Jerry Uhl's favorite quotations

I attended a very interesting talk at the AMS/MAA Joint Mathematics Meetings in January of the year 2000.  The speaker was Jerry Uhl, a professor I've known since I was a graduate student at the University of Illinois in the late 80's.  It was titled Following Descartes' advice and teaching to the gut, which didn't mean a lot to me, but it was provocative, and in the middle of the AMS Special Session on Philosophies in Mathematics Education.  His talks are always interesting and thought provoking, so I went.

Jerry started his talk this way, (and I am paraphrasing) "I wrote down a list of things I wanted to say today.  I realized then that other people have already said them, and better than I could.  So I would like to share their words with you."   The rest of the talk consisted of Jerry reading other people's words, occasionally commenting or challenging us to guess their authors.  The talk worked; and afterwards I asked him if I could put the quotations up on my website to provoke others.  He assented.

One caveat:  I don't agree with every quotation below.  In fact, I disagree with many of them.  But what you will see below gave me a lot to think about, and I hope it gives other educators a lot to think about as well. 

Jerry's Top Ten

1:    Calculus as currently taught is, alas, full of inert material . . . The real crisis is that at present [calculus] is badly taught; the syllabus has remained stationary, and modern points of view, especially those having to do with the roles of applications and computing are poorly represented. . .

Peter Lax, Past President of the American Mathematical Society

2:    The facts of mathematics are verified and presented by the axiomatic method. One must guard, however, against confusing the presentation of mathematics with the content of mathematics. An axiomatic presentation of a mathematical fact differs from the fact that is being presented as medicine differs from food. It is true that this particular medicine is necessary to keep mathematicians from self-delusions of the mind. Nonetheless, understanding mathematics means being able to forget the medicine and enjoy the food.

Gian - Carlo Rota

3:    The student needs to develop an understanding, however partial and imperfect, by descriptions rather than definitions, by typical examples rather than grandiose theorems....

Gian - Carlo Rota

4:    Mathematical skills are like any other kind.... If you are learning to play the piano, you usually start by practicing under supervision; you don't begin with theoretical lectures on acoustical vibrations and the internal structure of the instrument. Similarly for mathematical skills.

Ralph Boas

5:    Authors of textbooks need to remember that they are supposed to be addressing students, not the teachers... Contemporary prose style is simpler and more direct than the style of the 19th century- except in textbooks of mathematics. ....I blame the authors of textbooks for not realizing that contemporary students speak a different language.

Ralph Boas

6:    Mathematicians have developed habits of communication that are often dysfunctional. We go through the motions of saying for the record what the students 'ought' to learn while students grapple with the more fundamental issues of learning our language and guessing at our mental models. Books compensate by giving samples of how to solve every type of homework problem. Professors compensate by giving homework and tests that are much easier than the material 'covered' in the course, and then grading the homework and tests on a scale that requires little understanding. We assume the problem is with students rather than communication: that the students either don't have what it takes, or else just don't care.  Outsiders are amazed at this phenomenon, but within the mathematical community, we dismiss it with shrugs.

Bill Thurston

7:    Many conventional academic skills amount to the ability to select and apply. . . procedures rapidly and correctly....[Computer] courseware can concentrate on one skill at a time, in a manner impossible for a textbook and hardly available to the classroom teacher, namely by asking the student to handle only that part of a procedure on which pedagogical stress is to be laid, while other aspects of the same procedure are handled automatically by the computer. . . . This scheme, which combines student interaction with computer assistance, has the merit of focusing attention on the key strategic and conceptual decisions needed to handle a problem. . . This should be of significance to both the strong student . . . and the weak student.

Jacob T. Schwartz

8:    Civilization advances by extending the number of operations we can do without thinking about them. Operations of thought are like cavalry charges in a battle - they are strictly limited in number, they require fresh horses and must be made only at decisive moments.

