On Teaching (1990-1999)

"True Dialogue occurs when teachers ask questions to which they do not presume to already know the 'correct answer'." (Jay Lemke, "Talking Science: Language, Learning and Values", 1990)

"One of the lessons that the history of mathematics clearly teaches us is that the search for solutions to unsolved problems, whether solvable or unsolvable, invariably leads to important discoveries along the way. (Carl B Boyer & Uta C Merzbach, "A History of Mathematics", 1991)

"Probability does pervade the universe, and in this sense, the old chestnut about baseball imitating life really has validity. The statistics of streaks and slumps, properly understood, do teach an important lesson about epistemology, and life in general. The history of a species, or any natural phenomenon, that requires unbroken continuity in a world of trouble, works like a batting streak. All are games of a gambler playing with a limited stake against a house with infinite resources. The gambler must eventually go bust. His aim can only be to stick around as long as possible, to have some fun while he's at it, and, if he happens to be a moral agent as well, to worry about staying the course with honor!" (Stephen J Gould, 1991)

"[...] the first reason for teaching science to non scientists is that many of these nonscientists have a genuine desire to learn about science, and this, after all, is the best reason for teaching anything to anyone." (Jeremy Bernstein, "Cranks, Quarks, and the Cosmos: Writings on Science", 1993)

"We have to teach non-statisticians to recognize where statistical expertise is required. No one else will. We teach students how to solve simple statistical problems, but how often do we make any serious effort to teach them to recognize situations that call for statistical expertise that is beyond the technical content of the course." (Christopher J Wild, "Embracing the ‘Wider view’ of Statistics", The American Statistician 48, 1994)

"There are aspects of statistics other than it being intellectually difficult that are barriers to learning. For one thing, statistics does not benefit from a glamorous image that motivates students to persist through tedious and frustrating lessons[...]there are no TV dramas with a good-looking statistician playing the lead, and few mothers’ chests swell with pride as they introduce their son or daughter as 'the statistician'." (Chap T Le & James R Boen, "Health and Numbers: Basic Statistical Methods", 1995)

"Whoever teaches learns in the act of teaching, and whoever learns teaches in the act of learning." (Paulo Freire, "Pedagogy of Freedom", 1996)

"Because no one becomes statistically self-sufficient after one semester of study, I try to prepare students to become intelligent consumers of the assistance that they will inevitably seek. Service courses train future clients, not future statisticians." (Michael W Tosset, "Statistical Science", 1998)

"A teacher who cannot explain any abstract subject to a child does not himself thoroughly understand his subject; if he does not attempt to break down his knowledge to fit the child's mind, he does not understand teaching." (Fulton J Sheen, "Life Is Worth Living", 1999)

On Teaching (1979-1979)

 "By showing us the extreme diversity of the factors involved in scientific creativity, the history of science teaches us that we should open the doors of our laboratories more widely. If we put that lesson into practice, our reflection on the past will have had a beneficial effect on the future." (Jean Rostand, "Humanly Possible: A Biologist’s Note on the Future of Mankind", 1970)

"Many teachers and textbook writers have never recognized the power of sheer intellectual curiosity as a motive for the highest type of work in mathematics, and as a consequence they have failed to organize and present the work in a manner designed to stimulate the student’s interest through a challenge to his curiosity." (Charles H Butler & F Lynwood Wren, "The Teaching of Secondary Mathematics" 5th Ed., 1970)

"Each generation has its few great mathematicians, and mathematics would not even notice the absence of the others. They are useful as teachers, and their research harms no one, but it is of no importance at all. A mathematician is great or he is nothing." (Alfred Adler, "Mathematics and Creativity", New Yorker Magazine, 1972) 

"There are two subcategories of holist called irredundant holists and redundant holists. Students of both types image an entire system of facts or principles. Though an irredundant holist's image is rightly interconnected, it contains only relevant and essential constitents. In contrast, redundant holists entertain images that contain logically irrelevant or overspecific material, commonly derived from data used to 'enrich' the curriculum, and these students embed the salient facts and principles in a network of redundant items. Though logically irrelevant, the items in question are of great psychological importance to a 'redundant holist', since he uses them to access, retain and manipulate whatever he was originally required to learn." (Gordon Pask, "Learning Strategies and Individual Competence", 1972)

"A professor’s enthusiasm for teaching introductory courses varies inversely with the likelihood of his having to do it." (Thomas L Martin Jr, "Malice in Blunderland", 1973)

