School is an institution built on the axiom that learning is the result of teaching. And institutional wisdom continues to accept this axiom, despite overwhelming evidence to the contrary. -Ivan Illich

Whether you think you can or whether you think you can't, you're right! -Henry Ford

The goal of instruction should be to allow students to deal sensibly with problems that often involve evidence, quantitative consideration, logical arguments, and uncertainty; without the ability to think critically and independently, citizens are easy prey to dogmatists, flimflam artists, and purveyors of simple solutions to complex problems. -American Association for the Advancement of Science, 1989

We think in generalities, but we live in details. -Alfred North Whitehead

Mathematics as a science, commenced when first someone, probably a Greek, proved propositions about "any" things or about "some" things, without specifications of definite particular things. -Alfred North Whitehead

There is a tradition of opposition between adherents of induction and of deduction. In my view it would be just as sensible for the two ends of a worm to quarrel. -Alfred North Whitehead

It is a profoundly erroneous truism, repeated by all copy books and by eminent people when they are making speeches, that we should cultivate the habit of thinking of what we are doing. The precise opposite is the case. Civilization advances by extending the number of important operations which we can perform without thinking about them. -Alfred North Whitehead, An Introduction to Mathematics

If you ask mathematicians what they do, you always get the same answer. They think. They think about difficult and unusual problems. They do not think about ordinary problems: they just write down the answers. -M. Egrafov

We are fallible, and prone to error; but we can learn from our mistakes. -Karl Popper, Objective Knowledge

...it is of the utmost importance to give up cocksureness, and become open to criticism. Yet it is also of the greatest importance not to mistake this discovery, this step towards criticism, for a step towards relativism. If two parties disagree, this may mean that one is wrong, or the other, or both: this is the view of the criticist. It does not mean, as the relativist will have it, that both may be equally right. ...As two wrongs don't make a right, two wrong parties to a dispute do not make two right parties -Karl Popper, The Open Society and Its Enemies, Volume 2, p. 387

Next to the right to life itself, the most fundamental of all human rights is the right to control our own minds and thoughts. That means, the right to decide for ourselves how we will explore the world around us, think about our own and other persons' experiences, and find and make the meaning of our own lives. Whoever takes that right away from us, by trying to 'educate' us, attacks the very center of our being and does us a most profound and lasting injury. He tells us, in effect, that we cannot be trusted even to think, that for all our lives we must depend on others to tell us the meaning of our world and our lives, and that any meaning we may make for ourselves, out of our own experience has no value. -John Holt, Instead of Education

High achievement always takes place in the framework of high expectation. -Jack Kinder

How I hated this school, and what a life of anxiety I lived there for more than two years. I made very little progress at my lessons, and none at all at games. I counted the days and the hours to the end of every term, when I should return home from this hateful servitude and range my soldiers in line of battle on the nursery floor. The greatest pleasure I had in those days was reading. When I was nine and a half my father gave me Treasure Island, and I remember the delight with which I devoured it. My teachers saw me at once backward and precocious, reading books beyond my years and yet at the bottom of the Form. They were offended. They had large resources of compulsion at their disposal, but I was stubborn. Where my reason, imagination or interest were not engaged, I would not or I could not learn. -Winston Churchill, My Early Life, 1930

We lay down a fundamental principle of generalization by abstraction: "The existence of analogies between central features of various theories implies the existence of a general theory which underlies the particular theories and unifies them with respect to those central features...." -E.H. Moore, 1862-1932

To state a theorem and then to show examples of it is literally to teach backwards. -E. Kim Nebeuts

One had to cram all this stuff into one's mind, whether one liked it or not. This coercion had such a deterring effect that, after I had passed the final examination, I found the consideration of any scientific problems distasteful to me for an entire year ... It is in fact nothing short of a miracle that the modern methods of instruction have not yet entirely strangled the holy curiosity of enquiry; for this delicate little plant, aside from stimulation, stands mainly in need of freedom; without this it goes to wrack and ruin without fail. It is a very grave mistake to think that the enjoyment of seeing and searching can be promoted by means of coercion and a sense of duty. -Albert Einstein

