The traditional mathematics professor of the popular legend is absentminded. He usually appears in public with a lost umbrella in each hand. He prefers to face a blackboard and to turn his back on the class. He writes

"In order to solve this differential equation you look at it till a solution occurs to you."

"This principle is so perfectly general that no particular application of it is possible."

"Geometry is the science of correct reasoning on incorrect figures."

"My method to overcome a difficulty is to go round it."

"What is the difference between method and device? A method is a device which you used twice."

Even fairly good students, when they have obtained the solution of the problem and written down neatly the argument, shut their books and look for something else. Doing so, they miss an important and instructive phase of the work. ... A good teacher should understand and impress on his students the view that no problem whatever is completely exhausted.

One of the first and foremost duties of the teacher is not to give his students the impression that mathematical problems have little connection with each other, and no connection at all with anything else. We have a natural opportunity to investigate the connections of a problem when looking back at its solution.

In 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.

If there is a problem you can't solve, then there is an easier problem you can solve: find it.

A GREAT discovery solves a great problem, but there is a grain of discovery in the solution of any problem. Your problem may be modest, but if it challenges your curiosity and brings into play your inventive faculties, and if you solve it by your own means, you may experience the tension and enjoy the triumph of discovery.

The first rule of discovery is to have brains and good luck. The second rule of discovery is to sit tight and wait till you get a bright idea.

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 the theorem is true, you can start proving it.

**All the above are from ***How to Solve It* (Princeton 1945).

Mathematics consists of proving the most obvious thing in the least obvious way.

Quoted in N Rose *Mathematical Maxims and Minims* (Raleigh N C 1988).

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).

There are many questions which fools can ask that wise men cannot answer.

Quoted in H Eves *Return to Mathematical Circles* (Boston 1988).

When introduced at the wrong time or place, good logic may be the worst enemy of good teaching.

*The American Mathematical Monthly* **100** (3).

A mathematician who can only generalise is like a monkey who can only climb up a tree, and a mathematician who can only specialise is like a monkey who can only climb down a tree. In fact neither the up monkey nor the down monkey is a viable creature. A real monkey must find food and escape his enemies and so must be able to incessantly climb up and down. A real mathematician must be able to generalise and specialise.

Quoted in D MacHale, *Comic Sections * (Dublin 1993)

Mathematics is being lazy. Mathematics is letting the principles do the work for you so that you do not have to do the work for yourself.
Quoted in M Walter, T O'Brien, Memories of George Pólya, *Mathematics Teaching* **116** (1986)

Look around when you have got your first mushroom or made your first discovery: they grow in clusters.

A mathematics teacher is a midwife to ideas.

John von Neumann was the only student I was ever afraid of.

The apex and culmination of modern mathematics is a theorem so perfectly general that no particular application of it is feasible.

I am too good for philosophy and not good enough for physics. Mathematics is in between.

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.

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).

JOC/EFR February 2006

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