How to learn how to solve problems

   Firstly, you must learn to analyze the problems themselves.

   This means that you need to be able to dissect the problem into the basic situation and requirements. In every elementary statement you should see the object and its characteristics; if there is more than one object in the situation, you should find out their relation (link). You also need to establish the nature of each requirement (question), and thus determine the type of problem. It is useful to stick to the following rule: unless a complete, in-depth analysis of the problem is conducted, unless, if necessary, its schematic notation is formulated, you should not proceed to the solution.
    Secondly, we should well understand that the solution to any problem is based on the consistent application of some knowledge (mainly mathematical) to the statement of this problem in order to obtain conclusions from these statements (intermediate solutions), until conclusions are reached that satisfy all the requirements (questions) of the problem. In order to obtain these conclusions, it is necessary to remember all the knowledge (definitions, formulas, theorems) from the course of mathematics.
   Thirdly, you must be able to use basic methods for solving problems. There are only three of them: partitioning the problem into subproblems, modelling (transformation) of the problem and the method of auxiliary members.
   Upon receiving the problem, you should analyze it, build its schematic notation (if needed), and then act in the following order:
1. If you can, break down the problem into smaller subproblems. In a number of cases it is possible to partition it sequentially, isolating subproblems from the problem one by one.
2. If  it is not possible to break down a complex problem into subproblems, you should, if possible, transform it into a simpler, more familiar form.
3. If it is not possible either to break the problem into subproblems, or to transform it into a simpler form, you have to introduce some auxiliary elements in order to obtain a problem that can be broken down into subtasks, or transformed into a simpler form .


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