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What humans do with the language of mathematics is to describe patterns.
Sep 10, 2025
Teaching is the royal road to learning.
It is the duty of all teachers, and of teachers of mathematics in particular, to expose their students to problems much more than to facts.
It is impossible to be a mathematician without being a poet in soul.
I'm sorry to say that the subject I most disliked was mathematics. I have thought about it. I think the reason was that mathematics leaves no room for argument. If you made a mistake, that was all there was to it.
What science can there be more noble, more excellent, more useful for men, more admirably high and demonstrative, than this of mathematics?
A GREAT discovery solves a great problem but there is a grain of discovery in any problem.
Geometry is the science of correct reasoning on incorrect figures.
Don't just read it; fight it! Ask your own question, look for your own examples, discover your own proofs. Is the hypothesis necessary? Is the converse true? ... Where does the proof use the hypothesis?
I had done quite a bit of research about math education when I spoke before Congress in 2000 about the importance of women in mathematics. The session of Congress was all about raising more scholarships for girls in college. I told them I felt that it's too late by college.
It is not knowledge, but the act of learning, not possession but the act of getting there, which grants the greatest enjoyment.
Numbers constitute the only universal language.
Logic is invincible, because in order to combat logic it is necessary to use logic.
America ranks 21st when it comes to math education. We rank 25th when it comes to science. We used to be number one in the proportion of college graduates. We now rank ninth. And at an age where knowledge, skills, are the determinant of how successful we're going to be, unless we reverse that we're going to keep slipping behind economically to a lot of other countries.
One of the big misapprehensions about mathematics that we perpetrate in our classrooms is that the teacher always seems to know the answer to any problem that is discussed. This gives students the idea that there is a book somewhere with all the right answers to all of the interesting questions, and that teachers know those answers. And if one could get hold of the book, one would have everything settled. That's so unlike the true nature of mathematics.
Learning without thought is labor lost; thought without learning is perilous.
It is clear that the chief end of mathematical study must be to make the students think.
The title which I most covet is that of teacher. The writing of a research paper and the teaching of freshman calculus, and everything in between, falls under this rubric. Happy is the person who comes to understand something and then gets to explain it.
In the fall of 1972, President Nixon announced that the rate of increase of inflation was decreasing. This was the first time a president used the third derivative to advance his case for reelection.
Mathematics knows no races or geographic boundaries; for mathematics, the cultural world is one country.
The best teacher is not the one who knows most but the one who is most capable of reducing knowledge to that simple compound of the obvious and wonderful.
Lottery: A tax on people who are bad at math.
In a completely rational society, the best of us would be teachers and the rest of us would have to settle for something else.
The mind is not a vessel to be filled but a fire to be kindled.
The whole art of teaching is only the art of awakening the natural curiosity of young minds for the purpose of satisfying it afterwards.
I hear and I forget. I see and I remember. I do and I understand.
Music is a more potent instrument than any other for education.
Mathematics is the queen of the sciences.
If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is.
Pure mathematics is, in its way, the poetry of logical ideas.
Beauty is the first test: there is no permanent place in the world for ugly mathematics.
A teacher affects eternity; he can never tell where his influence stops.
Education is the key to unlock the golden door of freedom.
Opportunity is missed by most people because it is dressed in overalls and looks like work.
Standard mathematics has recently been rendered obsolete by the discovery that for years we have been writing the numeral five backward. This has led to reevaluation of counting as a method of getting from one to ten. Students are taught advanced concepts of Boolean algebra, and formerly unsolvable equations are dealt with by threats of reprisals.
The aim (of education) must be the training of independently acting and thinking individuals who, however, see in the service to the community their highest life problem.
Education is what remains after one has forgotten what one has learned in school.
Warmth is the vital element for the growing plant and for the soul of the child.
One looks back with appreciation to the brilliant teachers, but with gratitude to those who touched our human feelings. The curriculum is so much necessary raw material, but warmth is the vital element for the growing plant and for the soul of the child.
I never teach my pupils, I only attempt to provide the conditions in which they can learn.
Who dares to teach must never cease to learn.
Reductio ad absurdum, which Euclid loved so much, is one of a mathematician's finest weapons. It is a far finer gambit than any chess play: a chess player may offer the sacrifice of a pawn or even a piece, but a mathematician offers the game.
In most sciences one generation tears down what another has built and what one has established another undoes. In mathematics alone each generations adds a new story to the old structure.
Apply yourself. Get all the education you can, but then, by God, do something. Don't just stand there, make it happen.
Great spirits have always encountered violent opposition from mediocre minds.
Happiness is not so much in having as sharing. We make a living by what we get, but we make a life by what we give.
We make a living by what we get, but we make a life by what we give.
It is the mark of an educated mind to be able to entertain a thought without accepting it.
Everything that is really great and inspiring is created by the individual who can labor in freedom.
Keep away from people who try to belittle your ambitions. Small people always do that, but the really great make you feel that you, too, can become great.