sum of any three consecutive numbers is divisible by 3 (for any integer value of a).” Sequential numbers are such numbers, when each of the subsequent ones is one more than the previous one, so it seems? How can you prove it? 2, 3, and 4 do indeed add up to 3; 12, 13, 14 also add up to 39. You can prove this: the sum of three identical numbers, of course, is divisible by 3. Moreover, 3 units are added (the second number is one, and the third is two units more than the first), which are also divisible by 3. It is also possible to prove algebraically: x+(x+1)+(x+2)=3x+3=3(x+1). The last expression can always be divided by 3, whatever the original number x is.
This is how an incompetent student copes with such a task.
Task. Think of any number, multiply it by a number greater than planned by 6 and add 9. Prove that the result is a square.
Teacher: What does “is a square” mean? What number square?
Exp.: There are numbers that are not the square of a number, such as 13 or 20. And there are numbers that are the result of squaring a number, such as 9 (i.e. 3).
Teacher: Understood. And here how to prove?
Exp.: Think. Apply the method of algebraic proof. It is said: "Think of any number." What is the meaning of "any number" in algebra?
Teacher: And now I know: x(x+6)+9=x2+6x+9. Here x2 is the square of the intended number.
Exp.: You took only part of the result. And you need to prove that the whole result is the square of some number. Which expression is the square of your result? Do you remember the abbreviated multiplication formulas?
Teacher: I know. It will turn out (x + 3) 2. (gives no answer immediately).
Exp.: But is the result always a square?
Teacher: I don't know.
Only after a lengthy explanation did the experimenter answer: "In my opinion, always, since we took any number."
Ability to quickly and broadly generalize mathematical objects
Ability characteristic. The ability to generalize mathematical material is considered in two ways: 1) as a person’s ability to see in a particular, concrete already known to him the general (subsuming a particular case under a known general concept) and 2) the ability to see in a single, particular, as yet unknown general (to derive the general from special cases, to form a concept). It is one thing to see the possibility of applying a formula already known to the student to a given particular case, and another thing to deduce a formula still unknown to the student on the basis of particular cases.
The actions represented by this ability. With this mathematical ability, students perform the following actions:
they see a similar situation in the field of numerical and symbolic symbolism (where to apply);
own a generalized type of solution, a generalized scheme of proof, reasoning (what to apply).
In both cases, it is necessary to abstract from the specific content and highlight the similar, common and essential in the structures of objects, relations or actions.
Features of the implementation of the second stage of solving problems by students with this ability. To identify this ability, V.A. Krutetsky offers a series of tasks that have already been used to test mathematical ability - the ability to formalize the perception of mathematical material.
Let us give an example of solving one of the problems of this series. After solving the example on the application of the formula "square of the sum", a capable student is given an example to solve: (C + D + E) (E + C + D). The student applies the formula and writes (C + D + E) 2 and connects two terms - (C + (D + E)) 2 and then directly applies the formula and opens the brackets.
A student incapable of mathematics, having mastered the formula (a + b) 2 and the principle of reasoning, proceeds to solve the example (1 + a3b2) 2.
Exp.: Can this example be solved using the abbreviated multiplication formula?.
Teacher: There is something different here - both a and b are on the right and are not separated by a plus ... (writes: ". Exp.:" Where did the unit go?. The student is silent.
Exp.: Well, solve the following example: (2x+y)2.
The student writes by repeating the formula aloud: 4x2+22xy+y2=4x2+4x+y2.
Exp.: Right. Do the same for the previous problem.
Teacher: And here is something else... the square of the first one is this.
Exp.: Let's reason together. To apply the formula, we need to make sure that we are dealing with the square of the sum of two numbers. Do you understand that this is a square?
Teacher: Here (shows) the number 2 shows that something in brackets must be multiplied by itself.
Exp.: Right. What about a binomial in brackets? Show where the first term, the first "number".
Teacher: ...or not, what am I saying...there should be a plus sign between members. There is no first term, only the second.
In the future, the student still solves this example with the help of the experimenter.
The ability to curtail the process of mathematical reasoning and the system of corresponding actions. Ability to think in folded structures
Ability characteristic. Along with detailed inferences in the mental activity of schoolchildren, when solving problems, folded inferences also occupy a certain place, when the student is not aware of the rules, the general situation, in accordance with which he actually acts.
The actions represented by this ability. With this mathematical ability, students perform an action - folding inferences.
That is, in the process of solving problems, the student does not fulfill the whole chain of considerations and conclusions that form a complete, detailed structure of the solution.
Features of the implementation of the second stage of solving problems by students with this ability. To identify this ability, a series of "System of different types of tasks" is used. Let us give an example of how a capable student solved one of the problems in this series.
Task. A car traveled from A to B at a speed of 20 km/h and back at a speed of 30 km/h. What is the average speed of the car for the entire trip?
Teacher: It is clear that at a speed of 30 km per hour he walked less time than at a speed of 20 km per hour (with the same path). And if so, then the average speed will not be equal to 25 km per hour. How to decide? (We break the further course of the solution into separate links.) I will decide according to reasoning.
Speed is the result of dividing the distance by the time. So, you need to know the total path and the total time spent on the entire path, and divide the total path by the total time.
Now it is clear how to solve. You need to know the whole path. If the path to one end is denoted by x, then the entire path is 2x.
Now we need to know the time. It is different. To find out the time, you need to divide the distance by the speed.
Spent on the way there
And the way back is spent
And it took all the way, so =
We now divide the total path by the total number of hours:
2x: km per hour.
As for the incapable, they did not notice any noticeable curtailment even as a result of many exercises. At the first stages of mastering, they are constantly entangled in a cumbersome chain of inferences, which is fixed with difficulty, with the help of the experimenter, and gradually turns into a relatively harmonious system. There can be no question of any curtailment at these stages, since the process of reasoning itself is still in its infancy. And in the future, they needed only the full composition of the reasoning.
Flexibility of thought processes in mathematical activity
Ability characteristic. This mathematical ability is expressed in easy and free switching from one mental operation to another, in the variety of aspects of approaches to solving problems, in the ease of restructuring existing thought patterns and action systems.
The actions represented by this ability. With this mathematical ability, students perform the following action - they switch to a new mode of action, i.e. from one mental operation to another.
Features of the implementation of the second stage of solving problems by students with this ability. A series of tests "Tasks that lead to "self-restraint"" are aimed at this ability. This series includes reasoning problems that differ in the following abilities: either their condition is usually perceived with a restriction that does not actually exist, or in the process of solving, the solver involuntarily limits himself to some possibilities, illegally excluding each other. In both cases, an involuntary constraint leads to the idea of the impossibility of solving the problem.
A capable student solves the problem "In a right-angled triangle, one leg is 7 cm. Determine the other two sides if they are expressed as integers."
