Fragment of the book “How to solve it”, by G. Polya.
Teachers
A good teacher should understand and impress on his students the view that no problem whatever is completely exhausted. There remains always something to do; with sufficient study and penetration, we could improve any solution, and, in any case, we can always improve our understanding of the solution.
Problems to Find
Solving “problems to find” of any kind, we can follow its pattern: Keep only a part of the condition, drop the other part. Doing so, we weaken the condition of the proposed problem, we restrict less the unknown. How far is the unknown then determined, how can it vary? By asking this, we set, in fact, a new problem.
Heuristics
Heuristic, as an adjective, means “serving to discover”.
Heuristic reasoning is reasoning not regarded as final and strict but as provisional and plausible only, whose purpose is to discover the solution of the present problem. We are often obliged to use heuristic reasoning. We may need the provisional before we attain the final.
Notations
There is always something arbitrary and artificial about notation; to learn a new notation is a burden for the memory. The intelligent student refuses to assume the burden if he does not see any compensation for it. The intelligent student is justified in his aversion for algebra if he is not given ample opportunity to convince himself by his own experience that the language of mathematical symbols assists the mind.
Working Backwards
Retracing our steps, we finally succeed in deriving what was required.
There is certainly something in the method that is not superficial. There is a certain psychological difficulty in turning around, in going away from the goal, in working backwards, in not following the direct path to the desired end. When we discover the sequence of appropriate operations, our mind has to proceed in an order which is exactly the reverse of the actual performance.
Quite primitive problems may lead naturally to working backwards.
Working backwards is a common-sense procedure within the reach of everybody.