Everything should be made as simple as possible, but not simpler. -Albert Einstein

Imagination is more important than knowledge. -Albert Einstein

How can it be that mathematics, being after all a product of human thought independent of experience, is so admirably adapted to the objects of reality? -Albert Einstein

The grand aim of all science is to cover the greatest number of empirical facts by logical deduction from the smallest number of hypotheses or axioms. -Albert Einstein

Insanity is continuing to do the same thing over and over and expecting different results. -Albert Einstein

The words printed here are concepts. You must go through the experiences. -Carl Frederick

Man cannot discover new oceans unless he has the courage to lose sight of the shore. -Andre Gide

I do not know what I may appear to the world; but to myself I seem to have been only like a small boy playing on the sea-shore, diverting myself in now and then finding a smoother pebble or a prettier shell than the ordinary, whilst the great ocean of truth lay all undiscovered before me. -Isaac Newton

If I have seen further [than certain other men] it is by standing upon the shoulders of giants. [With reference to his dependency on Galileo’s and Kepler’s work in physics and astronomy.] -Isaac Newton, 1657

Learning to solve problems is the principal reason for studying mathematics. -National Council of Teachers of Mathematics

Euclid taught me that without assumptions there is no proof. Therefore, in any argument, examine the assumptions. -Eric Temple Bell

I hear and I forget, I see and I remember. I do and I understand. -Chinese Proverb

"When I use a word," Humpty Dumpty said, in a rather scornful tone, "it means just what I choose it to mean - neither more nor less." -Lewis Carroll, Through the Looking Glass

You can only find truth with logic if you have already found truth without it. -G.K. Chesterton

[When asked about his age.] I was x years old in the year x². -Augustus De Morgan, 1806-1871

The measure of our intellectual capacity is the capacity to feel less and less satisfied with our answers to better and better problems. -C.W. Churchman

When you have eliminated the impossible, what ever remains, however improbable must be the truth. -Sir Arthur Conan Doyle, 1859-1930

It's not the situation ... It's your reaction to the situation -Robert Conklin

Cogito Ergo Sum. "I think, therefore I am." -René Descartes, Discours de la Méthode, 1637

Each problem that I solved became a rule which served afterwards to solve other problems. -Rene Descartes

What happens is not as important as how you react to what happens. -Thaddeus Golas

The discovery in 1846 of the planet Neptune was a dramatic and spectacular achievement of mathematical astronomy. The very existence of this new member of the solar system, and its exact location, were demonstrated with pencil and paper; there was left to observers only the routine task of pointing their telescopes at the spot the mathematicians had marked. -James Newman, The World of Mathematics, 1956

God, grant me the serenity to accept the things I cannot change, the courage to change the things I can, and the wisdom to know the difference. -Reinhold Niebuhr

It is not because things are difficult that we do not dare; it is because we do not dare that they are difficult. -Seneca

I have learned from experience that the greater part of our happiness or misery depends on our dispositions and not on our circumstances. -Martha Washington

Success does not consist in never making mistakes but in never making the same one a second time. -George Bernard Shaw

Millions saw the apple fall, but Newton asked why. -Bernard Baruch

To assert that the earth revolves around the sun is as erroneous as to claim that Jesus was not born of a virgin. -Cardinal Bellarmine, 1615, during the trial of Galileo

Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the human mind will never penetrate. -Leonhard Euler, 1707-1783

We are usually convinced more easily by reasons we have found ourselves than by those which have occurred to others. -Blaise Pascal, 1623-1662

...there is a marvellous anecdote from the occasion of Russell's ninetieth birthday that best serves to summarize his attitude toward God and religion. A London lady sat next to him at this party, and over the soup she suggested to him that he was not only the world's most famous atheist but, by this time, very probably the world's oldest atheist. "What will you do, Bertie, if it turns out you're wrong?" she asked. "I mean, what if-uh-when the time comes, you should meet Him? What will you say?" Russell was delighted with the question. His bright, birdlike eyes grew even brighter as he contemplated this possible future dialogue, and then he pointed a finger upward and cried, "Why, I should say, 'God, you gave us insufficient evidence.' -Al Seckel, Preface to Bertrand Russell: On God and Religion