“Build a triangle on one side? Something strange ... True, another angle is given - a straight line, but still it is impossible ... (draws). Well, here you can see - the side and the angle are constant, but here are how many different triangles. Maybe the problem is not solved? (Exp.: No. The problem is being solved.) It's strange… (draws) Well, it's clearly seen that there are an infinite number of solutions (still draws). I am not so much solving something as trying to prove that it cannot be solved ... Maybe there are many options, but they are all expressed in fractional numbers (reads the condition again). Can there be only one case when they are expressed as integers? Probably so - the condition does not say this, but you can understand ... But then it must be proved ... If the hypotenuse is a, and the unknown leg is b, then a2 = 49 + b2 according to Pythagoras, and 49 = a2-b2 ... Well, what's next ? a+b=49/a-b. I feel that this will give something ... If a and b are integers, then their sum is an integer ... Well, everything is clear: it means that 49 is divisible by a-b without a remainder. And 49 is divisible only by 7... But a-b cannot be equal to 7, because then there will be no triangle (the hypotenuse is exactly equal to two legs - two sides are equal to the third)... Somewhere there is a solution, I missed it... But after all, 49 is divisible not only by 7, but also by 1 and 49. Well, now the solution is in your pocket: 49 cannot be either - the hypotenuse will be greater than the sum of the legs. One thing remains: a-b=1, a a+b=49. It will turn out 25 cm. The hypotenuse and 24 cm legs.
Incapable students are distinguished by inertia, inertness, stiffness of thought in the sphere of mathematical relations and actions, a stable, stereotyped nature of actions, an obsessive retention in the minds of the previous principle of decisions, a method of action that has an inhibitory effect, if necessary, rebuild the action, which determines a pronounced difficulty and switching from one mental operation to another, qualitatively different.
Striving for clarity, simplicity of solution, economy and rationality of solution
Ability characteristic. This feature of mathematical thinking of students capable of mathematics is closely related to the previous one. It is very typical for capable students to strive for the most rational solutions to problems, to search for the clearest, shortest, and, consequently, the most “elegant” path to the goal. This looks like a peculiar tendency towards economy of thought, expressed in the search for the most economical ways of solving problems.
The actions represented by this ability. With this mathematical ability, students perform the following action - they find the most rational solution to the problem.
Features of the implementation of the second stage of solving problems by students with this ability. Vadim Andreevich found out this ability with the help of the "Problem for the consideration of logical reasoning." To do this, he compared the real process of the student's reasoning with the most detailed one. I compared the number and nature of the "links" in both cases, they are compared with the nature and number of links of a really unfolded structure.
For example, a capable student solved the problem: "Find the smallest number that when divided by 3 gives a remainder of 1, when divided by 4 gives a remainder of 2, when divided by 5 gives a remainder of 3 and when divided by 6 gives a remainder of 4" A capable student before in total, he found the least common multiple of these numbers (60) and said: “60-2=58. That number is 58." At the request of the experimenter, he explained: “I presented all the numbers and the remainder in a column and immediately saw that in all cases the difference between the divisor and the remainder is 2. So, if you add 2 to the desired number, then it will be divided into all numbers without a remainder. The smallest of these numbers is 60. But now we remove the deuce - we will be 58.
Incapable students do not pay much attention to the quality of the solution. They stop working after the task and do not ask themselves the question: “Is it possible to solve it easier, more clearly?”.
The ability to quickly and freely restructure the direction of the thought process, switch from direct to reverse thought (reversibility of the thought process in mathematical reasoning)
Ability characteristic. The reversibility of the thought process is understood as the restructuring of its direction in the sense of switching from the direct to the reverse course of thought. This concept combines two different, albeit related, processes.
First, it is the establishment of two-way (or reversible) AB associations (links) as opposed to one-way AB-type links that function only in one direction.
Secondly, it is the reversibility of the thought process in reasoning, the reverse direction of thought from the result, the product to the initial data, which takes place, for example, when moving from a direct to an inverse theorem.
The actions represented by this ability. With this mathematical ability, students perform the following action - to rebuild the thought process from direct to reverse thinking.
Features of the implementation of the second stage of solving problems by students with this ability. To clarify this ability, V.A. Krutetsky proposed a series of problems "Direct and inverse problems". This series includes paired problems - direct and inverse. Inverse problems are conditionally called those that, in comparison with the original (direct) problems, while maintaining the plot, the desired is included in the condition, and one or more elements of the condition become the desired.
Here is an example of how capable and incapable students solved these problems:
A capable student mastered the type of solution according to the formula "the product of the sum of two numbers and their difference is equal to the difference of the squares of these numbers."
He is invited to factorize the expression (x-y)2-25y8. He says here that this problem is the opposite and there is already a difference of squares and writes down the expression (x-y + 5y4) (x-y-5y4). He explains his decision that you need to think about what the squares were made of and take the sum of these numbers and multiply by the difference.
An incapable student with difficulty, after a large number of exercises, mastered the method of solving problems according to this formula.
Exp.: Solve the problem 55=(the student gives the correct answer). Now solve this: what numbers need to be multiplied to get 25 (the student gives the correct answer). Now see 55=25 and 25=55. The second task is the reverse of the first. Solve the problem (2x+y)(2x-y)= (the student gives the correct answer). Right. But if (2x+y)(2x-y)=4x2-4y2, can we say vice versa that 4x2-4y2= (2x+y)(2x-y)? (Student answers in the affirmative). And what does 9x2-4y2 equal to?
Teacher: I don't know. These are some weird jobs. We didn't decide that.
Exp.: Yes, we didn't decide, but we are learning to solve. Think about it: what is the product of the sum of two numbers and their difference? This you know.
Teacher: The product of the sum of two numbers and their difference is the square of the first minus the square of the second.
Exp.: Right. Can you say the opposite? What is the difference of squares? What is a2-b2 equal to?.
Account: a2-b2=(a+b)(a-b).
Exp.: What is 9x2-4y2 equal to?
Teacher: (9x+4y)(9x-4y)…
We omit the rest of the conversation. Only after repeated explanations and exercises did the student learn to solve problems of this type, and only the simplest ones.
Capabilities required to store mathematical information
Mathematical memory (generalized memory for mathematical relations, typical characteristics, reasoning and proof schemes, problem solving methods and principles of approach to them)
Ability characteristic. The essence of mathematical memory lies in the generalized memorization of typical schemes of reasoning and actions. As for the memory for specific data, numerical parameters, it is "neutral" in relation to mathematical abilities.
The actions represented by this ability. With this mathematical ability, students perform the following actions:
remember typical features of tasks and generalized methods for solving them, reasoning schemes, main lines of evidence, logical schemes;
keep in memory typical features of tasks and generalized methods for solving them, reasoning schemes, main lines of evidence, logical schemes.
Features of the implementation of the III stage of solving problems by students with this ability. Capable students in most cases remember for a long time the type of problem they solved in their time, the general nature of the actions, but they do not remember the specific data of the problem, numbers. The incapable, on the contrary, remember only specific numerical data or specific facts related to the task. If an incapable person remembers that he was solving “some problem with cages and rabbits”, or “something about a fish that weighs 2 pounds”, then an able person usually remembers the type of problem much more often: “I solved a problem for various combinations of parts of a whole - about a fish whose tail and head weigh so much, and the head and body weigh so much, and the tail and body weigh so much more.
The distinguished abilities are closely related, influence each other and form in their totality a single system, an integral structure, a kind of syndrome of mathematical talent, a mathematical mindset.
Those abilities are not included in the structure of mathematical giftedness, the presence of which in this system is not necessary (although useful). In this sense, they are neutral in relation to mathematical giftedness. However, their presence or absence in the structure (more precisely, the degree of their development) determines the type of mathematical mindset. The following components are not mandatory in the structure of mathematical talent:
The speed of thought processes as a temporal characteristic.
