Polya George. 1887-1985.

The elegance of a mathematical theorem is directly proportional to the number of independent ideas one can see in the theorem and inversely proportional to the effort it takes to see them.
Mathematical discovery (New York, 1981).



To teach effectively a teacher must develop a feeling for his subject; he cannot make his students sense its vitality if he does not sense it himself. He cannot share his enthusiasm when he has no enthusiasm to share. how he makes his point may be as important as the point he makes; he must personally feel it to be important.
Mathematical discovery (New York, 1981).



Mathematics is the cheapest science. Unlike physics or chemistry, it does not require any expensive equipment. All one needs for mathematics is a pencil and paper.
Quoted in D J Albers, G L Alexanderson and C Reid, Mathematical People (Boston 1985).


 
When introduced at the wrong time or place, good logic may be the worst enemy of good teaching.
The American Mathematical Monthly 100.



If there is a problem you can't solve, then there is an easier problem you can solve: find it.
How to Solve It (Princeton, 1945).



If 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 theorem is true, you can start proving it.
How to Solve It (Princeton, 1945).