In the book of life, the answers aren't in the back. -Charlie Brown

Confidence comes not from always being right but from not fearing to be wrong. -Peter McIntyre

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

The mind is not a vessel to be filled but a fire to be kindled. -Plutarch

The invention of variables was, perhaps, the most important event in human evolution. The command of their use remains the most significant achievement in the history of the individual human being. -Percy Nunn, 1919

The symbolism of algebra is its glory. But it is also its curse. -William Betz, 1930

We have a habit in writing articles published in scientific journals to make the work as finished as possible, to cover up all the tracks, to not worry about the blind alleys or describe how you had the wrong idea first, and so on. So there isn't any place to publish, in a dignified manner, what you actually did in order to get to do the work. -Richard, Feynman, Nobel Lecture, 1966

[The universe] cannot be read until we have learnt the language and become familiar with the characters in which it is written. It is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. -Galileo Galilei, 1564-1642

I have had my results for a long time, but I do not yet know how I am to arrive at them. -Karl Friedrich Gauss, 1777-1855

A great part of its [higher arithmetic] theories derives an additional charm from the peculiarity that important propositions, with the impress of simplicity on them, are often easily discovered by induction, and yet are of so profound a character that we cannot find the demonstrations till after many vain attempts; and even then, when we do succeed, it is often by some tedious and artificial process, while the simple methods may long remain concealed. -Karl Friedrich Gauss, 1777-1855

Don't just read it; fight it! Ask your own questions, look for your own examples, discover your own proofs. Is the hypothesis necessary? Is the converse true? What happens in the classical special case? What about the degenerate cases? Where does the proof use the hypothesis? -Paul Halmos, I Want to be a Mathematician, 1985

Mathematics is not a deductive science - that's a cliche. 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, I Want to be a Mathematician, 1985

...the source of all great mathematics is the special case, the concrete example. It is frequent in mathematics that every instance of a concept of seemingly great generality is in essence the same as a small and concrete special case. -Paul Halmos, I Want to be a Mathematician, 1985

What does mathematics really consist of? Axioms (such as the parallel postulate)? Theorems (such as the fundamental theorem of algebra)? Proofs (such as Gödel's proof of undecidability)? Definitions (such as the Menger definition of dimension)? Theories (such as category theory)? Formulas (such as Cauchy's integral formula)? Methods (such as the method of successive approximations)? Mathematics could surely not exist without these ingredients; they are all essential. It is nevertheless a tenable point of view that none of them is at the heart of the subject, that the mathematician's main reason for existence is to solve problems, and that, therefore, what mathematics really consists of is problems and solutions. -Paul Halmos, 1980

A mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas. ... The mathematician's patterns, like the painter's or the poet's must be beautiful; the ideas, like the colors or the words must fit together in a harmonious way. Beauty is the first test: there is no permanent place in this world for ugly mathematics. -G.H. Hardy, A Mathematician's Apology, 1941

One cannot escape the feeling that these mathematical formulas have an independent existence and an intelligence of their own, that they are wiser that we are, wiser even than their discoverers, that we get more out of them than was originally put into them. -Heinrich Hertz

The errors of definitions multiply themselves according as the reckoning proceeds; and lead men into absurdities, which at last they see but cannot avoid, without reckoning anew from the beginning. -Thomas Hobbes, 1956

... learners are conceptualized as active mathematical thinkers, who try to construct meaning and make sense for themselves of what they are doing, on the basis of their personal experience ... and who are developing their ways of thinking as their experience broa-dens, always building on the knowledge which they have already constructed. -Hilary Shuard, 1986

Ideas and thoughts cannot be communicated in the sense that meaning is packaged into words and "sent" to another who unpacks the meaning from the sentences. That is, as much as we would like to, we cannot put ideas in students' heads, they will and must construct their own meanings. Our attempts at communication do not result in conveying meaning but rather our expression evoke meaning in another, different meanings for each person. -Grayson Wheatley, 1991

The empiricist believes you understand what you see. The constructivist believes you see what you understand.