Calculating ability (the ability to quickly and accurately calculate, often in the mind).
Memory for numbers, numbers, formulas.
The ability to spatial representations.
The ability to visualize abstract mathematical relationships and dependencies.
The work experience of a primary school teacher of MOAU "Secondary School No. 15 of Orsk" Vinnikova L.A.
Development of mathematical abilities of primary school students in the process of solving text problems.
The work experience of a primary school teacher of MOAU "Secondary School No. 15 of Orsk" Vinnikova L.A. Compiled by: Grinchenko I. A., methodologist of the Orsk branch of IPKiPPRO OGPU
Theoretical base of experience:
Theories of developmental learning (L.V. Zankov, D.B. Elkonin)
Psychological and pedagogical theories of R. S. Nemov, B. M. Teplov, L. S. Vygotsky, A. A. Leontiev, S. L. Rubinstein, B. G. Ananiev, N. S. Leites, Yu. D. Babaeva, V. S. Yurkevich about the development of mathematical abilities in the process of specially organized educational activities.
Krutetsky V. A. Psychology of mathematical abilities of schoolchildren. M.: Publishing house. Institute of Practical Psychology; Voronezh: Publishing House of NPO MODEK, 1998. 416 p.
The development of mathematical abilities of students is consistent and purposeful.
All researchers involved in the problem of mathematical abilities (A. V. Brushlinsky, A. V. Beloshistaya, V. V. Davydov, I. V. Dubrovina, Z. I. Kalmykova, N. A. Menchinskaya, A. N. Kolmogorov, Yu. M. Kolyagin, V. A. Krutetsky, D. Poya, B. M. Teplov, A. Ya. Khinchin), with all the variety of opinions, note first of all the specific features of the psyche of a mathematically capable child (as well as a professional mathematician), in particular, flexibility, depth, purposefulness of thinking. A. N. Kolmogorov, I. V. Dubrovina proved by their research that mathematical abilities appear quite early and require continuous exercise. V. A. Krutetsky in the book “Psychology of mathematical abilities of schoolchildren” distinguishes nine components of mathematical abilities, the formation and development of which takes place already in the primary grades.
Using the material of the textbook "My Mathematics" by T.E. Demidova, S. A. Kozlova, A. P. Tonkikh allows to identify and develop the mathematical and creative abilities of students, to form a steady interest in mathematics.
Relevance:
In elementary school age there is a rapid development of the intellect. The possibility of developing abilities is very high. The development of mathematical abilities of younger students today remains the least developed methodological problem. Many educators and psychologists are of the opinion that the elementary school is a “high-risk zone”, since it is at the stage of primary education, due to the primary orientation of teachers to the assimilation of knowledge, skills and abilities, that many children block the development of abilities. It is important not to miss this moment and find effective ways to develop the abilities of children. Despite the constant improvement of the forms and methods of work, there are significant gaps in the development of mathematical abilities in the process of solving problems. This can be explained by the following reasons:
Excessive standardization and algorithmization of problem solving methods;
Insufficient inclusion of students in the creative process of solving the problem;
The imperfection of the teacher's work in developing the ability of students to conduct a meaningful analysis of the problem, put forward hypotheses for planning a solution, rationally determining the steps.
The relevance of the study of the problem of developing the mathematical abilities of younger students is explained by:
Society's need for creative thinking people;
Insufficient degree of development in practical methodological terms;
The need to generalize and systematize the experience of the past and present in the development of mathematical abilities in a single direction.
As a result of purposeful work on the development of mathematical abilities in students, the level of academic performance and quality of knowledge increases, and interest in the subject develops.
Fundamental principles of the pedagogical system.
Progress in the study of the material at a rapid pace.
The leading role of theoretical knowledge.
Training at a high level of difficulty.
Work on the development of all students.
Students' awareness of the learning process.
Development of the ability and need to independently find a solution to previously unseen educational and extracurricular tasks.
Conditions for the emergence and formation of experience:
Erudition, high intellectual level of the teacher;
Creative search for methods, forms and techniques that provide an increase in the level of mathematical abilities of students;
The ability to predict the positive progress of students in the process of using a set of exercises to develop mathematical abilities;
The desire of students to learn new things in mathematics, to participate in olympiads, competitions, intellectual games.
The essence of experience is the activity of the teacher to create conditions for the active, conscious, creative activity of students; improving the interaction between the teacher and students in the process of solving text problems; the development of mathematical abilities of schoolchildren and the education of their industriousness, efficiency, exactingness to themselves. By identifying the causes of success and failure of students, the teacher can determine what abilities or inability affect the activities of students and, depending on this, purposefully plan further work.
To carry out high-quality work on the development of mathematical abilities, the following innovative pedagogical products of pedagogical activity are used:
Optional course "Non-standard and entertaining tasks";
Use of ICT technologies;
A set of exercises for the development of all components of mathematical abilities that can be formed in primary grades;
A cycle of classes on the development of the ability to reason.
Tasks contributing to the achievement of this goal:
Constant stimulation and development of the student's cognitive interest in the subject;
Activation of the creative activity of children;
Development of the ability and desire for self-education;
Cooperation between the teacher and the student in the learning process.
Extracurricular work creates an additional incentive for the creativity of students, the development of their mathematical abilities.
The novelty of the experience lies in the fact that:
The specific conditions of activity that contribute to the intensive development of the mathematical abilities of students have been studied, reserves for increasing the level of mathematical abilities for each student have been found;
The individual abilities of each child in the learning process are taken into account;
The most effective forms, methods and techniques aimed at developing the mathematical abilities of students in the process of solving text problems are identified and described in full;
A set of exercises is proposed for the development of the components of the mathematical abilities of primary school students;
Requirements for exercises have been developed that, by their content and form, would stimulate the development of mathematical abilities.
This makes it possible for students to master new types of tasks with less time and more efficiency. Part of the tasks, exercises, some tests to determine the progress of children in the development of mathematical abilities were developed in the course of work, taking into account the individual characteristics of students.
Productivity.
The development of mathematical abilities of students is achieved through consistent and purposeful work by developing methods, forms and techniques aimed at solving text problems. Such forms of work provide an increase in the level of mathematical abilities of most students, increase productivity and creative direction of activity. The majority of students increase the level of mathematical abilities, develop all the components of mathematical abilities that can be formed in the primary grades. Students show a steady interest and a positive attitude towards the subject, a high level of knowledge in mathematics, successfully complete tasks of the Olympiad and creative nature.
Labor intensity.
The complexity of the experience is determined by its rethinking from the standpoint of the creative self-realization of the child's personality in educational and cognitive activity, the selection of optimal methods and techniques, forms, means of organizing the educational process, taking into account the individual creative capabilities of students.
Possibility of implementation.
Experience solves both narrow methodological and general pedagogical problems. The experience is interesting for primary and secondary school teachers, university students, parents and can be used in any activity that requires originality, unconventional thinking.
Teacher work system.
The teacher's work system consists of the following components:
1. Diagnosis of the initial level of development of mathematical abilities of students.
2. Predicting the positive results of students' activities.
3. Implementation of a set of exercises to develop mathematical abilities in the educational process within the framework of the School 2100 program.