Alfred North Whitehead

9:    Mathematics-speaking machines are about to sweep the campuses....The ready availability of powerful computers will enable students to set new ground rules for college mathematics... Teachers will be forced to change their approach and their assignments. They will no longer be able to teach as they were taught in the pencil and paper era... Undergraduate mathematics will become more like real mathematics . . . By using machines to expedite calculations, students can experience mathematics as it really is - as a tentative, exploratory discipline in which risks and failures yield clues to success. Computers change our perceptions of what is possible and what is valuable. . . Weakness in algebra skills will no longer prevent students students from pursuing studies that require college mathematics . . .

Lynn Arthur Steen, Past President of the Mathematical Association of America

10:    Our teaching does not make full use of that historic event, which is probably the most important event in the history of science, namely, the invention of the decimal system of numeration.

Henri Lebesgue

Jerry's favorites from the modern era

The student needs to develop an understanding, however partial and imperfect, by descriptions rather than definitions, by typical examples rather than grandiose theorems....

The facts of mathematics are verified and presented by the axiomatic method. One must guard, however, against confusing the presentation of mathematics with the content of mathematics. An axiomatic presentation of a mathematical fact differs from the fact that is being presented as medicine differs from food. It is true that this particular medicine is necessary to keep mathematicians from self-delusions of the mind. Nonetheless, understanding mathematics means being able to forget the medicine and enjoy the food.

The best introduction to mathematics is not achieved by rigorous presentation. What one wishes for is a feeling for a piece of mathematics. Let the student work with unrigorous concepts that lead as quickly as possible to a half-baked understanding of the main results and their applications. Rigorous presentation can occur later....to be given only to those who develop a genuine interest in mathematics. . .

The axiomatic method of presentation has reached a pinnacle of fanaticism in our time. The pretense of identifying mathematics with a style of exposition is having a corrosive effect on the way mathematics is viewed by scientists in other disciplines. . . . The mistaken identification of mathematics with the axiomatic method has led to a widespread prejudice among scientists that mathematics is nothing but a pedantic grammar, suitable only for belaboring the obvious and for producing marginal counterexamples to useful facts that are by and large true.

Most mathematicians who teach mathematics fail. They bask in the illusion that the majority of their students should become mathematicians or their teaching is wasted...

Nowhere in the sciences does one find a wide a gap as that between the written version of a mathematical result and the discourse that is required to understand the same result.

It has been an acknowledged fact, since Poincare pointed an accusing finger at the Twentieth Century, that much of the mathematics of our time has had negative overtones.

If mathematics were formally true but in no way enlightening, then mathematics would be a curious game played by weird people. ,,, Mathematicians seldom explicitly acknowledge the phenomenon of enlightenment.....

Anyone who is about to teach the undergraduate curriculum should come down to earth by looking through the Schaum's outline series before burdening students with those well-printed, many colored, highly advertised hardcover volumes that are pathetically passed off as text books. Most elementary probability texts books are carbon copies of each other.

Gian - Carlo Rota

Why do we speak and write about mathematics in ways that interfere so dramatically with what we ostensibly want to accomplish? When you want a young child to learn about cats, do you explain that a cat is a relatively small,primarily carnivorous mammal with rectile claws and distinctive sonic output. Or do you arrange for the child to play with lots of kitties and let them form their own first impressions?

When you want a student to learn about math, do you give a great lecture, introducing new ideas, new terminology, new notation Or do you arrange for the student to play with lots of math kitties and let them form their own first impressions?

Authors of textbooks need to remember that they are supposed to be addressing students, not the teachers...

Contemporary prose style is simpler and more direct than the style of the 19th century- except in textbooks of mathematics. ....I blame the authors of textbooks for not realizing that contemporary students speak a different language.

Mathematical skills are like any other kind.... If you are learning to play the piano, you usually start by practicing under supervision; you don't begin with theoretical lectures on acoustical vibrations and the internal structure of the instrument. Similarly for mathematical skills. The point that a definition is satisfactory only if the students understand it was already made by Poincare in 1909, but teachers of mathematics seem not to have paid much attention to it.

As a means of instruction, lectures ought to have become obsolete when the printing press was invented. We had a second chance when the Xerox machine was invented, but we muffed it.

Only professional mathematicians learn anything from proofs. Other people learn from explanations. A great deal can be accomplished with arguments that fall short of formal proofs.

