A characteristic feature of our time is the desire of many of teachers to rebuild the educational process, to strengthen the students, interest them, to accustom to independent work. Some of the teachers there has been serious progress.

- Every teacher wants to have students who can think. Logical thinking is not an intrinsic condition for successful learning. Unfortunately, students rarely try to think. Effect included in last years in school practice the habit of doing (whatever it is!) «faster than all»: who made quickly and raised his hand (the first!), he is the hero of the day. And if you think to raise his hand in no time, you will outperform others. Or announce reply, the teacher of the Vedas he «needs» to do all the planned work, then, delay not, to think — once. One task was solved, move on to the next... [1].

The belief of teachers that the lesson would «need» to execute the pre-planned volume of work that you think students should be fast, and only fast — a dangerous delusion. In this formulation, the training, the student is forced to solve the problem only «in the image and likeness» previous tasks. And the results of such a formulation of the training can't be good.

Often students enforceby addressing several dozen of the same tasks. Yes, of course, after that almost all will pass this test where you will be offered a similar task. But if after «exploring» this way to the six themes to offer a test which will consist of three different tasks, the result will be much less favorable.

So, the solution of each task required to teach a student to think, to to summarize, to analyze, to consider the options to build counterexamples, to make their tasks — not only dismantled the same, but it follows naturally from rules, formulas, theorems, properties of the functions... Much more useful to parse several ways to accomplish the same (and very simple) task, than hastily to solve three or four are similar to each other.

It is necessary to systematically equip students and techniques of proof (or solve problems on the proof). Proof by induction, «on the contrary», cycle through the options etc. Are a powerful means of enhancing the cognitive activity is not only student — they are helpful and Mature, and can be used not only in the context of learning.

An important requirement of school reform — the development of logical thinking — not getting exercise, examining only standard tasks, even if be solved again from scratch. Of course, another extreme, focusing only on tasks requiring ingenuity and innovative solutions, may not give the desired results of study: not owning the basic knowledge (calculations, transformations, skills in solving basic types of the equation and the simplest scheme), students will not cope with many of the proposed tasks (for example, task reference works).

In the described experience, it is possible to allocate following basic directions of this activity in the classroom.

1) Lessons and lectures with the purpose of studying new topics large unit, to improve the minds of students when learning a new, time saving for further creative work.

2) Lessons to address key challenges on the topic.The teacher (with the students), extracts a minimal number of tasks, which is implemented studied the theory, taught to recognize and solve key challenges.

3) Lessons — consultations at which the questions are asked by students and answers them is a teacher.

4) Transcripts of the lessons, the purpose of which is to organize individual work, assistance older students for junior, gradual preparation for solving more complex problems.

- Lesson-lecture — the most difficult type of lesson, even for experienced teachers. On the one hand, the teacher must be a brilliant lecturer, awakens in each the desire for reflection, and the actor's own skills, on the other — to keep an eye on each student in the class and manage its activities.

For 45–90 minutes the teacher presents all the most important information on the topic. It's not just a retelling, not a «chewing» of the textbook, as it were, the transformation of the topic through personal experience of teachers, the interpretation of the themes of the teacher.

In the process of preparation for the lecture the teacher will decide not only what to present but how to present: what material to cover yourself, some leave students to independent work that to discuss in detail, less detail. You can, for example, a question to direct to report one of the disciples, as if to intersperse his mini-lecture in his lecture. You can provide the success, delight, joy, and it is possible — and obviously erroneous reasoning, that would be after establishing its falsity (for example, by constructing a counterexample) to direct the attention of students differently and to offer students tasks practical content on the topic of study.

During the lectures students should be encouraged to ask questions, make suggestions and not be afraid to make a mistake. It is even possible to resolve the issue is to invoke the participation of the entire class. Better when the student makes as many sentences, even if erroneous, than remained silent, remains aloof from the discussion.

Learning math is primarily a training problem solving. Should we aspire to student decided as much as possible? Not at all.

Many of the tasks published in the textbooks, problem books, teaching guides, to a large extent duplicate each other, differing only in numerical values, physical content, marks or other not very important items, while their mathematical essence is the same.

Wizard can perform there work for 20 days and the student is 30 days. How many days they execute this work together?The faucet with cold water can fill empty tank in 20 min, and faucet with hot — 30 min. how much time will the empty tank be filled if you open both faucet?

From point A to point In the train, which goes all the way for 20 h; at the same time from paragraph To paragraph And sent by another train, end up in here is the path for 30 h. after how many hours the trains will meet?

The solution to each of these tasks can be obtained by the same arithmetic:

A) 1/20+1/30=1/12) 1∶ 1/12=12

- So the key tasks are learned, the necessary base of fundamental knowledge and skills in the topic lies. However this knowledge only of the algorithms for the solution of key tasks may not satisfy those students who show interest in mathematics. In working with them, it is important to go to the tutorial custom.
- The purpose of the lessons — consultations — to teach students to think about the problem, to understand — for myself — what are having difficulty in getting acquainted with a certain topic and to resolve these difficulties is to formulate the questions on which he wished to answer[2].

Noticed that younger students often ask for clarification from the teacher, to elders, to family members. To 5–7 classes, they usually stop asking questions. Are they not facing any difficulties?

Difficulties certainly arise, as before. Changing the psychology. Not received times of a response from parents (not all parents are able to satisfactorily answer the increasingly complex questions) after hearing from teachers, «then» or «another time», students become isolated, lose interest in solving problems, and sometimes for learning.

After addressing the key issues, after the acquisition some skill to task one of the key thinking students questions arise:

– How is solving a more General, more private tasks?