4. Creation of conditions for inclusion in the activities of each student.
5. Fulfillment and compilation by students and the teacher of tasks of an Olympiad and creative nature.
The system of work that helps to identify children who are interested in mathematics, teach them to think creatively and deepen their knowledge includes:
Preliminary diagnostics to determine the level of mathematical abilities of students, making long-term and short-term forecasts for the entire course of study;
The system of mathematics lessons;
Diverse forms of extracurricular activities;
Individual work with schoolchildren capable of mathematics;
Independent work of the student himself;
Participation in olympiads, competitions, tournaments.
Work efficiency.
With 100% progress, a consistently high quality of knowledge in mathematics. Positive dynamics of the level of mathematical abilities of students. High educational motivation and motivation for self-realization in the performance of research work in mathematics. Increase in the number of participants in Olympiads and competitions at various levels. Deeper awareness and assimilation of program material at the level of application of knowledge, skills in new conditions; increased interest in the subject. Increasing the cognitive activity of schoolchildren in the classroom and extracurricular activities.
The leading pedagogical idea of the experiment is to improve the process of teaching schoolchildren in the process of lesson and extracurricular work in mathematics for the development of cognitive interest, logical thinking, and the formation of students' creative activity.
The prospects of the experience are explained by its practical significance for increasing the creative self-realization of children in educational and cognitive activities, for the development and realization of their potential.
Experience technology.
Mathematical abilities are manifested in the speed with which, how deeply and how firmly people learn mathematical material. These characteristics are most easily detected in the course of solving problems.
The technology includes a combination of group, individual and collective forms of learning activity of students in the process of solving problems and is based on the use of a set of exercises to develop the mathematical abilities of students. Skills develop through activity. The process of their development can go spontaneously, but it is better if they develop in an organized learning process. Conditions are created that are most favorable for the purposeful development of abilities. At the first stage, the development of abilities is characterized to a greater extent by imitation (reproductivity). Gradually, elements of creativity, originality appear, and the more capable a person is, the more pronounced they are.
The formation and development of the components of mathematical abilities takes place already in the primary grades. What characterizes the mental activity of schoolchildren capable of mathematics? Capable students, perceiving a mathematical problem, systematize the given values in the problem, the relationship between them. A clear holistically dissected image of the task is created. In other words, capable students are characterized by a formalized perception of mathematical material (mathematical objects, relations and actions), associated with a quick grasp of their formal structure in a specific task. Pupils with average abilities, when perceiving a task of a new type, determine, as a rule, its individual elements. It is very difficult for some students to comprehend the connections between the components of the task, they hardly grasp the totality of the diverse dependencies that make up the essence of the task. To develop the ability to formalize the perception of mathematical material, students are offered exercises [Appendix 1. Series I]:
1) Tasks with an unformulated question;
2) Tasks with an incomplete composition of the condition;
3) Tasks with redundant composition of the condition;
4) Work on the classification of tasks;
5) Drawing up tasks.
The thinking of capable students in the process of mathematical activity is characterized by fast and broad generalization (each specific problem is solved as a typical one). For the most capable students, such a generalization occurs immediately, by analyzing one individual problem in a series of similar ones. Capable students easily move on to solving problems in literal form.
The development of the ability to generalize is achieved by presenting special exercises [Appendix 1. Series II.]:
1) Solving problems of the same type; 2) Solving problems of various types;
3) Solving problems with a gradual transformation from a concrete to an abstract plan; 4) Drawing up an equation according to the condition of the problem.
The thinking of capable students is characterized by a tendency to think in folded conclusions. For such students, the curtailment of the reasoning process is observed after solving the first problem, and sometimes after the presentation of the problem, the result is immediately given. The time to solve the problem is determined only by the time spent on the calculations. A folded structure is always based on a well-founded reasoning process. Average students generalize the material after repeated exercises, and therefore the curtailment of the reasoning process is observed in them after solving several tasks of the same type. In students with low ability, curtailment can begin only after a large number of exercises. The thinking of capable students is distinguished by great mobility of thought processes, a variety of aspects in the approach to solving problems, easy and free switching from one mental operation to another, from direct to reverse thought. For the development of flexibility of thinking, exercises are proposed [Appendix 1. Series III.]
1) Tasks that have several ways to solve.
2) Solving and compiling problems that are inverse to this one.
3) Solving problems in reverse.
4) Solving problems with an alternative condition.
5) Solving problems with uncertain data.
It is typical for capable students to strive for clarity, simplicity, rationality, economy (elegance) of the solution.
The mathematical memory of capable students is manifested in the memorization of types of problems, methods for solving them, and specific data. Able students are distinguished by well-developed spatial representations. However, when solving a number of problems, they can do without relying on visual images. In a sense, logicality replaces "figurativeness" for them; they do not experience difficulties in operating with abstract schemes. While completing the learning tasks, students at the same time develop their mental activity. So, when solving mathematical problems, the student learns analysis, synthesis, comparison, abstraction and generalization, which are the main mental operations. Therefore, for the formation of abilities in educational activities, it is necessary to create certain conditions:
A) positive motives for learning;
B) students' interest in the subject;
C) creative activity;
D) a positive microclimate in the team;
D) strong emotions;
E) providing freedom of choice of actions, variability of work.
It is more convenient for the teacher to rely on some purely procedural characteristics of the activity of capable children. Most children with mathematical abilities tend to:
Increased propensity for mental action and a positive emotional response to any mental load.
The constant need to renew and complicate the mental load, which leads to a constant increase in the level of achievements.
The desire for independent choice of affairs and planning of their activities.
Increased performance. Prolonged intellectual loads do not tire this child, on the contrary, he feels good in a situation where there is a problem.
The development of mathematical abilities of students involved in the program "School 2100" and the textbooks "My Mathematics" by the authors: T. E. Demidova, S. A. Kozlova, A. P. Tonkikh takes place in every mathematics lesson and in extracurricular activities. Effective development of abilities is impossible without the use of intelligence tasks, joke tasks, and mathematical puzzles in the educational process. Students learn to solve logical problems with true and false statements, compose algorithms for transfusion, weighing problems, use tables and graphs to solve problems.
In the search for ways to more effectively use the structure of lessons for the development of mathematical abilities, the form of organization of educational activities of students in the lesson is of particular importance. In our practice we use frontal, individual and group work.
In the frontal form of work, students perform a common activity for all, compare and summarize its results with the whole class. Due to their real capabilities, students can make generalizations and conclusions at different levels of depth. The frontal form of organization of learning is implemented by us in the form of a problematic, informational and explanatory-illustrative presentation and is accompanied by reproductive and creative tasks. All textual logical tasks, the solution of which must be found using a chain of reasoning, proposed in the 2nd grade textbook, are analyzed frontally in the first half of the year, since their independent solution is not available to all children of this age. Then these tasks are offered for independent solution to students with a high level of mathematical abilities. In the third grade, logical problems are first given to all students for independent solution, and then the proposed options are analyzed.
The application of acquired knowledge in changed situations is best organized using individual work. Each student receives a task for independent completion, specially selected for him in accordance with his training and abilities. There are two types of individual forms of organizing tasks: individual and individualized. The first one is characterized by the fact that the student’s activity in fulfilling tasks common to the whole class is carried out without contact with other students, but at the same pace for all, the second allows using differentiated individual tasks to create optimal conditions for the realization of the abilities of each student. In our work, we use the differentiation of educational tasks according to the level of creativity, difficulty, volume. When differentiated by the level of creativity, the work is organized as follows: students with a low level of mathematical abilities (Group 1) are offered reproductive tasks (work according to the model, performing training exercises), and students with an average (Group 2) and high level (Group 3) are offered creative tasks. tasks.