A sweater is what a child puts on when its parent feels cold, but a proof is what students have to listen to when the teacher feels shaky about a theorem. 

Experienced parents know that when a child says Why? it doesn't necessarily mean that it wants a reason; it just wants more conversation.

If you are going to explain to an average class how to find the distance from a point to a plane, you should first find the distance from {2,-3,1} to the plane x - 2y - 4 z = 7. After that, the general procedure will seem almost obvious. Textbooks used to be written this way.

At the beginning, a lot of the students' effort has to go into memorizing words, when it could go more advantageously into learning mathematics. Paying more attention to vocabulary than content obscures the content.

Ralph Boas, Past President of the Mathematical Association of America

If a field ever needed to be brought out of mystery to reality, it is calculus. . . Calculus is really exciting stuff, yet [the traditional course is] not presenting it as an exciting subject. . . Calculus must become a pump instead of a filter in the pipeline.

Robert White, Past President of the National Academy of Engineering

Calculus as currently taught is, alas, full of inert material . . . The real crisis is that at present [calculus] is badly taught; the syllabus has remained stationary, and modern points of view, especially those having to do with the roles of applications and computing are poorly represented. . . There is too much preoccupation with what might be called the magic in [traditional] calculus. For instance, too much time is spent in pulling exact integrals out of a hat, and, what is worse, in drilling students how to perform this parlor trick. Summing infinite series is another topic that has the aura of a magic trick, and is overemphasized at the expense of the concept of approximation.. . . I feel that rigor at this level is misplaced; it appears as an arid game to those who understand it, and mumbo jumbo to those who don't.. . . Many students have difficulty in grasping the idea that the integral of a function over an interval is a number. The reason is that this number is difficult to produce by traditional methods, i.e. by antidifferentiation, and so the central idea is lost. Numerical methods have the great virtue that they apply universally. When special methods are introduced to deal with each one of the pitifully small class of [differential] equations that can be handled analytically, students are apt to lose sight of the general idea that every differential equation has a solution and that this solution is uniquely determined by initial data.   That today we can use computers to explore the solutions of [differential] equations is truly revolutionary; we are only beginning to glimpse the consequences.

Peter Lax, Past President of the American Mathematical Society

[The traditional calculus course deals with] precisely the ability to do [by hand] the kinds of things that calculators and computers are now doing. . . There is a great deal of concern about making sure that freshman courses . . . make a significant contribution to broaden the aims of undergraduate education - that they help students learn to think clearly, to communicate, . . . etc. There is very, very little in the [traditional] calculus course of today that does any of these things. . . .

Mathematics-speaking machines are about to sweep the campuses....The ready availability of powerful computers will enable students to set new ground rules for college mathematics... Teachers will be forced to change their approach and their assignments. They will no longer be able to teach as they were taught in the pencil and paper era... Undergraduate mathematics will become more like real mathematics . . . By using machines to expedite calculations, students can experience mathematics as it really is - as a tentative, exploratory discipline in which risks and failures yield clues to success. Computers change our perceptions of what is possible and what is valuable. . . Weakness in algebra skills will no longer prevent students students from pursuing studies that require college mathematics . . . Mathematics learning will become more active and hence more effective. By carrying most of the calculational burden of mathematics homework, computers will enable students to explore a wider variety of examples, to study graphs of a quantity and variety unavailable with pencil-and-paper methods, to witness the dynamic nature of mathematical processes, and to encourage realistic applications using typical-not oversimplified-data. Students will be able to explore mathematics on their own without, constant advice from their instructors. Although computers will not compel students to think for themselves, these machines can provide an environment in which student-generated mathematical ideas can survive.  Study of mathematics will build long-lasting knowledge, not just short-lived strategies for calculation. Most students take only one or two terms of college mathematics, and quickly forget what little they learned of memorized methods for calculation. Innovative instruction using a new symbiosis of machine calculation and human thinking can shift the balance of mathematical learning to understanding, insight, and mathematical intuition.