– How to change the decision, if you change the condition in one or another part?

– How to make this more complicated key?

– Whether the converse is true (theorem)?

The lesson-consultations, of course, also within the school timetables, issues and challenges offer students and responsible teacher. However, at first, students often have no idea what questions they might ask: because most students are accustomed only to reproductive activities, that is, to «separate» solution of problems similar to what you disassembled. Therefore, at the beginning of lessons-consulting teacher helping to formulate questions.So, after studying in 7th grade of the theorem on inscribed angle, the teacher asks one of students more time to formulate it and then to put the question on the measure of the angle whose apex lies inside of the circle, and the sides intersect the circle. Then asks other probing questions, and in the end, the students of «themselves» for putting «new» issues: measurement of the angle with vertex outside the circle angle between tangent and secant, of the angle between two tangents to the same circle, etc.

In every class there are students who formulated the questions and tasks, but feel free to Express them in the presence of the class. Sometimes they ask questions at break or after school. Roman every time he praises them for the interesting question and promise to respond in detail to the next lesson. This, he believes, first, it increases the response code, which perceives the whole class, and secondly, allows you to publicly praise the one who asked the question and thereby stimulate its activity further. Note that Roman loves children, trying to find the slightest reason to praise the pupil, in no case does not raise accusations, shows his dissatisfaction with the student.

Gradually students get accustomed to find issues and challenges, using not only textbooks, but also journals and other supporting literature. For each lesson, with advice students prepare cards — the usual sheets of paper with questions and tasks.

High school students sometimes include a card so many problems and issues, to analyse them all in a single lesson is impossible. Then the teacher tries to combine related tasks into 5–6 groups so that when one task was to identify ways to address remaining challenges.

The most effective finds a way to answer: he formulates a new, more General problem, which covers several proposed by students tasks as special cases, parses her decision and outlines the decision options, relevant tasks from the cards.

It is clear that students usually do not have time to write out solutions to all the parsed task. But at home, they repeat the arguments and, if necessary, record the solutions in their notebooks. Many of parsed tasks, perhaps in other wording will be included in the student's record card — individual assignments offered for credit lessons.

- Test lessons is lessons individual work, which serve for monitoring and evaluation of knowledge, and to an even greater extent for the purposes of training, education and development.

In fact, in a survey at the Board many students are not interviewed on this topic; others having two, «fix» her answer to a completely different subject — «the tail of ignorance» is growing. Examination identifies gaps but does not allow to provide differentiated assistance. Using traditional forms of survey particularly affected the most able students: the teacher has to spend class time on repetition and explanation of the material that caused difficulties for the weak, and the remaining students are bored and are losing interest in the subject.

The scoring system takes teachers care about the accumulation of assessments; lessons, for example lessons to address key challenges, there is the possibility of more creative communication; discussion of issues becomes more free; you can Express any thoughts, even «blurt out» stupid is a bad evaluation or reprimand will follow. The student, originally decided in the difficult task of not waiting for «the payment» a good rating, but certainly gets the aesthetic pleasure, and the recognition of friends.

The scoring is performed on each studied topic and promotes robust enough learning themes. A huge benefit and the host gets the credit: he repeats the theme as a whole at a higher level compared to the previous year. There has been a rethinking of the material, systematization, comparison of old and new — and thereby develops the thinking of students of the senior class.

Set-off usually have a lesson or two». After receiving the card, the student within 45 minutes preparing: formulates answers to questions,preparing the proof of the theorem, a formula solution to the problem, but doesn't spend much time on the design and rewriting. For the next 45 minutes, he responds, and receives three evaluations: for theory, for the solution of tasks, for the maintenance of a workbook. Each estimate is motivated. In case of unsatisfactory evaluation, the credit is re-during after school hours.

In one of the scoring cards (a regular sheet of paper) on which a pupil of the ninth class took the offset pupil of the 8th class, entered tasks, for example:

1) Find the first and the fifth members of the geometric series, the denominator of which is equal to 3, and the sum of the first six members is 1820.

2) Between the numbers 60 and 15/16 to paste these 5 numbers together with the data numbers would be in geometric progression.

3) Find the sum of the first seven members of a geometric progression, if the denominator is equal to 2/3, and the seventh member 64/81.

4) Find the number of the member of geometric progression 0,1; 0,3......, different 218,7 as well as a number of theory questions[3].

A scoring card for the pupils of the seventh class on the theme «Signs of equality of triangles» included tasks for the proof of the equality of triangles:

- On the side of мedina, held to that side and the angles which it forms with the side of мedina;
- At the corners of мedina and that мedina divides the angle of the triangle;
- The мedina and the two parties emanating from the same vertex, аnd tasks:

In triangle ABC side AC points M and E so that AM=EU. It is known that MV=VE (given in the card drawing). To prove that the triangle ABC is equal to thigh.

The two-hour competition (and the examiner spends on the standings only 1 hour, not counting, of course, time spent at home on preparation of the record cards) makes a significant contribution to the development and education of six to seven dozen students of the two classes. So solved one of the major problems of mass — and at the same time, individual training to decide which traditional methods are not possible.

References:

- Khazankin R. G., Silberberg N. The experience of the organization and work of the Scientific society of students. //The Teacher оf Bashkortostan, 1984, № 1.
- Khazankin R. G., Silberberg N. Lesson: lecture at the school.//The Teacher оf Bashkortostan, 2000, № 1.
- Khazankin R. G. Mathematics education and secondary school.//Mathematical education, 2002, № 3.

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