I have observed, not only with other people but also with myself ... that sources of insight can be clogged by automatisms. One finally masters an activity so perfectly that the question of how and why is not asked any more, cannot be asked any more, and is not even understood any more as a meaningful and relevant question. -Hans Freudenthal, 1983

From the constructivist point of view, there can be no doubt that reflective ability is a major source of knowledge on all levels of mathematics. That is the reason why ... it (is) important that students be led to talk about their thoughts, to each other, to the teacher, or to both. To verbalize what one is doing ensures that one is examining it. -Ernst von Glasersfeld, 1991

It is not that a teacher's approval and pat on the head have no effect on the student, but the effect is to strengthen the student's inclination to please the teacher rather than to buid up understanding of the conceptual area in which the task was situated. Thus students are prevented from experiencing the rewarding elation that follows upon having found one's own way and recognizing it as a good way. If students are not oriented or led towards autonomous intellectual satisfaction, we have no right to blame them for their lack of proper motivation. The motivation to please superiors without understanding why they demand what they demand, may be required in an army - in an institution that purports to serve the propagation of knowledge, it is out of place. -Ernst von Glasersfeld, 1991

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. -George Polya

... a teacher of mathematics has a great opportunity. If he fills his allotted time with drilling his students in routine operations he kills their interest, hampers their intellectual development, and misuses his opportunity. But if he challenges the curiosity of his students by setting them problems proportionate to their knowledge, and helps them to solve their problems with stimulating questions, he may give them a taste for, and some means of, independent thinking. -George Polya, 1957

Geometry is the science of correct reasoning on incorrect figures. -George Polya, How to Solve It, 1945

The first rule of teaching is to know what you are supposed to teach. The second rule of teaching is to know a little more than what you are supposed to teach. . . . Yet it should not be forgotten that a teacher of mathematics should know some mathematics, and that a teacher wishing to impart the right attitude of mind toward problems to his students should have acquired that attitude himself. -George Polya, How to Solve it, 1945

...in our search for the solution to the problems of educational inequality, our focus was almost exclusively on the characteristics of the children themselves. We looked for sources of educational failure in their homes, their neighborhoods, their language, their cultures, even in their genes. In all our searching we almost entirely overlooked the possibility that what happens within schools might contribute to unequal educational opportunities and outcomes. -Jeannie Oakes

...the unquestioned assumptions that drive school practice and the basic features of schools may themselves lock schools into patterns that make it difficult to achieve either excellence or equality. -Jeannie Oakes

...we have overcome the notion that mathematical truths have an existence independent and apart from our own minds. It is even strange to us that such a notion could ever have existed. -E. Kasner & J. Newman, Mathematics and the Imagination, New York: Simon and Schuster, 1940

When the mathematician says that such and such a proposition is true of one thing, it may be interesting, and it is surely safe. But when he tries to extend his proposition to everything, though it is much more interesting, it is also much more dangerous. In the transition from one to all, from the specific to the general, mathematics has made its greatest progress, and suffered its most serious setbacks. -E. Kasner & J. Newman, Mathematics and the Imagination, New York: Simon and Schuster, 1940

The chief aim of all investigations of the external world should be to discover the rational order and harmony which has been imposed on it by God and which He revealed to us in the language of mathematics. -Johannes Kepler, 1571-1630

He who loves practice without theory is like the sailor who boards ship without a rudder and compass and never knows where he may cast. -Leonardo da Vinci, 1452-1519

I have often noticed that when people come to understand a mathematical proposition in some other way than that of the ordinary demonstration, they promptly say, "Oh, I see. That's how it must be." This is a sign that they explain it to themselves from within their own system. -Georg Christoph Lichtenberg, 1742-1799

God made the integers, all else is the work of man. -Leopold Kronecker, 1823-1891