Lynn Arthur Steen, Past President of the Mathematical Association of America

Mathematicians have developed habits of communication that are often dysfunctional. We go through the motions of saying for the record what the students 'ought' to learn while students grapple with the more fundamental issues of learning our language and guessing at our mental models. Books compensate by giving samples of how to solve every type of homework problem. Professors compensate by giving homework and tests that are much easier than the material 'covered' in the course, and then grading the homework and tests on a scale that requires little understanding. We assume the problem is with students rather than communication: that the students either don't have what it takes, or else just don't care.  Outsiders are amazed at this phenomenon, but within the mathematical community, we dismiss it with shrugs.

However, I think the real question for communicating mathematics is how language is used; there are many styles of writing, and I think much of modern mathematical writing overuses symbols in a way that significantly interferes (at least for me) with geometric thinking. Any language can be used to evoke geometry. We don't need to spell out the equation for a `spiral staircase' to get the geometric image, since spiral staircases are part of a common assumed culture. Cultural assumptions can be mistaken, but for any given audience, there are enough geometric ideas that can be evoked by reference, so that if used with imagination they can go pretty far in describing mathematical phenomena geometrically.

Bill Thurston

[Pedagogical Content Knowledge is the] particular form of content knowledge that embodies the aspects of content most germane to its teachability.... the ways of representing and formulating the subject that make it comprehensible to others.

Lee Shulman

The job of the teacher is to translate the textbook into the vernacular.

Paul Halmos

Intuition-trial-error-speculation-conjecture-proof is a sequence for understanding of mathematics.

Saunders MacLane

There's another unrecognized cause of the failure [of mathematics education]: misconception of the nature of mathematics. A philosophy of mathematics that obscures the teachability of mathematics is unacceptable.

Reuben Hersh

We no longer ask students to understand. Now it is [algebraic] manipulation pure and simple. . . .What we're teaching is not only the wrong thing- in that it is not what students will use [outside the calculus classroom]- it is obsolete [because of computers and calculators]. It is like spending all your time in elementary school adding and subtracting and never being told what addition and subtraction are for.

Ronald Douglas, Committee on the Mathematical Sciences in the year 2000

Many conventional academic skills amount to the ability to select and apply. . . procedures rapidly and correctly....[Computer] courseware can concentrate on one skill at a time, in a manner impossible for a textbook and hardly available to the classroom teacher, namely by asking the student to handle only that part of a procedure on which pedagogical stress is to be laid, while other aspects of the same procedure are handled automatically by the computer. . . . This scheme, which combines student interaction with computer assistance, has the merit of focusing attention on the key strategic and conceptual decisions needed to handle a problem. . . This should be of significance to both the strong student . . . and the weak student.

Jacob T. Schwartz

When I started off doing mathematics, I wasn't very good at it. I never learned my multiplication tables and it was certainly the conclusion of my teachers at that time that there was no way I would ever go on and do anything . . . mathematically oriented. As it turned out, I found out about computers and found out that you could make computers do these kinds of things.

Stephen Wolfram

Mathematical concepts may be communicated easily in a format which combines visual, verbal, and symbolic representations in tight coordination.

Ralph Abraham

Before Gutenburg, illustration and type were one in the same; they were inseparable. But afterward, the two disciplines became separate and diverged. Now that we've got the [graphic computer], I can see a medium where they come back together again.

Scott Kim

Perhaps, . . . one should consider teaching mathematics backwards, that is, teaching the images and patterns first and teaching the conventional symbol system, rules and rigorous processes later. This kind of fundamental change . . . may be worth the trouble, especially if it can be shown that some of those who could not do well with elementary mathematics may still do superbly well with the more advanced forms. It may require entirely different ways of teaching, but it may, in the end be more appropriate for a new era in mathematics and work, when all the easy things will be done by machine and the hard things will be the only things of value left for humans to do.

Thomas West

There's a terrible problem that I run into in teaching, which is that when you tell people something, you keep them from knowing it. If they find it on their own, they'll know it in a way they never will if you tell them. What I try to more and more is to bring students to my studio and get them really working.

