If the proof starts from axioms, distinguishes several cases, and takes thirteen lines in the text book ... it may give the youngsters the impression that mathematics consists in proving the most obvious things in the least obvious way.
George PolyaThe result of the mathematician's creative work is demonstrative reasoning, a proof; but the proof is discovered by plausible reasoning, by guessing.
George PolyaThe teacher can seldom afford to miss the questions: What is the unknown? What are the data? What is the condition? The student should consider the principal parts of the problem attentively, repeatedly, and from from various sides.
George PolyaIf you have to prove a theorem, do not rush. First of all, understand fully what the theorem says, try to see clearly what it means. Then check the theorem; it could be false. Examine the consequences, verify as many particular instances as are needed to convince yourself of the truth. When you have satisfied yourself that the theorem is true, you can start proving it.
George PolyaThe principle is so perfectly general that no particular application of it is possible.
George PolyaIn order to translate a sentence from English into French two things are necessary. First, we must understand thoroughly the English sentence. Second, we must be familiar with the forms of expression peculiar to the French language. The situation is very similar when we attempt to express in mathematical symbols a condition proposed in words. First, we must understand thoroughly the condition. Second, we must be familiar with the forms of mathematical expression.
George Polya