Math & Teaching Quotations


Many of these quotes come from my ever-growing collection of recreational math books; others were found on my internet travels. I am always on the lookout for math cartoons, jokes, puzzles, and quotations. These quotes have been used to stimulate student interest in math and can be tied into the lesson for the day, a current event, or a related subject of interest. Questions can be posed about the quotes, and short biographical essays about the authors can be made as assignments in the history of math. I like these quotes also because they remind me of great thoughts and keep my teaching interest keen.

Another source of quotations can be found at my companion website, Teacher to Teacher Press. For those wanting larger online sources of math quotes, see the Furman University Mathematics Quotation Server or the Mathematical and Educational Quotation Server at Westfield State College. A good source of quotations in book form can be found in Out of the Mouths of Mathematicians: A Quotation Book for Philomaths, by Rosemary Schmalz.

The quotations below are listed alphabetically by author. If you have a favorite quote, please email me the quote and the source. I appreciate sharing great ideas; they’re a way to stimulate the thinking and feel great about mathematics and life.


Mr. L


The arithmetic of life does not always have a logical answer.

– Inshirah Abdur-Rauf

Mathematical knowledge adds vigour to the mind, frees it from prejudice, credulity, and superstition.

– John Arbuthnot

Pedagogy, like language itself, can either liberate or imprison ideas, inspire of suffocate constructive thinking.

– Hyman Bass, from “Mathematicians as Educators”

Mathematics is one of the deepest and most powerful expressions of pure human reason, and, at the same time, the most fundamental resource for description and analysis of the experiential world.

– Hyman Bass

To see the world in a grain of sand,
And a heaven in a wild flower;
Hold infinity in the palm of your hand,
And eternity in an hour.

– William Blake

Somebody came up to me after a talk I had given, and say, “You make mathematics seem like fun.” I was inspired to reply, “If it isn’t fun, why do it?”

– Ralph P. Boas

I have discovered such wonderful things that I was amazed… Out of nothing I have created a strange new universe.

– Janos Bolyai, speaking about his discovery of non-Euclidean Geometry

I advise my students to listen carefully the moment they decide to take no more mathematics courses. They might be able to hear the sound of closing doors.

– James Caballero

What distinguishes a mathematical model from, say, a poem, a song, a portrait or any other kind of “model,” is that the mathematical model is an image or picture of reality painted with logical symbols instead of with words, sounds or watercolors.

– John Casti, from Reality Rules: 1.

Statistics are no substitute for judgment.

– Henry Clay

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.

– Marshall Cohen

One of the endlessly alluring aspects of mathematics is that its thorniest paradoxes have a way of blooming into beautiful theories.

– Philip J. Davis

Though the structures and patterns of mathematics reflect the structure of, and resonate in, the human mind every bit as much as do the structures and patterns of music, human beings have developed no mathematical equivalent to a pair of ears. Mathematics can only be “seen” with the “eyes of the mind”. It is as if we had no sense of hearing, so that only someone able to sight-read music would be able to appreciate its patterns and harmonies.

– Keith Devlin, from Mathematics: the Science of Patterns.

Indeed, nowadays no electrical engineer could get along without complex numbers, and neither could anyone working in aerodynamics or fluid dynamics.

– Keith Devlin, from Mathematics: The New Golden Age.

We only think when confronted with a problem.

– John Dewey, quoted in Harper’s Quotes.

The life of a mathematician is dominated by an insatiable curiosity, a desire bordering on passion to solve the problems he is studying.

– Jean Dieudonne, from Mathematics – The Music of Reason.

Geometry is the foundation of all painting.

– Albrecht Durer

Although he may not always recognize his bondage, modern man lives under a tyranny of numbers.

– Nicholas Eberstadt, The Tyranny of Numbers: Mismeasurement and Misrule

It’s not that I’m so smart, it’s just that I stay with problems longer.

– Albert Einstein

Not everything that counts can be counted. Not everything that can be counted counts.

– Albert Einstein

Great spirits have always encountered violent opposition from mediocre minds.

– Albert Einstein

Teaching should be such that what is offered is perceived as a valuable gift rather than a hard duty.

– Albert Einstein

Pure mathematics is, in its way, the poetry of logical ideas.

– Albert Einstein

Pure mathematics is, in its way, the poetry of logical ideas.