Richard Benson

Recently I met with an engineering administrator at a fine university - a liberal, progressive institution where one would expect the best sorts of things to be happening. To my surprise, he complained bitterly about the mathematics department. The math professors refused to say even a few words to engineering students about applicability of mathematics to engineering, refused to include in their classes a few examples of how engineers actually use math. He wanted them to help his students see that mathematics was not a mere obstacle to be overcome, but a vital element of the profession to which they were committed. But apparently the mathematics professors were determined to teach their subject straight pure on its own terms, as if each student were destined to become a Ph. D. in that field. It was a matter of pride in their specialty and honor for their department. The engineering administrator was utterly disheartened. And if at a liberal institution the math department resists co-operation with the engineering school, I can only imagine how frustrating things are in other places and in other circumstances. It is difficult enough for engineering deans to achieve change within the realm they nominally control. When they go outside their domain - as they must do to revamp the curriculum in a meaningful way - the problems seem overwhelming.... At the core of the problem is a combination of both disciplinary hubris and academic territoriality, both militating against cooperative ventures across disciplinary and administrative boundaries.

Instead of trying to interest the students in engineering, to nurture enthusiasm for the profession to be their life's calling, the [engineering] program was designed to be an obstacle course. The first two years were dedicated almost exclusively to mathematics and basic sciences, with no effort made to show how these often brutally difficult studies would help the future engineer to do constructive work. . . . The Spartan legacy dies hard - " If I had to do it, you can do it." . . . If the course of study that once produced intrepid engineers is now perceived as suitable for nerds, then none but nerds will pursue it.

With a recrafted curriculum, introducing real engineering at an early stage and showing it to best advantage, the situation can be improved. Instead of a medicinal dose of math and science, let freshmen get a taste of engineering design and how engineering fits into society at large. Math and science will then take on added meaning and appeal.

Samuel Florman

If textbook cloning represented the discovery of a true educational optimum . . . I would not object...Good teaching requires fresh thought and genuine excitement... Rote copying can only indicate boredom...A carelessly cloned work will not excite students however pretty the pictures. We will not have great texts if authors cannot shape content but must serve a commercial master as one cog in an ultimately powerless consortium with other packagers.

Stephen Jay Gould

[If] logic is the hygiene of the mathematician, it is not his source of food.

Andrew Weil

There are two kinds of rigor: Intellectual rigor and mathematical rigor. They are not always the same.

Andrew Gleason,Past President of the American Mathematical Society

Classical proof must must move over and share the educational stage with other means of arriving at mathematical evidence. Mathematical textbooks must modify the often deadening rigidity of the Euclidean model of exposition. The capabilities of all mathematicians are elevated by their association with computation. ... mathematics develops in such a way that the role of the mathematician is always manifest. . .

In connection with computer visualization, I have argued for the recognition of visual theorems where what the eye sees need not be verbalized let alone formalized in traditional formal mathematical language.

Philip Davis

The traditional course on differential equations that I took many years ago and taught up until the past couple of years dealt almost entirely with derivation of formulas for solutions of various kinds of differential equations,... Most of the algebraic manipulations featured in the traditional course can now easily be relegated to a computer. ... A good deal of this traditional material can be dispensed with and replaced by experiences more valuable for the student... A decrease of time spent on symbol manipulation by hand should provide an opportunity for more emphasis on conceptual understanding,...Details of mathematical procedures and algorithms are rapidly forgotten unless they are used frequently, but underlying concepts and ideas become part of an individual's mindset.

William Boyce 1995

I can't say too strongly how unimportant symbolic manipulation is in engineering. We see the effects of MSM (mindless symbolic manipulation) every day: Students who can integrate t^3 Cos[t] but have no idea what they are doing and why.... Not only is the material of an MSM course useless and outdated, but the message it sends to students is bad. It basically suggests that learning math is mastering a certain list of stimulus-response behaviors.....No wonder we in EE and CS and other engineering fields rely less and less on people in math to deliver the basic training needed in our fields.

Stephen Boyd

Mathematics courses are so locked in that even the suggestion of change precipitates a crisis.

Jeff Knisley

Visual mathematics: no magic bullets, but such tantalizing potential!