(Grade 2. Lesson No. 36. Problem No. 7. 36 yachts participated in the race of sailing ships. How many yachts reached the finish line if 2 yachts returned to the start due to a breakdown, and 11 due to a storm?
Task for the 1st group. Solve the problem. Consider whether it can be solved in another way.
Task for the 2nd group. Solve the problem in two ways. Come up with a problem with a different plot so that the solution does not change.
Task for the 3rd group. Solve the problem in three ways. Make a problem inverse to this one and solve it.
It is possible to offer productive tasks to all students, but at the same time, children with low abilities are given tasks with elements of creativity in which they need to apply knowledge in a changed situation, and the rest are given creative tasks to apply knowledge in a new situation.
(Grade 2. Lesson No. 45. Task No. 5. There are 75 budgerigars in three cages. There are 21 parrots in the first cage, 32 parrots in the second. How many parrots are in the third cage?
Task for the 1st group. Solve the problem in two ways.
Task for the 2nd group. Solve the problem in two ways. Come up with a problem with a different plot, but so that its solution does not change.
Task for the 3rd group. Solve the problem in three ways. Change the question and the condition of the problem so that the data on the total number of parrots become redundant.
Differentiation of educational tasks according to the level of difficulty (the difficulty of a task is a combination of many subjective factors depending on personality characteristics, for example, such as intellectual capabilities, mathematical abilities, degree of novelty, etc.) involves three types of tasks:
1. Tasks, the solution of which consists in the stereotypical reproduction of learned actions. The degree of difficulty of the tasks is related to how complex the skill of reproducing actions is and how firmly it is mastered.
2. Tasks, the solution of which requires some modification of the learned actions in changing conditions. The degree of difficulty is related to the number and heterogeneity of elements that must be coordinated along with the features of the data described above.
3. Tasks, the solution of which requires the search for new, still unknown methods of action. Tasks require creative activity, a heuristic search for new, unknown patterns of action or an unusual combination of known ones.
Differentiation in terms of the volume of educational material assumes that all students are given a certain number of tasks of the same type. At the same time, the required volume is determined, and for each additionally completed task, for example, points are awarded. Creative tasks can be offered for compiling objects of the same type and it is required to compose the maximum number of them for a certain period of time.
Who will make more tasks with different content, the solution of each of which will be a numerical expression: (54 + 18): 2
As additional tasks, creative or more difficult tasks are offered, as well as tasks that are not related in content to the main one - tasks for ingenuity, non-standard tasks, exercises of a game nature.
When solving problems independently, individual work is also effective. The degree of independence of such work is different. First, students perform tasks with a preliminary and frontal analysis, imitating a model, or according to detailed instruction cards. [Annex 2]. As learning skills are mastered, the degree of independence increases: students (especially with an average and high level of mathematical abilities) work on general, non-detailed tasks, without the direct intervention of a teacher. For individual work, we offer worksheets developed by us on topics, the deadlines for which are determined in accordance with the desires and capabilities of the student [Appendix 3]. For students with a low level of mathematical abilities, a system of tasks is compiled, which contains: samples of solutions and tasks to be solved on the basis of the studied sample, various algorithmic prescriptions; theoretical information, as well as all kinds of requirements to compare, compare, classify, generalize. [Appendix 4, fragment of lesson No. 1] Such an organization of educational work enables each student, by virtue of his abilities, to deepen and consolidate the knowledge gained. The individual form of work somewhat limits the communication of students, the desire to transfer knowledge to others, participation in collective achievements, so we use a group form of organizing educational activities. [Appendix 4. Fragment of lesson No. 2]. Tasks in the group are carried out in a way that takes into account and evaluates the individual contribution of each child. The size of the groups is from 2 to 4 people. The composition of the group is not permanent. It varies depending on the content and nature of the work. The group consists of students with different levels of mathematical abilities. Often we are preparing students with a low level of mathematical ability in extracurricular activities for the role of consultants in the lesson. The fulfillment of this role is sufficient for the child to feel himself the best, his significance. The group form of work makes clear the abilities of each student. In combination with other forms of education - frontal and individual - the group form of organizing the work of students brings positive results.
Computer technologies are widely used in mathematics lessons and optional courses. They can be included at any stage of the lesson - during individual work, with the introduction of new knowledge, their generalization, consolidation, for the control of ZUNs. For example, when solving problems for obtaining a certain amount of liquid from a large or infinite volume of a vessel, reservoir or source using two empty vessels, setting different volumes of vessels, various required amounts of liquid, you can get a large set of tasks of different levels of complexity for their hero " Overflows". The volume of liquid in the conditional vessel A will correspond to the volume of the drained liquid, the volumes B and C will correspond to the given volumes according to the condition of the problem. An action denoted by a single letter, for example, B, means filling a vessel from a source.
Task. Breeding instant mashed potatoes "Green Giant" requires 1 liter of water. How, having two vessels with a capacity of 5 and 9 liters, pour 1 liter of water from a tap?
Children look for a solution to a problem in different ways. They come to the conclusion that the problem is solved in 4 moves.
Action
For the development of mathematical abilities, we use the wide possibilities of auxiliary forms of organization of educational work. These are optional classes on the course "Non-standard and entertaining tasks", home independent work, individual lessons on the development of mathematical abilities with students of low and high levels of their development. In optional classes, part of the time was devoted to learning how to solve logical problems according to the method of A. Z. Zak. Classes were held once a week, the duration of the lesson was 20 minutes and contributed to an increase in the level of such a component of mathematical abilities as the ability to correct logical reasoning.
In the classroom of the optional course "Non-standard and entertaining tasks", a collective discussion is held on solving a problem of a new type. Thanks to this method, children develop such an important quality of activity as awareness of their own actions, self-control, the ability to report on the steps taken in solving problems. Most of the time in the classroom is occupied by students independently solving problems, followed by a collective verification of the solution. In the classroom, students solve non-standard tasks, which are divided into series.
For students with a low level of development of mathematical abilities, individual work is carried out after school hours. The work is carried out in the form of a dialogue, instruction cards. With this form, students are required to speak out loud all the ways of solving, searching for the right answer.
For students with a high level of ability, after-hours consultations are provided to meet the needs for in-depth study of the issues of the mathematics course. Classes in their form of organization are in the nature of an interview, consultation or independent performance of tasks by students under the guidance of a teacher.
For the development of mathematical abilities, the following forms of extracurricular work are used: olympiads, competitions, intellectual games, thematic months in mathematics. Thus, during the thematic month "Young Mathematician", held in elementary school in November 2008, the students of the class participated in the following activities: the release of mathematical newspapers; competition "Entertaining tasks"; exhibition of creative works on mathematical topics; meeting with the associate professor of the department of SP and PPNO, defense of projects; Olympiad in mathematics.
Mathematical Olympiads play a special role in the development of children. This is a competition that allows capable students to feel like real mathematicians. It was during this period that the first independent discoveries of the child take place.
Extracurricular activities are held on mathematical topics: "KVN 2 + 3", the Intellectual game "Choosing an heir", the Intellectual marathon, "Mathematical traffic light", "Pathfinders" [Appendix 5], the game "Funny Train" and others.
