Onmatu- General Math Learning Article

Solving a math problem II

        What can I do with an incomplete idea when solving a problem? Well you can check it. If you have the impression it is advantageous for you, then you need to take a closer look on it. If it seems a reliable idea it's good to make sure how far will lead you and then reconsider the situation. The situation changed because of the useful idea. Examine the new situation under different aspects and look for relations with the things you already know.
What can I gain by doing it so? You can have some luck and a new idea comes into your mind. Eventually, this new idea will lead you to the final solution. Maybe you will need some more useful ideas after this new idea. Maybe one of these ideas will lead you into an error. With all of this, you need to be grateful for all the new ideas, even for those less good, even for those nebulous, also for the additional ideas that add clarifications to a nebulous idea or try to improve an idea which is less happy. Even that a while you don't come with a good idea, you need to be satisfied if the way you are conceiving the problem becomes more clear and coherent, more homogeneous or balanced.
        4. Making the plan. 

        From where to start? Start from the happy idea which will lead you to the solution. Go ahead when you are sure on the way you are controlling the main relation and when you trust yourself that you can complete the secondary details which are missing.
        What can I do? Do whatever you can to control the problem at your best.
Perform in detail all algebraic and geometric operations which you figure out that they are possible. Convince yourself of the correctness of each step, through a formal thinking or through intuition or, if it's possible, through both. If the problem is too complicated, you can make a difference between the "big" steps and the "small" steps, every big step being made from small steps. Verify first the big steps and get down gradually to steps which are smaller and smaller.
What can I gain by doing it so? A presentation of the solution in which every step it is undeniable correct.

        5. A retrospective look.

From where to start? Start from the complete and correct resolution in all its details.
        What can I do? Consider the solution under different aspects and look for connections with the things you already know.
        Examine the details of the solution and try to make them as simple as you can; cover the more extensive parts of the problem and try to make them shorter; try to grasp from a single look the whole solution. Try to modify in an advantageous way smaller or bigger parts of the solution, try to perfect the structure of the solution, to make it intuitive, to fit it as natural as you can in what you already know.
Try to make shorter the method that led you to the solution, to show its stronger points and use it in other problems.
What can I gain by doing it so? You can find a new and better idea, you can discover new and interesting facts. Howsoever, If you make the habit of examining and shorten in this way the solutions, you will get well ordered knowledge and ready to be used, you will develop the ability of solving problems. Thank you  
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