David Olson

So far as differential equations are concerned, from looking back at what I did 40 years ago, compared to what I practice now, it seems to me that I'd eliminate a large amount of DE solving and categorizing tricks I learned in those days. My current view is: If Mathematica knows how to solve it analytically, then I don't need to know. If Mathematica doesn't know how to solve it analytically, hell, I'll just solve it numerically.

Tony Siegman, Stanford Electrical Engineer

The axiomization and algebraization of mathematics, after more than 50 years, has led to the illegibility so such a large number of mathematical texts that the threat of complete loss of contact with physics and the natural sciences has been realized.

V. I. Arnold

Many mathematicians seem to be frozen in the past. Many of today's math courses actually look to others a lot like 1860 understanding of [math] looks to us. We better learn to acknowledge that math is incredibly sloppy today, just as it was in 1860. For me, the daily use of computer algebra systems has opened my eyes; others may find other ways of waking up.

Matthias Kawski

Jerry's favorites from the masters

The more it approaches intuition, the more reliable the deduction is. Intuition has two distinctive features - it is an instantaneous act and it consists of clear grasp of an idea. Intuition and deduction should be trustworthy processes which we can use to lead to genuine knowledge.  There are two ways of arriving at knowledge - through experience and deduction.

Rene Descartes

It is unworthy of excellent persons to lose hours like slaves in the labor of calculation.

Gottfried Wilhelm von Leibniz

Some defects ordinarily found in the method used by mathematicians:

Defect 1: To pay more attention to proof than to evidence, and to try to convince rather than enlighten the mind.

Defect 2: To prove things that do not require proof.

Antoine Arnaud and Pierre Nicole

[The normal student attempts to fix formal terminology] in his memory because it means nothing to his intelligence.

Blaise Pascal

Never undertake to prove things that are so evident in themselves that one has nothing clearer by which to prove them.

[Some facts] can be seen more clearly by an example than by a proof.

Leonhard Euler

[A good calculating machine could] weave algebraic patterns just as the Jacquard loom weaves flowers and leaves.

Augusta Ada Byron

We must agree that the idea [of formal definition of limit] is not sufficiently clear to use as a basis for a science in which certainty must be founded on evidence, especially when presented to beginners.

J.L. Lagrange

All theory, dear friend, is gray, but the golden tree of actual life springs ever green.

Goethe

Thoughts die the moment they are embodied by words.

Schopenhauer

When a student commences seriously to study mathematics, he believes he knows what a fraction is, what continuity is, and what the area of a curved surface is; he considers evident, for example that a continuous function cannot change its sign without vanishing. If [the teacher] says to him: No that is not evident; I must prove it to you, . . . what would this unfortunate student think? He will think that the science of mathematics is only an arbitrary accumulation of useless subtleties; either he will be disgusted with it or he will amuse himself with it as a game and arrive at a state of mind analogous to that of the Greek sophists. Can one ever understand a theory if one builds it up right from the start in the definitive form that rigorous logic imposes? No; one does not really understand it . . .one retains it only by learning it by heart.

Henri Poincare

I am led to the idea which in substance is that the ordinary first course in calculus should be more largely graphical. The question that the beginner desires answered is what is what is the calculus about - -what does it do? . . . Let us be satisfied with a rough definition of the derivative, allowing intuition and not rigor to be the chief factor in supplying it and therefore drawing very largely on the geometrical side. . .

W. B. Ford

Elementary mathematics . . . must be purged of every element which can only be justified by reference to a more prolonged course of study. There can be nothing more destructive of true education than to spend long hours in the acquirement of ideas and methods that lead nowhere. . . . [The] elements of mathematics should be treated as the study of fundamental ideas, the importance of which the student can immediately appreciate; . . . every proposition and method which cannot pass this test, however important for a more advanced study, should be ruthlessly cut out. The solution I am urging is to eradicate the fatal disconnection of topics which kills the vitality of our modern curriculum. There is only one subject matter, and that is Life in all its manifestations. Instead of this single unity, we offer children Algebra, from which nothing follows. . .

Civilization advances by extending the number of operations we can do without thinking about them. Operations of thought are like cavalry charges in a battle - they are strictly limited in number, they require fresh horses and must be made only at decisive moments.

