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.
It is better to solve one problem five different ways, than to solve five problems one way.
Mathematics is not a spectator sport!
Solving problems is a practical art, like swimming, or skiing, or playing the piano: you can learn it only by imitation and practice.
Beauty in mathematics is seeing the truth without effort.
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.
A mathematics teacher is a midwife to ideas.
Look around when you have got your first mushroom or made your first discovery: they grow in clusters.
It may be more important in the mathematics class how you teach than what you teach.
An idea which can be used once is a trick. If it can be used more than once it becomes a method.
My method to overcome a difficulty is to go round it.
The open secret of real success is to throw your whole personality into your problem.
A GREAT discovery solves a great problem but there is a grain of discovery in any problem.
The world is anxious to admire that apex and culmination of modern mathematics: a theorem so perfectly general that no particular application of it is feasible.
The 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.
If you cannot solve the proposed problem, try to solve first some related problem.
Hilbert once had a student in mathematics who stopped coming to his lectures, and he was finally told the young man had gone off to become a poet. Hilbert is reported to have remarked: ‘I never thought he had enough imagination to be a mathematician.’
I am intentionally avoiding the standard term which, by the way, did not exist in Euler’s time. One of the ugliest outgrowths of the “new math” was the premature introduction of technical terms.