– Albert Einstein

Only two things are infinite, the universe and human stupidity, and I’m not sure about the former.

– Albert Einstein

It will be another million years, at least, before we understand the primes.

– Paul Erdos

By keenly confronting the enigmas that surround us, and by considering and analyzing the observations that I had made, I ended up in the domain of mathematics. Although I am absolutely without training in the exact sciences, I often seem to have more in common with mathematicians than with my fellow-artists.

– M.C. Escher

For me it remains an open question whether [this work] pertains to the realm of mathematics or to that of art.

– M.C. Escher

It is impossible to overstate the imporance of problems in mathematics. It is by means of problems that mathematics develops and actually lifts itself by its own bootstraps… Every new discovery in mathematics, results from an attempt to solve some problem.

– Howard Eves

There is a distinction between what may be called a problem and what may be considered an exercise. The latter serves to drill a student in some technique or procedure, and requires little if any, original thought… No exercise, then, can always be done with reasonbable dispatch and with a miniumum of creative thinking. In contrast to an exercise, a problem, if it is a good one for its level, should require though on the part of the student.

– Howard Eves

The journey for an education starts with a childhood question.

– David L. Finn

A math student’s best friend is BOB (the Back Of the Book), but remember that BOB doesn’t come to school on test days.

– Josh Folb

Mathematics compares the most diverse phenomena and discovers the secret analogies that unite them.

Joseph Fourier

The profound study of nature is the most fertile source of mathematical discovery.

– Joseph Fourier

The universe stands continually open to our gaze, but it cannot be understood unless one first learns to comprehend the language and interpret the characters in which it is written. It is written in the language of mathematics, and its characters are triangles, circles, and other geometric figures, without which it is humanly impossible to understand a single word of it; without these, one is wandering about in a dark labyrinth.

– Galileo

[Paradoxes of the infinite arise] only when we attempt, with our finite minds, to discuss the infinite, assigning to it those properties which we give to the finite and limited.

– Galileo

If present trends continue, our country may soon find itself far behind many other nations in both science and technology–nations where, if you inform strangers that you are a mathematician, they respond with admiration and not by telling you how much they hated math in school, and how they sure could use you to balance their checkbooks.

– Martin Gardner

Biographical history, as taught in our public schools, is still largely a history of boneheads; ridiculous kings and queens, paranoid political leaders, compulsive voyagers, ignorant general – the flotsam and jetsam of historical currents. The men who radically altered history, the great scientists and mathematicians, are seldom mentioned, if at all.

– Martin Gardner

All mathematicians share… a sense of amazement over the infinite depth and the mysterious beauty and usefulness of mathematics.

– Martin Gardner

The essence of mathematics is not to make simple things complicated but to make complicated things simple.

– S. Gudder

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.

– Paul Halmos

The only way to learn mathematics is to do mathematics.

– Paul Halmos

Mathematics is not a deductive science – that’s a cliché. When you try to prove a theorem, you don’t just list the hypotheses, and then start to reason. What you do is trial and error, experimentation, guesswork.”

– Paul Halmos

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.

– Hermann Hankel

The value of a problem is not so much coming up with the answer as in the ideas and attempted ideas it forces on the would be solver.

– I. N. Herstein

One cannot escape the feeling that these mathematical formulas have an independent existence and an intelligence of their own, that they are wiser than we are, wiser even than their discoverers…

– Heinrich Hertz

The discoveries of Newton have done more for England and for the race, than has been done by whole dynasties of British monarchs.

– Thomas Hill

Just as any sensitive human being can be brought to appreciate beauty in art, music or literature, so that person can be educated to recognize the beauty in a piece of mathematics. The rarity of that recognition is not due to the “fact” that most people are not mathematically gifted but to the crassly utilitarian manner of teaching mathematics and of deciding syllabi and curricula, in which tedious, routine calculations, learned as a skill, are emphasized at the expense of genuinely mathematical ideas, and in which students spend almost all their time answering someone else”s questions rather than asking their own.

– Peter Hilton

No wonder that Churchill described this effort [the British codebreakers working at Bletchley Park] as “Britian”s secret weapon,” a weapon far more effective than the buzz bombs and the rockets that Werner von Braun designed for a German victory, a weapon absolutely decisive, in the judgement of many, in winning the war for the Allies.

– Peter Hilton

The hardest arithmetic to master is that which enables us to count our blessings.