Alfred North Whitehead

There is a real hypocrisy, quite frequent in the teaching of mathematics. The teacher takes verbal precautions, which are valid in the sense he gives them, but that the students most assuredly will not understand the same way.

Unfortunately competitive examinations often encourage [an educational] deception. The teachers must train their students to answer little fragmentary questions quite well, and they give them model answers that are often veritable masterpieces and that leave no room for criticism. To achieve this, the teachers isolate each question from the whole of mathematics and create for this question alone a perfect language without bothering about its relationships to other questions. Mathematics is no longer a monument but a heap.

If the limit of the [polygonal areas] had been designated as the 'tarababump' of the circle, one would not be permitted to derive from it the value of the tarababump of the sector. . . We allow ourselves to do it because instead of the word tarababump, we used the word area! . . . imagine the inevitable confusion that will be caused by making students identify this new area with the areas to which they are accustomed.

Henri Lebesgue

From the beginning Nature has led the way and established the pattern which mathematics, the language of nature, must follow.

George D. Birkhoff

Personally, I am always ready to learn, although I do not always like being taught.

Winston Churchill

The talent [of being able to rapidly make very complicated calculations in the head] is, in reality, distinct from mathematical ability. Very few known mathematicians are said to have possessed it. . .

I insist that words are totally absent from my mind when I really think. .

. . Even after reading or hearing a question, every word disappears at the very moment I am beginning to think it over; words do not reappear in my consciousness before I have accomplished or given up the research.

. . . and I fully agree with Schopenhauer when he writes, Thoughts die the moment they are embodied by words I think it is also essential to emphasize that I behave in this way not only about words, but even about algebraic signs. I use them when dealing with easy calculations; but whenever the matter looks more difficult, they become too heavy baggage for me.

Jacques Hadamard

The words or language, as they are written or spoken do not seem to play any role in my mechanism of thought. The psychical entities which seem to serve as elements in thought are certain signs and more or less clear images which can be voluntarily reproduced and combined. We will never solve the problems of the world from the level of thinking we were at when we created them.

Albert Einstein

At a great distance from its empirical source. . ., a mathematical subject is in danger of degeneration. At the inception the style is usually classical; when it shows signs of becoming baroque, then the danger sign is up.

John von Neumann

[Mathematics] must remain . . . as a unified vital strand in the broad stream of science and must be prevented from becoming a little side brook that might disappear in the sand . . .

[The student] refuses to be bored by diffuseness and general statements which convey nothing. . . [The student] will not tolerate a pedantry which makes no distinction between the essential and the non-essential, and which for the sake of a systematic set of axioms, deliberately conceals the facts to which the growth of the subject [of calculus] is due.

Richard Courant

The purpose of computing is insight, not numbers.

Richard Hamming

In mathematics, you can be as formal and rigorous as you want to be; but without insight, you'll get nowhere.

Richard Duffin

It's all visual. It's hard to explain. . . . Ordinarily I try to get the pictures clearer, but in the end mathematics can take over and be more efficient at communicating the idea of the picture. In certain problems that I have done it was necessary to continue the development of the picture as the method before the mathematics could really be done.

Many of the math books . . . are full of . . . nonsense - of carefully and precisely defined words that are used by pure mathematicians in their most subtle and and difficult analyses, and are used by nobody else . . .. The real problem in speech is not precise language. The problem is clear language.

Richard Feynman

Mathematics today is a vital, vibrant discipline composed of many parts which in mysterious ways influence and enrich each other . . .. It is beginning to emerge from a self imposed isolation [from its sister disciplines] and listen with attention to the voices of nature.

Mark Kac

As societies become more complex in structure and resources, the need of formal or intentional teaching and learning increases. As formal teaching and training grow in extent, there is the danger of creating an undesirable split between the experience gained in more direct associations and what is acquired in school. This danger was never greater than the present time, on account of the rapid growth in the last few centuries of knowledge and technical modes of skill."

John Dewey

Is it life, I ask, is it even prudence,
To bore thyself and bore the students?

Johann Wolfgang von Goethe

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