Mathematical ability can be identified and assessed based on how a child solves certain problems. The very solution of these problems depends not only on abilities, but also on motivation, on existing knowledge, skills and abilities. Making a forecast of the results of development requires knowledge of precisely the abilities. The results of observations allow us to conclude that the prospects for the development of abilities are available for all children. The main thing that should be paid attention to when improving the abilities of children is the creation of optimal conditions for their development.
^ Tracking the results of research activities:
For the purpose of practical substantiation of the conclusions obtained during the theoretical study of the problem: what are the most effective forms and methods aimed at developing the mathematical abilities of schoolchildren in the process of solving mathematical problems, a study was conducted. Two classes took part in the experiment: experimental 2 (4) "B", control - 2 (4) "C" of secondary school No. 15. The work was carried out from September 2006 to January 2009 and included 4 stages.
Stages of experimental activity
I - Preparatory (September 2006). Purpose: determination of the level of mathematical abilities based on the results of observations.
II - Ascertaining series of experiment (October 2006) Purpose: to determine the level of formation of mathematical abilities.
III - Formative experiment (November 2006 - December 2008) Purpose: to create the necessary conditions for the development of mathematical abilities.
IV - Control experiment (January 2009) Purpose: to determine the effectiveness of forms and methods that contribute to the development of mathematical abilities.
At the preparatory stage, students of the control - 2 "B" and experimental 2 "C" classes were observed. Observations were carried out both in the process of studying new material and in solving problems. For observations, those signs of mathematical abilities that are most clearly manifested in younger students were identified:
1) relatively fast and successful mastery of mathematical knowledge, skills and abilities;
2) the ability to consistently correct logical reasoning;
3) resourcefulness and ingenuity in the study of mathematics;
4) flexibility of thinking;
5) the ability to operate with numerical and symbolic symbols;
6) reduced fatigue during mathematics;
7) the ability to shorten the process of reasoning, to think in collapsed structures;
8) the ability to switch from direct to reverse course of thought;
9) the development of figurative-geometric thinking and spatial representations.
In October, teachers filled out a table of schoolchildren's mathematical abilities, in which they rated each of the listed qualities in points (0-low level, 1-average level, 2-high level).
At the second stage, diagnostics of the development of mathematical abilities was carried out in the experimental and control classes.
For this, the "Problem Solving" test was used:
1. Compose compound problems from these simple problems. Solve one compound problem in different ways, underline the rational one.
The cow of the cat Matroskin on Monday gave 12 liters of milk. Milk was poured into three-liter jars. How many cans did the cat Matroskin get?
Kolya bought 3 pens for 20 rubles each. How much money did he pay?
Kolya bought 5 pencils at a price of 20 rubles. How much do pencils cost?
Matroskin's cow gave 15 liters of milk on Tuesday. This milk was poured into three-liter jars. How many cans did the cat Matroskin get?
2. Read the problem. Read the questions and expressions. Match each question with the correct expression.
IN
a + 18
class 18 boys and a girls.
How many students are in the class?
How many more boys than girls?
How many fewer girls than boys?
3. Solve the problem.
In his letter to his parents, Uncle Fyodor wrote that his house, the house of the postman Pechkin and the well were on the same side of the street. From the house of Uncle Fyodor to the house of the postman Pechkin 90 meters, and from the well to the house of Uncle Fyodor 20 meters. What is the distance from the well to the house of the postman Pechkin?
With the help of the test, the same components of the structure of mathematical abilities were checked as during observation.
Purpose: to establish the level of mathematical abilities.
Equipment: student card (sheet).
table 2
The test tests skills and mathematical abilities:
The skills required to solve the problem.
Abilities manifested in mathematical activity.
The ability to distinguish the task from other texts.
^ APPENDIX #1.
1) Tasks with an unformulated question:
The mass of a box of oranges is 28 kg, and the mass of a box of apples is 27 kg. Two boxes of oranges and one box of apples were brought to the school cafeteria.
One vase has 15 flowers and the other has 6 flowers more.
The fishermen pulled out a net with 30 fish. Among them there were 17 breams, and the rest were perches.
2) Tasks with an incomplete composition of the condition:
There are 4 more pencils in the box than in the pencil case. How many fewer pencils are in the pencil case than in the box?
Which question can you answer and which can't? Why?
Think! How to supplement the condition of the problem to answer both questions?
3) Problems with redundant composition of the condition:
Task. At the feeder there were 6 gray and 5 white pigeons. One white dove flew away. How many white doves were at the feeder?
Text analysis shows that one of the data is redundant - 6 gray doves. It is not needed to answer the question. After answering the question of the problem, the teacher suggests making changes to the text of the problem so that this data is needed, which leads to a compound problem. At the feeder there were 6 gray and 5 white pigeons. One dove flew away. How many pigeons are left at the feeder?
These changes will require you to do two things.
(6 + 5) - 1 or (6 - 1) + 5 or (5 - 1) + 6
4) Work on the classification of tasks.
Break these tasks into two so that you can make one out of them:
1. At labor lessons, students sewed 7 bunnies and 5 bears. How many toys did the students make in total?
2.1 Psychological structure of mathematical abilities
ability student math sports
Mathematics is a tool of knowledge, thinking, development. It is rich in opportunities for creative enrichment. Not a single school subject can compete with the possibilities of mathematics in the education of a thinking person. The special importance of mathematics in mental development was noted back in the 18th century by M.V. Lomonosov: "Mathematics should be taught later, that it puts the mind in order."
There is a generally accepted classification of abilities. According to it, abilities are divided into general and special, which determine a person’s success in certain types of activity and communication, where a special kind of inclinations and their development are needed (mathematical, technical, literary and linguistic, artistic and creative, sports, etc.).
Mathematical abilities are determined not only by good memory and attention. For a mathematician, it is important to be able to grasp the order of elements, and the ability to operate with these data. This peculiar intuition is the basis of mathematical ability.
Such scientists in psychology as A. Binet, E. Thorndike and G. Reves, and such outstanding mathematicians as A. Poincare and J. Hadamard contributed to the study of mathematical abilities. A wide variety of directions also determines a wide variety in approaches to the study of mathematical abilities. Of course, the study of mathematical abilities should begin with a definition. Attempts of this kind have been made repeatedly, but there is still no established, satisfying definition of mathematical abilities. The only thing that all researchers agree on is, perhaps, the opinion that one should distinguish between ordinary, "school" abilities for mastering mathematical knowledge, for their reproduction and independent application, and creative mathematical abilities associated with the independent creation of an original and of social value. product.
Back in 1918, in the work of A. Rogers, two aspects of mathematical abilities were noted, reproductive (associated with the function of memory) and productive (associated with the function of thinking). W. Betz defines mathematical abilities as the ability to clearly understand the internal connection of mathematical relations and the ability to think accurately in mathematical concepts.