– Eric Hoffer, Reflections On The Human Condition

In a completely rational society, the best of us would aspire to be teachers and the rest of us would have to settle for something less, because passing civilization along from one generation to the next ought to be the highest honor and highest responsibility anyone could have.

– Lee Iacocca

Fall seven times, stand up eight.

– Japanese proverb

I’ve missed 3000 shots. Twenty-six times the game-winning shot has been trusted to me, and I’ve missed. I’ve lost over 300 games. I’ve failed over and over and over again, and that is why I’ve succeeded.

– Michael Jordan

The infinite in mathematics is always unruly unless it is properly treated.

– Edward Kasner and James Newman

…it is the greatest achievement of a teacher to enable his students to surpass him.

– John Kemeny

One’s intellectual and aesthetic life cannot be complete unless it includes an appreciation of the power and the beauty of mathematics. Simply put, aesthetic and intellectual fulfillment requires that you know about mathematics.

Jerry P. King, from The Art of Mathematics

The tantalizing and compelling pursuit of mathematical problems offers mental absorption, peace of mind amid endless challenges, repose in activity, battle without conflict, refuge from the goading urgency of contingent happenings, and the sort of beauty changeless mountains present to senses tried by the present- day kaleidoscope of events.

– Morris Kline

Many who have never had the occasion to discover more about mathematics consider it a dry and arid science. In reality, however, it is a science which demands the greatest imagination.

– Sofia Kovalevskaia

There is no branch of mathematics, however abstract, which may not someday be applied to phenomena of the real world.

– Nicolai Lobachevsky

The teacher who is attempting to teach without inspiring the pupil to learn is hammering on cold iron.

– Horace Mann

We encourage children to read for enjoyment, yet we never encourage them to “math” for enjoyment. We teach kids that math is done fast, done only one way and if you don’t get the answer right, there’s something wrong with you. You would never teach reading this way.

– Rachel McAnallen, from “Math?  No Problem”

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 …

–  H.L. Mencken

It seems quite unrealistic to judge a curriculum by its general outline, or to judge a course by its syllabus. We can “cover” very impressive material, if we are willing to turn the student into a spectator. But if you cast the student in a passive role, then saying that he has “studied” your course may mean no more than saying of a cat that he has looked at a king. Mathematics is something that one does.

– Edwin E. Moise

Concepts are the substance of mathematical knowledge.

– National Council of Teachers of Mathematics

Unless we want mathematics to continue to be viewed as something distinct and separate from the mainstream of culture and consisting of a bag of clever tricks or skills, we must change the way we relate to the general public and the way we teach mathematics.

– Harald M. Ness, Jr., in “Mathematics: an integral part of our culture”,

Children need to do what “real” mathematicians do – explore and invent for the rest of their lives.

– Susan Ohanian, from Garbage Pizzas, Patchwork Quilts and Math Magic

For what is important when we give children a theorem to use is not that they should memorize it. What matters most is that by growing up with a few very powerful theorems one comes to appreciate how certain ideas can be used as tools to think with over a lifetime. One learns to enjoy and to respect the power of powerful ideas. One learns that the most powerful idea of all is the idea of powerful ideas.

– Seymour Papert, from Mindstorms

Bees … by virtue of a certain geometrical forethought … know that the hexagon is greater than the square and the triangle, and will hold more honey for the same expenditure of material.

– Pappas

Mathematics is no more computation than typing is literature.

– John Allen Paulos

There are too many people who get degrees and think that they’re educated. In order to be a truly knowledgeable person one has got to be engaged in serious, systematic, lifelong learning.

– Benjamin Payton

I love mathematics…principally because it is beautiful; because man has breathed his spirit of play into it, and because it has given him his greatest game – the encompassing of the infinite.

– Rozso Peter

To most outsiders, modern mathematics is unknown territory.  Its borders are protected by dense thickets of technical terms; its landscapes are a mass of indecipherable equations and incomprehensible concepts.  Few realize that the world of modern mathematics is rich with vivid images and provocative ideas.

– Ivars Peterson

Students must learn that mathematics is the most human of endeavors. Flesh and blood representatives of their own species engaged in a centuries long creative struggle to uncover and to erect this magnificent edifice. And the struggle goes on today. On the very campuses where mathematics is presented and received as an inhuman discipline, cold and dead, new mathematics is created. As sure as the tides.