Mathematics is the science which draws necessary conclusions. -Benjamin Peirce, 1809-1880

Science is built up with facts, as a house is with stones. But a collection of facts is no more a science than a heap of stones is a house. -Jules Henri Poincaré, 1854-1912

Mathematicians do not study objects, but relations between objects. Thus, they are free to replace some objects by others so long as the relations remain unchanged. Content to them is irrelevant: they are interested in form only. -Jules Henri Poincaré, 1854-1912

To avoide the tediouse repetition of these woordes: is equalle to, I will settle as I doe often in woorke use, a paire of paralleles, or gemowe [twin] lines of one lengthe: =, bicause noe .2. thynges, can be moare equalle. -Robert Recorde, 1557, the first to use the =-notation

In ... 1972 President Nixon announced that the rate of increase of inflation was decreasing. This was the first time a sitting president used the third derivative to advance his case for reelection. -Hugo Rossi, Notices of the AMS, 1996

A habit of basing convictions upon evidence, and of giving to them only that degree or certainty which the evidence warrants, would, if it became general, cure most of the ills from which the world suffers. -Bertrand Russell, 1872-1970

Aristotle maintained that women have fewer teeth than men; although he was twice married, it never occurred to him to verify this statement by examining his wives' mouths. -Bertrand Russell, The Impact of Science on Society, 1952

Mathematics, rightly viewed, possesses not only truth, but supreme beauty. —Bertrand Russell

Attaching significance to invariants is an effort to recognize what, because of its form or colour or meaning or otherwise, is important or significant in what is only trivial or ephemeral. A simple instance of failing in this is provided by the [student] who learned perfectly how to factorize a² - b² but was floored because the examiner unkindly asked for the factors of p² - q² . -H.W. Turnbull in J.R. Newman (Ed.): The World of Mathematics, 1956.

The art of remembering is the art of thinking; ...when we wish to fix a new thing in either our own mind or a pupil's, our conscious effort should not be so much to impress and retain it as to connect it with something else already there. The connecting is the thinking ... -William James, 1888

The mathematical meaning of proof carries three senses. The first is verification or justification, concerned with the truth of the proposition; the second is illumination, in that a good proof is expected to convey an insight into why the proposition is true; ... The third sense of proof is the most characteristically mathematical, that of systematisation, i.e. the organisation of results into a deductive system of axioms, major concepts and theorems, and minor results derived from these. -Alan Bell, 1976

Theorem: For algebra enthusiasts, x is always negative.
Proof: For algebra enthusiasts, nothing is better than x. Therefore, 0 > x. QED. -Helen Chick, 2001

The problem now is that the economy has changed much faster than the schools. People used to say, `You know, the schools just aren't what they used to be.' The problem may be that too many of our schools are too much like they `used to be,' but the world the children move out into is not at all as it used to be... We've got the give the schools the tools they need to do the job. -Bill Clinton, 1999

Culture is activity of thought, and receptiveness to beauty and humane feeling... A merely well-informed man is the most useless bore on God's earth. -Alfred North Whitehead, Aims of Education and Other Essays, 1929

The greatest obstacle to discovery is not ignorance - it is the illusion of knowledge. -Daniel J Boorstin

... becoming a good mathematical problem solver - becoming a good thinker in any domain - may be as much a matter of acquiring the habits and dispositions of interpretation and sense-making as of acquiring any particular set of skills, strategies, or knowledge. If this is so, we may do well to conceive of mathematics education less as an instructional process (in the traditional sense of teaching specific, well-defined skills or items of knowledge), than as a socialization process. -Resnick, 1989

One’s conception of what mathematics is affects one’s conception of how it should be presented. One’s manner of presenting is an indication of what one believes is most essential in it ... The issue, then, is not, What is the best way to teach? but, What is mathematics really all about? -Reuben Hersh, 1986