Of the works of Russian authors, it is necessary to mention the original article by D. Mordukhai-Boltovsky "Psychology of Mathematical Thinking", published in 1918. The author, a specialist mathematician, wrote from an idealistic position, giving, for example, special significance to the “unconscious thought process”, arguing that “the thinking of a mathematician is deeply embedded in the unconscious sphere, now surfacing to its surface, now plunging into depth. A mathematician is not aware of each step of his thought, like a virtuoso of the movement of the bow" [op. to 13, p. 45]. The sudden appearance in consciousness of a ready-made solution to a problem that we cannot solve for a long time, - the author writes, - we explain by unconscious thinking, which continued to deal with the task, and the result pops up beyond the threshold of consciousness [cit. to 13, p. 48]. According to Morduchai-Boltovsky, our mind is capable of performing painstaking and complex work in the subconscious, where all the "rough" work is done, and the unconscious work of thought is even less error than the conscious one.
The author notes the completely specific nature of mathematical talent and mathematical thinking. He argues that the ability to do mathematics is not always inherent even in brilliant people, that there is a significant difference between the mathematical and non-mathematical mind. Of great interest is Mordukhai-Boltovsky's attempt to isolate the components of mathematical abilities. He refers to these components in particular:
* "strong memory", memory for "objects of the type with which mathematics deals", memory rather than for facts, but for ideas and thoughts.
* "wit", which is understood as the ability to "embrace in one judgment" concepts from two loosely connected areas of thought, to find in the already known something similar to the given, to look for something similar in the most distant seemingly completely heterogeneous objects.
* speed of thought (speed of thought is explained by the work that unconscious thinking does to help the conscious). Unconscious thinking, according to the author, proceeds much faster than conscious.
D. Mordukhai-Boltovsky also expresses his views on the types of mathematical imagination that underlie different types of mathematicians - "geometers" and "algebraists". Arithmeticians, algebraists, and analysts in general, whose discovery is made in the most abstract form of breakthrough quantitative symbols and their interrelationships, cannot imagine like a "geometer".
D.N. Bogoyavlensky and N.A. Menchinskaya, speaking of individual differences in the learning ability of children, introduces the concept of psychological properties that determine success in learning, all other things being equal. They do not use the term "ability", but in essence the corresponding concept is close to the definition given above.
Mathematical abilities are a complex structural mental formation, a kind of synthesis of properties, an integral quality of the mind, covering its various aspects and developing in the process of mathematical activity. This set is a single qualitatively original whole - only for the purposes of analysis, we single out individual components, by no means considering them as isolated properties. These components are closely connected, influence each other and form in their totality a single system, the manifestations of which we conditionally call the "syndrome of mathematical giftedness".
Speaking about the structure of mathematical abilities, it should be noted the contribution to the development of this problem by V.A. Krutetsky. The experimental material collected by him allows us to speak about the components that occupy a significant place in the structure of such an integral quality of the mind as mathematical talent.
General scheme of the structure of mathematical abilities at school age
1. Obtaining mathematical information
A) The ability to formalize the perception of mathematical material, covering the formal structure of the problem.
2. Processing of mathematical information.
A) The ability for logical thinking in the field of quantitative and spatial relations, numerical and symbolic symbolism. The ability to think in mathematical symbols.
B) The ability to quickly and broadly generalize mathematical objects, relationships and actions.
C) The ability to curtail the process of mathematical reasoning and the system of corresponding actions. The ability to think in folded structures.
D) Flexibility of thought processes in mathematical activity.
E) Striving for clarity, simplicity, economy and rationality of decisions.
E) The ability to quickly and freely restructure the direction of the thought process, switching from direct to reverse thought (reversibility of the thought process in mathematical reasoning).
3. Storage of mathematical information.
A) Mathematical memory (generalized memory for mathematical relations, typical characteristics, reasoning and proof schemes, problem solving methods and principles of approach to them)
4. General synthetic component.
A) Mathematical orientation of the mind.
Not included in the structure of mathematical giftedness are those components whose presence in this structure is not necessary (although useful). In this sense, they are neutral in relation to mathematical giftedness. However, their presence or absence in the structure (more precisely, the degree of development) determines the types of mathematical mentality.
1. The speed of thought processes as a temporal characteristic.
The individual pace of work is not critical. A mathematician can think slowly, even slowly, but very thoroughly and deeply.
2. Computational abilities (the ability to quickly and accurately calculate, often in the mind). It is known that there are people who are able to perform complex mathematical calculations in their minds (almost instantaneous squaring and cube of three-digit numbers), but who are not able to solve any complex problems.
It is also known that there were and still are phenomenal "counters" that did not give anything to mathematics, and the outstanding mathematician A. Poincaré wrote about himself that even addition cannot be done without error.
3. Memory for numbers, formulas, numbers. As academician A.N. Kolmogorov, many outstanding mathematicians did not have any outstanding memory of this kind.
4. Ability for spatial representations.
5. The ability to visualize abstract mathematical relationships and dependencies.
It should be emphasized that the scheme of the structure of mathematical abilities refers to the mathematical abilities of the student. It cannot be said to what extent it can be considered a general scheme of the structure of mathematical abilities, to what extent it can be attributed to well-established gifted mathematicians.
Types of mathematical mindsets.
It is well known that in any field of science, giftedness as a qualitative combination of abilities is always diverse and unique in each individual case. But with the qualitative diversity of giftedness, it is always possible to outline some basic typological differences in the structure of giftedness, to single out certain types that differ significantly from one another, and come to equally high achievements in the corresponding field in different ways.
Analytic and geometric types are mentioned in the works of A. Poincaré, J. Hadamard, D. Mordukhai-Boltovsky, but with these terms they rather associate a logical, intuitive way of creativity in mathematics.
Among domestic researchers, N.A. Menchinskaya. She singled out students with a relative predominance of: a) figurative thinking over abstract; b) abstract over figurative and c) harmonious development of both types of thinking.
One cannot think that the analytic type appears only in algebra, and the geometric type in geometry. The analytical warehouse can manifest itself in geometry, and the geometric one - in algebra. V.A. Krutetsky gave a detailed description of each type.
Analytic type.
The thinking of representatives of this type is characterized by a clear predominance of a very well-developed verbal-logical component over a weak visual-figurative one. They easily operate with abstract schemes. They have no need for visual supports, for the use of subject or schematic visualization in solving problems, even those when the mathematical relationships and dependencies given in the problem “suggest” visual representations.
Representatives of this type do not differ in the ability of visual-figurative representation and, therefore, use a more difficult and complex logical-analytical solution path where reliance on an image gives a much simpler solution. They very successfully solve problems expressed in an abstract form, while problems expressed in a concrete-visual form try to translate them into an abstract plan as far as possible. Operations associated with the analysis of concepts are carried out by them easier than operations associated with the analysis of a geometric diagram or drawing.
Geometric type
The thinking of representatives of this type is characterized by a very well-developed visual-figurative component. In this regard, we can conditionally speak of predominance over a well-developed verbal-logical component. These students feel the need for a visual interpretation of the expression of abstract material and demonstrate great selectivity in this regard. But if they fail to create visual supports, use objective or schematic visualization in solving problems, then they hardly operate with abstract schemes. They stubbornly try to operate with visual schemes, images, ideas, even where the problem is easily solved by reasoning, and the use of visual supports is unnecessary or difficult.
harmonic type.
This type is characterized by a relative balance of well-developed verbal-logical and visual-figurative components, with the former playing the leading role. Spatial representations in representatives of this type are well developed. They are selective in the visual interpretation of abstract relationships and dependencies, but visual images and schemes are subject to their verbal-logical analysis. Using visual images, these students are clearly aware that the content of the generalization is not limited to particular cases. They also successfully implement a figurative-geometric approach to solving many problems.