– J. D. Philips, from “Mathematics as an Aesthetic Discipline”

Every child is an artist. The problem is how to remain an artist after he(she) grows up.

– Pablo Picasso

Mathematics is the abstract key which turns the lock of the physical universe.

– John Polkinghorne

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.

– George Polya

The real voyage of discovery consists not in seeking new landscapes but in having new eyes.

– Marcel Proust

Indeed, only a few are mathematically gifted in the sense that they are endowed with the talent to discover new mathematical facts. But by the same token, only a few are musically gifted in that they are able to compose music. Nevertheless there are many who can understand and perhaps reproduce music, or who at least enjoy it. We believe that the number of people who can understand simple mathematical ideas is not relatively smaller than the number of those who are commonly called musical, and that their interest will be stimulated if only we can eliminate the aversion toward mathematics that so many have acquired from childhood experiences.

– Hans Rademacher

The simplest schoolboy is now familiar with truths for which Archimedes would have sacrificed his life.

– Ernest Renan

Intellectually, perspective [drawing] is a breakthrough, because here, for the first time, the physical space we live in is being depicted as ifit were an abstract, mathematical space. A less obvious innovation due to perspective is that here, for the first time, people are actually drawing pictures of infinities.

– Rudy Rucker, from Mind Tools: The Five Levels of Mathematical Reality

We cannot hope that many children will learn mathematics unless we find a way to share our enjoyment and show them its beauty as well as its utility.

– Mary Beth Ruskai

Physics is mathematical not because we know so much about the physical world, but because we know so little; it is only its mathematical properties that we can discover.

– Bertrand Russell

Arithmetic is numbers you squeeze from your head to your hand to your pencil to your paper till you get the answer.

– Carl Sandburg, Arithmetic

In mathematics, if a pattern occurs, we can go on to ask, Why does it occur? What does it signify? And we can find answers to these questions. In fact, for every pattern that appears, a mathematician feels he ought to know why it appears.

– W. W. Sawyer

Mathematics is a linguistic activity; its ultimate area is preciseness of communication.

– William L. Schaff

Go down deep enough into anything and you will find mathematics.

– Dean Schlicter

One can invent mathematics without knowing much of its history. One can use mathematics without knowing much, if any, of its history. But one cannot have a mature appreciation of mathematics without a substantial knowledge of its history.

– Abe Shenitzer

Giving students a lot of worksheets to fill out is indicative of low expectations. It suggests that you don”t think they”re capable of deep thinking about mathematics.

– Midge Siegfried

Calculus is the most powerful weapon of thought yet devised by the wit of man.

– W. B. Smith

The lock-step approach of algebra, geometry, and then more algebra (but rarely any statistics) is still dominant in U. S. schools, but hardly anywhere else. This fragmented approach yields effective mathematics education not for the many but for the few – primarily those who are independently motivated and who will learn under any conditions.

– Lynn Arthur Steen, from Does Everybody Need to Study Algebra?

Mathematics, in the common lay view, is a static discipline based on formulas…But outside the public view, mathematics continues to grow at a rapid rate…the guid to this growth is not calculation and formulas, but an open ended search for pattern.

– Lynn Arthur Steen, from On the Shoulders of Giants, edited by Lynn A. Steen

What humans do with the language of mathematics is to describe patterns… To grow mathematically children must be exposed to a rich variety of patterns appropriate to their own lives through which they can see variety, regularity, and interconnections.

– Lynn Arthur Steen, from On the Shoulder of Giants

I used to love mathematics for its own sake, and I still do, because it allows for no hypocrisy and no vagueness….

– Stendhal (Henri Beyle), The Life of Henri Brulard

One of the biggest problems of mathematics is to explain to everyone else what it is all about. The technical trappings of the subject, its symbolism and formality, its baffling terminology, its apparent delight in lengthy calculations: these tend to obscure its real nature. A musician would be horrified if his art were to be summed up as “a lot of tadpoles drawn on a row of lines”; but that”s all that the untrained eye can see in a page of sheet music… In the same way, the symbolism of mathematics is merely its coded form, not its substance.

– Ian Stewart, from From Here to Infinity

As one reads mathematics, one needs to have an active mind, asking questions, forming mental connections between the current topic and other ideas from other contexts, so as to develop a sense of the structure, not just familiarity with a particular tour through the structure.