"Small wonder that students have trouble [with statistical hypothesis testing]. They may be trying to think." (W Edwards Deming, "On probability as a basis for action", American Statistician 29, 1975)

"We don’t teach our students enough of the intellectual content of experiments - their novelty and their capacity for opening new fields. [...] My own view is that you take these things personally. You do an experiment because your own philosophy makes you want to know the result. It’s too hard, and life is too short, to spend your time doing something because someone else has said it’s important. You must feel the thing yourself [...]" (Isidor Isaac Rabi, The New Yorker Magazine, October 13, 1975)

"I would [...] urge that people be introduced to [chaos] early in their mathematical education. [Chaos] can be studied phenomenologically by iterating it on a calculator, or even by hand. Its study does not involve as much conceptual sophistication as does elementary calculus. Such study would greatly enrich the student's intuition. Not only in research, but also in the everyday world of politics and economics, we would all be better off if more people realised that simple nonlinear systems do not necessarily possess simple dynamical properties." (Robert May, "Simple mathematical models with very complicated dynamics", Nature 26(5560), 1976)

On Teaching (1960-1969)

"Every mathematician agrees that every mathematician must know some set theory; the disagreement begins in trying to decide how much is some. [...] The student's task in learning set theory is to steep himself in unfamiliar but essentially shallow generalities till they become so familiar that they can be used with almost no conscious effort. In other words, general set theory is pretty trivial stuff really, but, if you want to be a mathematician, you need some, and here it is; read it, absorb it, and forget it [...] the language and notation are those of ordinary informal mathematics. A more important way in which the naive point of view predominates is that set theory is regarded as a body of facts, of which the axioms are a brief and convenient summary; in the orthodox axiomatic view the logical relations among various axioms are the central objects of study." (Paul R Halmos, "Naive Set Theory", 1960)

"The first [principle], is that a mathematical theory can only he developed axiomatically in a fruitful way when the student has already acquired some familiarity with the corresponding material - a familiarity gained by working long enough with it on a kind of experimental, or semiexperimental basis, i.e. with constant appeal to intuition. The other principle [...]  is that when logical inference is introduced in some mathematical question, it should always he presented with absolute honesty - that is, without trying to hide gaps or flaws in the argument; any other way, in my opinion, is worse than giving no proof at all." (Jean Dieudonné, "Thinking in School Mathematics", 1961)

"The teaching of probabilistic reasoning, so very common and important a feature of modern science, is hardly developed in our educational system before college." (Jerome S Bruner, , "The Process of Education", 1961)

"But both managed to understand mathematics and to make a 'fair' number of contributions to the subject. Rigorous proof is not nearly so important as proving the worth of what we are teaching; and most teachers, instead of being concerned about their failure to be sufficiently rigorous, should really be concerned about their failure to provide a truly intuitive approach.. The general principle, then, is that the rigor should be suited to the mathematical age of the student and not to the age of mathematics." (Morris Kline, "Mathematics: A Cultural Approach", 1962) 

"Creativity is the heart and soul of mathematics at all levels. The collection of special skills and techniques is only the raw material out of which the subject itself grows. To look at mathematics without the creative side of it, is to look at a black-and-white photograph of a Cezanne; outlines may be there, but everything that matters is missing." (Robert C Buck "Teaching Machines and Mathematics Programs",  The American Mathematical Monthly 69, 1962)

"The word model is used as a noun, adjective, and verb, and in each instance it has a slightly different connotation. As a noun 'model' is a representation in the sense in which an architect constructs a small-scale model of a building or a physicist a large-scale model of an atom. As an adjective 'model' implies a degree or perfection or idealization, as in reference to a model home, a model student, or a model husband. As a verb 'to model' means to demonstrate, to reveal, to show what a thing is like." (Russell L Ackoff, "Scientific method: optimizing applied research decisions", 1962)

"Determinants are often advertised to students of elementary mathematics as a computational device of great value and efficiency for solving numerical problems involving systems of linear equations. This is somewhat misleading, for their value in problems of this kind is very limited. On the other hand, they do have definite importance as a theoretical tool. Briefly, they provide a numerical means of distinguishing between singular and non-singular matrices (and operators)." (George F Simmons, "Introduction to Topology and Modern Analysis", 1963)

