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Let us return to the question how to teach problem-solving skills? In our article "Methodical feedbacks - what for are they needed?" we came to conclusion that a set of special exercises is required. In this article, we explain what are these special exercises are and how to choose them. 
The typical tasks, you should be capable to cope with at school, are: solve the equation, prove the identity, etc. In all these cases, you deal with symbolical records. While solving textual tasks, e.g. progressions, an additional step of transformation of textual information into symbolical is added.
What is a "formal" indicator that these tasks are solved? It is a set of consecutive calculations, proving and substantiating the solution. At each step, a number of questions should be answered, such as: what for do we carry out a given local transformation, what are the definitions and formulas used, etc. Not all these questions are formulated explicitly. Instead, this work is done "internally", on basis of information in our brains. In pedagogics, such information is referred to as a level of knowledge, abilities and skills. If this information lacks just one element, basic for decision-making, we will be not able to make the following step and, as a result, we will not cope with the task.
In this situation we need help of someone or something (remember cribs with formulas) who would help us to get this information. If the help is unprofessional, then we will be simply shown what the next step must be. This does not help to close a "hole" in our knowledge. A professional, on the other hand, will go into the whys and wherefores of the problem and will help us to close a vacancy in our basic knowledge, abilities and skills. Usually, the skilled teachers are familiar with typical errors and their causes. This is, so to speak, a stage of diagnostics. The therapy is a set of special exercises discussed in this article.
It seems enough to model a level of knowledge, abilities and skills, necessary for the solution of tasks, and then to construct a set of exercises. Teachers, surely, are engaged in this business, both theorists and practics. However, the "computer" between our ears is a too complex device. Centuries of teaching math yielded a multitude of special exercises but none of them guarantees that they suit personally you.
Below, I specify a few typical flaws in knowledge, abilities and skills, which limit ability to carry out mathematical transformations. This list of typical diseases is not complete at all; it is just one of many possible lists, with some "computer" specific.
- Lack of knowledge of general methods for solving the given type of tasks.
- Lack of understanding of purposes of the carried out transformations.
- Lack of knowledge of definitions and concepts concerning the parent calculation.
- Lack of knowledge of formulas and theorems used in the carried out transformation.
- Lack of skill to apply definitions.
- Lack of skill to apply formulas or theorems.
- Lack of ability to compare mathematical categories with each other while solving.
- Etc.
This article is not a theoretical paper and therefore we keep this list short. What is important to understand for the non-professionals is a character of problems. The professional can easily extend the list above by using the info from our list of "methodical schemes" on the page "Methods".
Any of the flaws from the list above can and, most likely, will result in "breaking" the process of solution. A possibility to oversee the next step, offered by our program, hardly helps. In a similar situation this "braking" will almost inevitably appear again. It is worthy to treat the cause of the disease instead of treating the consequence.
How to treat?
To build up knowledge, abilities and skills that are lacking. If there is the lack in knowledge of formulas, then it is necessary to force a student to learn them and to help him in learning. If the skill to apply a particular transformation is absent, then it is necessary to select similar transformations and focus on a choice of transformations, etc. Surely, the professional teacher should do this big work with a student. The real teacher, however, lacks time and strength to do this with each student. As a result, not all students manage. You probably know what happens further. The teacher starts talking about "inabilities" of a student, gives recommendations to parents to find for a student something "easier", that is to transfer him to classes with the "lowered" level of mathematical requirements etc. In a country where a factor is calculated which affects a chance to continue the education, such situations may result in tragedies. By the way, about abilities: I shall remind you that Einstein and Hilbert were classified at school as "incapable" and there are plenty of examples of such sort. As a teacher, I personally prefer a word "neglected". "Incapable" sounds as a God's burn while the "neglected" child can be helped if the appropriate efforts are applied.
What should be done if my child does not manage?
To employ a personal tutor (coach). This is, certainly, the best way, but it is rather expensive and not everyone has such opportunity. If you cannot employ a tutor, try to help your child yourself. It is your own child; it is worthy to fight for him.
Now let us consider whether such tutorial work can be imposed on computer. What is the output from work with the program? Can such program replace the professional teacher?
Is it possible to impose work of a teacher on the training program?
In part, it is certainly possible. We actually offer the program with functions aiming at solution of problems described above. The natural question arises: is it a good idea to allot a task of choice of specialized exercises to the student himself, considering that not every professional can cope with this task? It is certainly difficult, but studying is a difficult job in general. Let us try to understand what is difficult. (It is better, if you will read the text below with the program before your eyes). Thus, we open the list of contents, with themes, chapters and sections, and we choose an example. The next step is to choose a methodical scheme suitable for studying the chosen example. If you have no slightest idea where to begin, the best choice would be the simplest scheme "To view" the solution. Go through the course of solution. What is further? Try the same example with the other technique - "To solve independently". If you did not succeed, try an "easier" technique (any scheme above the current one in the list of schemes). Even if you succeeded, there remain things to do with the studied example. Namely, ask yourself a question: do I know all formulas and definitions related with this example? To learn them, select one of schemes "To learn" and go through the example with it. Well, and so on. By the way, didn't you notice that we were engaged in self-diagnostics? Was it difficult? Not at all. All questions, answers, hints and marks were under hand.
What is difficult is to get used to the program, to learn how to work with it. However, there is no way out; you have to learn how to work with any tool, even with a hammer. Do you need to go through each example with all available techniques? Certainly not. If you "went through" an example with one technique without any problem, then this means that you drilled a number of problems for which the given methodical scheme is intended. In some time, you will necessary get familiar with a set of methodical schemes, which suits you particularly. Eventually you will learn all formulas and definitions and their applications, and then you can set aside methodical schemes meant to build up these skills.
This is a long procedure; you cannot do this in a couple of hours. Remember, how many years are allocated to studying of elementary mathematics at schools. This is the centuries-old experience. Besides, none of training programs can learn mathematics instead of you. The program can help you to fill your head with necessary knowledge and skills. Well, this cannot be done without your active participation. One important recommendation: do write the solutions of examples on paper. Unfortunately, the fast and easy input of mathematical information to the computer is currently not possible. Therefore, for checking of your knowledge, it is necessary to use a system of tests. Each test, however, necessarily contains hints. I suppose that you yourself can provide the best check of your knowledge: just take a problem to be solved, a piece of paper, a pen - and go ahead!
Can such program replace the professional teacher?
At the present level of computer technologies, it is impossible to replace a good teacher by the program. Each program is a model of an aspect of activity, and each model is a small "piece" of the actual situation. Considering that each model contains many uncertainties (e.g., the model "student - teacher" contains more uncertainties than any model studied in NASA) - the result is clear.
(About NASA and pedagogics. At present, like many years ago, one needs 15-20 years of his life to learn math well. To go to space, on the other hand, one needs just 15-20 millions dollars - possibly borrowed; what is more - the prices are falling).
What is the payoff of work with the program?
The payoff will be substantial if methodical schemes offered by the program will cover "holes" in your knowledge and skills. If not - there will be no payoff. In order to help you in trying our programs "on", so that you can come to your own conclusions, we produce the free-of-charge "Light" versions of our programs. I repeat once again: there exists no universal method of teaching, and no universal "best in the world" program. The teaching program is a tool with a number of certain properties. If these properties facilitate your work, then you need it, but if not - then better look for another tool.
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