– William P. Thurston, from Three-Dimensional Geometry and Topology.

There is a difference between not knowing and not knowing yet.

– Shelia Tobias

A man is like a fraction whose numerator is what he is and whose denominator is what he thinks of himself. The larger the denominator, the smaller the fraction.

– Tolstoy

Pure mathematics is the world’s best game.  It is more absorbing than chess, more of a gamble than poker, and lasts longer than Monopoly.  It’s free.  It can be played anywhere – Archimedes did it in a bathtub.

– Richard J. Trudeau, Dots and Lines

Make your mistakes work for you by learning from them.

– Donald Trump

Figures don’t lie, but liars figure.

– Mark Twain

We could use up two Eternities in learning all that is to be learned about our own world and the thousands of nations that have arisen and flourished and vanished from it.  Mathematics alone would occupy me eight million years.

– Mark Twain

The human mind has never invented a labor-saving machine equal to algebra.

– Unknown

In the binary system we count on our fists instead of on our fingers.

– Unknown

A circle is a round straight line with a hole in the middle.

– Unknown

There are 10 kinds of people in the world, those who understand binary math, and those who don’t.

– Unknown

Life is complex: it has both real and imaginary components.

– Unknown

Statistics means never having to say you’re certain.

– Unknown

A statistician can have his head in an oven and his feet in ice, and he will say that on the average he feels fine.

– Unknown

Infinity is a floorless room without walls or ceiling.

– Unknown

3 out of 4 Americans make up 75% of the population.

– Unknown

Trigonometry is a sine of the times.

– Unknown

Statistically speaking, in China, even if you are a one in a million kind of guy, there are a thousand more just like you.

– Unknown

Natural numbers are better for your health.

– Unknown

It is a mathematical fact that fifty percent of all doctors graduate in the bottom half of their class.

– Unknown

Mathematics is much more than computation with pencil and a paper and getting answers to routine exercises. In fact, it can easily be argued that computation, such as doing long division, is not mathematics at all. Calculators can do the same thing and calculators can only calculate – they cannot do mathematics.

John A. Van de Walle

The calculus was the first achievement of modern mathematics and it is difficult to overestimate its importance. I think it defines more unequivocally than anything else the inception of modern mathematics; and the system of mathematical analysis, which is its logical development, still constitutes the greatest technical advance in exact thinking.

– John von Neumann

Symmetry, as wide or as narrow as you define is meaning, is one idea by which man through the ages has tried to comprehend and create order, beauty and perfection.

– Hermann Weyl

Besides language and music, it [mathematics] is one of the primary manifestations of the free creative power of the human mind, and it is the universal organ for world-understanding through theoretical construction. Mathematics must therefore remain an essential element of the knowledge and abilities which we have to teach, of the culture we have to transmit, to the next generation.

– Hermann Weyl

Algebra is the intellectual instrument which has been created for rendering clear the quantitative aspects of the world.

– Alfred North Whitehead

From the very beginning of his education, the child should experience the joy of discovery.

– Alfred North Whitehead

I put the relation of a fine teacher to a student just below the relation of a mother to her son and I don’t think I could say more than this.

– Thomas Wolfe

Five out of four people have trouble with fractions.

– Steven Wright

The excitement of learning separates youth from old age. As long as you’re learning you’re not old.

Rosalyn S. Yallow

  23 Responses to “Math & Teaching Quotations”

  1. Absolutely lovely!

    Keep up this beautiful effort!

  2. I echo Sikandra’s words. What a rich compilation!

  3. it’s very funny reading this many quotation…:)

  4. you’re such a great…..good job!

  5. I so much like this.

  6. Tools & Techniques for Math Teachers and Students that’s great sir


  7. I enjoyed the quotes. Some I have seen before and now they are all in one place.

    Thank you for sharing your collection

  8. Amazing article. Good work done ! I’ll add it to my list now. Thanks for sharing ! :-)

  9. Fall seven times stand up eight – is really cute. Japanese sayings are indeed very insightful and have a cute way of saying things. It was a delighful experience to go through this collection of Math quotations. As a teacher I like to use quotes and sayings while teaching my students as quotes express thoughts in an unusually cute and charming way. I have a special collection of cute quotes in my PC and I will surely add some from here in my compilation. I found the quotes by Albert Einstein and Mark Twain very inspiring. This is an awesome collection. Hats off to you.

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