"Science is a way to teach how something gets to be known, what is not known, to what extent things are known (for nothing is known absolutely), how to handle doubt and uncertainty, what the rules of evidence are, how to think about things so that judgments can be made, how to distinguish truth from fraud, and from show." (Richard P Feynman, "The Problem of Teaching Physics in Latin America", Engineering and Science, 1963)

"Creative activity could be described as a type of learning process where teacher and pupil are located in the same individual." (Arthur Koestler, "Drinkers of Infinity: Essays 1955-1967", 1967)

"Teaching is more difficult than learning because what teaching calls for is this: to let learn. The real teacher, in fact, let nothing else be learned than learning. His conduct, therefore, often produces the impression that we properly learn nothing from him, if by ‘learning’ we now suddenly understand merely the procurement of useful information." (Martin Heidegger, "What is called thinking?", 1968)

"But, really, mathematics is not religion; it cannot be founded on faith. And what was most important, the methods yielding such remarkable results in the hands of the great masters began to lead to errors and paradoxes when employed by their less talented students. The masters were kept from error by their perfect mathematical intuition, that subconscious feeling that often leads to the right answer more quickly than lengthy logical reasoning. But the students did not possess this intuition […]" (Naum Ya. Vilenkin, "Stories about Sets", 1968)

"Science does not exclude faith. […] Science does not teach a harsh materialism. It does not teach anything beyond its boundaries, and those boundaries have been severely limited by science itself." (Vannevar Bush, "Modern Arms and Free Men", 1968)

"Science progresses not only because it helps to explain newly discovered facts, but also because it teaches us over and over again what the word 'understanding' may mean." (Werner K Heisenberg, "Physics and Beyond: Encounters and Conversations", 1969)

On Teaching (-1849)

"If I am given a formula, and I am ignorant of its meaning, it cannot teach me anything, but if I already know it what does the formula teach me?" (Saint Augustine of Hippo, "De Magistro", cca. 5th century)

"For in such a case, the teacher has a double job: the first to erase the [effects of] previous faulty instruction, the second to give the student true and correct training." (John of Salisbury, "Metalogicon", 1159)

"Often I have considered the fact that most of the difficulties which block the progress of students trying to learn analysis stem from this: that although they understand little of ordinary algebra, still they attempt this more subtle art. From this it follows not only that they remain on the fringes, but in addition they entertain strange ideas about the concept of the infinite, which they must try to use." (Leonhard Euler, "Introduction to Analysis of the Infinite", 1748)

"To know how to suggest is the great art of teaching. To attain it we must be able to guess what will interest." (Henri-Frédéric Amiel, 1864)

"The first thing to be attended to in reading any algebraical treatise, is the gaining a perfect understanding of the different processes there exhibited, and of their connection with one another. This cannot be attained by a mere reading of the book, however great the attention which may be given. It is impossible, in a mathematical work, to fill up every process in the manner in which it must be filled up in the mind of the student before he can be said to have completely mastered it. Many results must be given of which the details are suppressed, such are the additions, multiplications, extractions of the square root, etc., with which the investigations abound. These must not be taken on trust by the student, but must be worked by his own pen, which must never be out of his hand, while engaged in any algebraical process." (Augustus de Morgan, "On the Study and Difficulties of Mathematics", 1830)

"There is no teaching until the pupil is brought into the same state or principle in which you are; a transfusion takes place; he is you, and you are he; then is a teaching; and by no unfriendly chance or bad company can he ever lose the benefit." (Ralph W Emerson, "Essays", 1841)

On Teaching (1850-1874)

"This science, Geometry, is one of indispensable use and constant reference, for every student of the laws of nature; for the relations of space and number are the alphabet in which those laws are written. But besides the interest and importance of this kind which geometry possesses, it has a great and peculiar value for all who wish to understand the foundations of human knowledge, and the methods by which it is acquired. For the student of geometry acquires, with a degree of insight and clearness which the unmathematical reader can but feebly imagine, a conviction that there are necessary truths, many of them of a very complex and striking character; and that a few of the most simple and self-evident truths which it is possible for the mind of man to apprehend, may, by systematic deduction, lead to the most remote and unexpected results." (William Whewell, "The Philosophy of the Inductive Sciences", 1858)

