Many students and parents struggle with the poor quality of knowledge they receive at school. What a child learns in school math classes is often not enough to confidently cope with most of the assignments and exercises offered and to get a good grade. And this is often true even for children who are quite developed and intelligent. Practice shows that learning at school does not always allow you to learn at the level of individual abilities and capabilities, - at their own level many children do not have the results that they could have. And even children who perform well often have places in the material that they could have known much better. This is what a tutor has to deal with in everyday practice.
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There is no mystery as to why this happens. Of course, it all depends on the particular teacher teaching your child's school - some do a brilliant job of teaching. Still, more often the emphasis is on the controlling and punitive function, rather than on the transmission of knowledge. As a rule, the learning process at school is built in such a way that it does not take into account both individual and universal features of perception and assimilation of material by each student. And these problems are especially acute in learning mathematics.
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Mathematics as a subject is "terrible" because it consists of a very large number of small skills and operations, the knowledge of which is absolutely necessary in solving various problems and examples. If you do not master even one of these skills, it is impossible to correctly solve a huge number of tasks. Neither schoolteachers nor parents are often able to track their adequate learning. It should be noted that for each of these skills to be firmly mastered, one needs practice; thorough practice in a wide variety of tasks and situations. Besides, the human memory is organized in such a way that it is necessary to return to each learned technique several times after certain intervals; otherwise, the acquired information will not be transformed into a confident knowledge. There is another peculiarity. The human brain is able to perceive only that information which is somehow related to something already known and familiar. For example, there is a rule that all good lecturers know: the amount of new material must not exceed 20%, otherwise the attention of the audience will be lost. In practice this is realized in the well-known principle of teaching "from the simple to the complex. If the new topic in the presentation is overloaded with techniques unknown to the child, and if the complexity of the tasks is not built up gradually, the student is simply not able to understand and perceive it. Here it is a matter of individual differences which are very difficult to take into account in any group teaching.
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Our textbooks, for the most part, do not implement these principles sufficiently. The same is true of the large number of different training manuals. The number of exercises in them, as a rule, is enough to master the material only for the most quick-thinking children. Such exercises are chaotically scattered throughout the textbook, disorienting children by giving them a limited number of exercises which do not allow them to properly understand and practice the new material, to "sink" into it, even if they are repeated more than once. As a result, every new technique that needs to be taught to students is taught too superficially, vaguely and fragmented, leaving no lasting knowledge in the mind. It is as if all of our textbooks and teaching materials are written for mathematically gifted students who can learn from a minimum of examples, or that their authors have never taught. Children, who are often just as talented, but "slower" in speed of perception, do not have time to understand, to penetrate into the material studied, and begin to accumulate incomplete or not understood at all parts of the program. When such "gaps" become too many, it is immediately disastrous for the knowledge and grades.
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Individual teaching has undoubted advantages over school group teaching, precisely because it takes into account all the above-mentioned aspects of learning. Can the results of learning be guaranteed? Any competent tutor after one or two lessons can assess the prospects of a given student. And they depend, of course, not so much on the level of knowledge, as on the "teachability" of the child. In mathematics, this implies, first of all, the ability to memorize the necessary techniques, the ability to reproduce them and use the acquired knowledge in practice. The category of students who are doing well with these abilities, and problems with performance exist solely because of the lack of knowledge, practice of problem solving - the most "grateful" in terms of "return" from the lessons. Existing gaps are quickly filled, at any level, even if one or two previous years of study have to be "picked up". In these cases the instructor can speak confidently about the results of the class. More difficult