The established types seem to have a general meaning. Their presence is confirmed by many studies [cit. by 10, p. 115].
Age features of mathematical abilities.
In foreign psychology, ideas about the age-related features of the mathematical development of a schoolchild, based on the early studies of J. Piaget, are still widespread. Piaget believed that a child only by the age of 12 becomes capable of abstract thinking. Analyzing the stages of development of a teenager’s mathematical reasoning, L. Schoanne came to the conclusion that in terms of visual-specific, a schoolchild thinks up to 12-13 years old, and thinking in terms of formal algebra, associated with mastering operations, symbols, develops only by 17 years.
A study of domestic psychologists gives different results. More P.P. Blonsky wrote about the intensive development in a teenager (11-14 years old) of generalizing and abstracting thinking, the ability to prove and understand evidence.
A legitimate question arises: to what extent can we talk about mathematical abilities in relation to younger students? Research led by I.V. Dubrovina, gives grounds to answer this question in the following way. Of course, excluding cases of special giftedness, we cannot speak of any formed structure of mathematical abilities proper in relation to this age. Therefore, the concept of "mathematical abilities" is conditional when applied to younger schoolchildren - children of 7-10 years old, when studying the components of mathematical abilities at this age, we can usually talk only about the elementary forms of such components. But individual components of mathematical abilities are already formed in the primary grades.
Experimental training, which was carried out in a number of schools by employees of the Institute of Psychology (D.B. Elkonin, V.V. Davydov), shows that with a special teaching method, younger students acquire a greater ability for distraction and reasoning than is commonly thought. However, although the age characteristics of the student to a greater extent depend on the conditions in which learning is carried out, it would be wrong to assume that they are entirely created by learning. Therefore, the extreme point of view on this question, when it is believed that there is no regularity in natural mental development, is wrong. A more effective teaching system can "become" the whole process, but up to certain limits, the sequence of development can change somewhat, but cannot give the line of development a completely different character.
There can be no arbitrariness here. For example, the ability to generalize complex mathematical relations and methods cannot be formed earlier than the ability to generalize simple mathematical relations.
Thus, the age features that are mentioned are a somewhat arbitrary concept. Therefore, all studies are focused on a general trend, on the general direction of development of the main components of the structure of mathematical abilities under the influence of learning.
Gender differences in the characteristics of mathematical abilities.
Do gender differences have any influence on the nature of the development of mathematical abilities and on the level of achievement in the relevant field? Are there qualitatively unique features of the mathematical thinking of boys and girls at school age?
In foreign psychology, there are works where an attempt is made to identify certain qualitative features of the mathematical thinking of boys and girls. V. Stern, speaks of his disagreement with the point of view, according to which the differences in the mental area of men and women are the result of unequal education. In his opinion, the reasons lie in various internal inclinations. Therefore, women are less prone to abstract thinking and less capable in this regard. Studies were also conducted under the guidance of Ch. Spearman and E. Thorndike, they came to the conclusion that "there is no big difference in terms of abilities", but at the same time they note a greater tendency for girls to detail, remember details.
Relevant research in domestic psychology was carried out under the guidance of I.V. Dubrovina and S.I. Shapiro, they did not find any qualitative specific features in the mathematical thinking of boys and girls. The teachers they interviewed did not point out these differences either.
Of course, in fact, boys are more likely to show mathematical ability.
Boys are more likely to win Mathematical Olympiads than girls. But this actual difference must be attributed to the difference in traditions, in the education of boys and girls, due to the widespread view of male and female professions.
This leads to the fact that mathematics is often outside the focus of the interests of girls.
1. Mathematical abilities are determined not only by good memory and attention. For a mathematician, it is important to be able to grasp the order of elements, and the ability to operate with these data. This peculiar intuition is the basis of mathematical ability.
2. Age features - this is a somewhat arbitrary concept. Therefore, all studies are focused on a general trend, on the general direction of development of the main components of the structure of mathematical abilities under the influence of learning.
3. Relevant studies in domestic psychology did not reveal any qualitative specific features in the mathematical thinking of boys and girls.
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The study of mathematical abilities in foreign psychology.
Such outstanding representatives of certain trends in psychology as A. Binet, E. Trondike and G. Reves, and such outstanding mathematicians as A. Poincare and J. Hadamard contributed to the study of mathematical abilities.
A wide variety of directions also determined a wide variety in the approach to the study of mathematical abilities, in methodological tools and theoretical generalizations.
The only thing that all researchers agree on is, perhaps, the opinion that one should distinguish between ordinary, “school” abilities for mastering mathematical knowledge, for their reproduction and independent application, and creative mathematical abilities associated with the independent creation of an original and of social value. product.
Foreign researchers show great unity of views on the question of innate or acquired mathematical abilities. If here we distinguish two different aspects of these abilities - "school" and creative abilities, then with respect to the latter there is complete unity - the creative abilities of a mathematician are an innate formation, a favorable environment is necessary only for their manifestation and development. With regard to "school" (educational) abilities, foreign psychologists are not so unanimous. Here, perhaps, the theory of the parallel action of two factors - the biological potential and the environment - dominates.
The main issue in the study of mathematical abilities (both educational and creative) abroad has been and remains the question of the essence of this complex psychological education. Three important issues can be identified in this regard.
1. The problem of the specificity of mathematical abilities. Do mathematical abilities proper exist as a specific education, different from the category of general intelligence? Or is mathematical ability a qualitative specialization of general mental processes and personality traits, that is, general intellectual abilities developed in relation to mathematical activity? In other words, is it possible to argue that mathematical talent is nothing more than general intelligence plus an interest in mathematics and an inclination to do it?
2. The problem of the structure of mathematical abilities. Is mathematical giftedness a unitary (single indecomposable) or an integral (complex) property? In the latter case, one can raise the question of the structure of mathematical abilities, of the components of this complex mental formation.
3. The problem of typological differences in mathematical abilities. Are there different types of mathematical giftedness or, on the same basis, are there differences only in interests and inclinations towards certain branches of mathematics?
7. Teaching ability
Pedagogical abilities are called a set of individual psychological characteristics of a teacher's personality that meet the requirements of pedagogical activity and determine success in mastering this activity. The difference between pedagogical abilities and pedagogical skills lies in the fact that pedagogical abilities are personality traits, and pedagogical skills are separate acts of pedagogical activity carried out by a person at a high level.
Each ability has its own structure, it distinguishes between leading and auxiliary properties.
The leading properties in pedagogical abilities are:
pedagogical tact;
observation;
love for children;
need for knowledge transfer.
Pedagogical tact is the observance by the teacher of the principle of measure in communicating with children in a wide variety of fields of activity, the ability to choose the right approach to students.
Pedagogical tact involves:
Respect for the student and exactingness to him;
development of independence of students in all types of activities and firm pedagogical guidance of their work;
attentiveness to the mental state of the student and the reasonableness and consistency of the requirements for it;
Trust in students and systematic verification of their academic work;
Pedagogically justified combination of business and emotional nature of relations with students, etc.
Pedagogical observation is the teacher's ability, manifested in the ability to notice the essential, characteristic, even subtle properties of students. In another way, we can say that pedagogical observation is a quality of the teacher's personality, which consists in a high level of development of the ability to concentrate attention on one or another object of the pedagogical process.
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