All too often we focus on a narrow collection of well-defined tasks and train students to execute those tasks in a routine, if not algorithmic fashion. Then we test the students on tasks that are very close to the ones they have been taught. If they succeed on those problems, we and they congratulate each other on the fact that they have learned some powerful mathematical techniques. In fact, they may be able to use such techniques mechanically while lacking some rudimentary thinking skills. To allow them, and ourselves, to believe that they "understand" the mathematics is deceptive and fraudulent. -Alan Schoenfeld

To instruct someone ... is not a matter of getting him to commit results to mind. Rather, it is to teach him to participate in the process that makes possible the establishment of knowledge. We teach a subject not to produce little living libraries on that subject, but rather to get a student to think mathematically for himself, to consider matters as an historian does, to take part in the process of knowledge-getting. Knowing is a process not a product. -Jerome Bruner, Toward a Theory of Instruction, 1966, p. 72

The purpose of computing is insight, not numbers. -R.W. Hamming

When we look at the history of mathematics, we see a kind of lifelike elemental rhythm. There are periods of exuberant untidy growth, when exciting, vital structures rise upon untried assumptions, and loose ends lie about all over the place. Logic and precision are not unduly honoured; because restlessness, enthusiasm, daring, and ability to tolerate a measure of confusion, are the appropriate qualities of mind at these times. Such periods are followed by pauses for consolidation, when the analysts and systematisers get to work; material is logically ordered, gaps are filled, loose ends are neatly tied up, and rigorous proofs supplied. Solemn commentators sit in judgment upon great innovators. Whole areas of mathematics are formed into deductive systems, based on sets of unproved, explicitly stated axioms. Work of this kind, at its best, is also creative: new ideas grow from the critical examination of old, and the cycle is renewed. -L.W.H. Hull, The superstition of educated men, 1969

... there is no human knowledge which cannot lose its scientific character when men forget the conditions under which it originated, the questions which it answered, and the function it was created to serve. A great part of the mysticism and superstition of educated men consists of knowledge which has broken loose from its historical moorings. -Benjamin Farrington, Greek Science: Its Meaning for Us. London: Penguin, 1953, p. 311

Whilst it is possible to teach arithmetic in a purely mechanical manner, it is certainly not desirable to do so. Even for the simplest operations, it is easier to remember what has to be done if one knows the reason. For anyone who wants to go on to the other branches of mathematics, mechanical learning is fatal. ... It is a pity that there are still schools ... where arithmetic is still taught on the lines of 'You do this, then you do that' - as though the subject were some form of religious ritual. -W.W. Sawyer, Mathematician's Delight, 1961, p. 55

In no case should one start with the deductive approach, even after students have come to know what this means. The deductive proof is the final step … [The student] should be allowed to accept and use any facts that are so obvious to him that he does not realize he is using them. … Proofs of whatever nature should be invoked only where the students think they are required. The proof is meaningful when it answers the student’s doubts, when it proves what is not obvious. -Morris Kline, Why Johnny Can't Add: The Failure of the New Math, 1973, p. 195

The capacity to appreciate mathematical rigor is a function of the age of the student and is independent of the age of mathematics. -Morris Kline, 1955

... rigorous proof is the polish on mathematics. It is the last stage of a development. ... the rigorous proof comes after the discovery and full intuitive understanding of the theorem to be proved. This is true historically of mathematics in the large and is true of the very process of creating individual theorems. Today we are trying to emphasize the gilt and the polish and we are leaving out the substance. We are throwing away the heritage of mathematics. -Morris Kline, 1955

Let me get this straight. We’re behind the rest of our class and we’re going to catch up to them by going slower? -Bart Simpson

The black doubt that lurks in the bottom of every honest pedagogue’s heart is not so much whether he is teaching correctly as whether what he is teaching is worth teaching at all. The real danger is not that we shall teach the right things inefficiently, but that we shall teach the wrong things more and more efficiently. -C.E. Beeby, 1935

No amount of observations of white swans can allow the inference that all swans are white, but the observation of a single black swan is sufficient to refute that conclusion. -Karl Popper

Logic is the hygiene the mathematician practices to keep his ideas healthy and strong. -Hermann Weyl (1885–1955)