"Few will deny that even in the first scientific instruction in mathematics the most rigorous method is to be given preference over all others. Especially will every teacher prefer a consistent proof to one which is based on fallacies or proceeds in a vicious circle, indeed it will be morally impossible for the teacher to present a proof of the latter kind consciously and thus in a sense deceive his pupils. Notwithstanding these objectionable so-called proofs, so far as the foundation and the development of the system is concerned, predominate in our textbooks to the present time. Perhaps it will be answered, that rigorous proof is found too difficult for the pupil’s power of comprehension. Should this be anywhere the case, - which would only indicate some defect in the plan or treatment of the whole, - the only remedy would be to merely state the theorem in a historic way, and forego a proof with the frank confession that no proof has been found which could be comprehended by the pupil; a remedy which is ever doubtful and should only be applied in the case of extreme necessity. But this remedy is to be preferred to a proof which is no proof, and is therefore either wholly unintelligible to the pupil, or deceives him with an appearance of knowledge which opens the door to all superficiality and lack of scientific method." (Hermann G Grassmann, "Stücke aus dem Lehrbuche der Arithmetik", 1861)

"However good lectures may be, and however extensive the course of reading by which they are followed up, they are but accessories to the great instrument of scientific teaching — demonstration." (Thomas H Huxley, "A Lobster; or, The Study of Zoology", 1861)

"[...] good methods can teach us to develop and use to better purpose the faculties with which nature has endowed us, while poor methods may prevent us from turning them to good account. Thus the genius of inventiveness, so precious in the sciences, may be diminished or even smothered by a poor method, while a good method may increase and develop it." (Claude Bernard,, "An Introduction to the Study of Experimental Medicine", 1865)

"Observation, then, is what shows facts.; experiment is what teaches about facts and gives experience in relation to anything." (Claude Bernard, "An Introduction to the Study of Experimental Medicine", 1865)

"Nothing can be more fatal to progress than a too confident reliance on mathematical symbols; for the student is only too apt to take the easier course, and consider the formula not the fact as the physical reality." (William T Kelvin & Peter G Tait, "Treatise on Natural Philosophy", 1867)

"It is very desirable to have a word to express the Availability for work of the heat in a given magazine; a term for that possession, the waste of which is called Dissipation. Unfortunately the excellent word Entropy, which Clausius has introduced in this connexion, is applied by him to the negative of the idea we most naturally wish to express. It would only confuse the student if we were to endeavour to invent another term for our purpose. But the necessity for some such term will be obvious from the beautiful examples which follow. And we take the liberty of using the term Entropy in this altered sense [...] The entropy of the universe tends continually to zero." (Peter G Tait, "Sketch Of Thermodynamics", 1868)

"Therefore, the great business of the scientific teacher is, to imprint the fundamental, irrefragable facts of his science, not only by words upon the mind, but by sensible impressions upon the eye, and ear, and touch of the student, in so complete a manner, that every term used, or law enunciated, should afterwards call up vivid images of the particular structural, or other, facts which furnished the demonstration of the law, or the illustration of the term." (Thomas H Huxley, "Lay Sermons, Addresses and Reviews", 1870)

"The teacher who is attempting to teach without inspiring the pupil to learn is hammering on cold iron." (Horace Mann, "Thoughts Selected from the Writings of Horace Mann", 1872

"The success of academic teaching is based on the teacher continuously inducing his student to engage in independent research. The teacher achieves this by the fact that the very layout of materials when stating the subject and demonstration of guiding ideas shows the student the way which would lead a mature thinker who possesses all observations to new results in the right sequence or to a better substantiation of the already known results." (Karl Weierstrass, [speech] 1873)

"Thought is symbolical of Sensation as Algebra is of Arithmetic, and because it is symbolical, is very unlike what it symbolises. For one thing, sensations are always positive; in this resembling arithmetical quantities. A negative sensation is no more possible than a negative number. But ideas, like algebraic quantities, may be either positive or negative. However paradoxical the square of a negative quantity, the square root of an unknown quantity, nay, even in imaginary quantity, the student of Algebra finds these paradoxes to be valid operations. And the student of Philosophy finds analogous paradoxes in operations impossible in the sphere of Sense. Thus although it is impossible to feel non-existence, it is possible to think it; although it is impossible to frame an image of Infinity, we can, and do, form the idea, and reason on it with precision. (George H Lewes "Problems of Life and Mind", 1873)

On Teaching (2000-2009)

"Two types of graphic organizers are commonly used for comparison: the Venn diagram and the comparison matrix [...] the Venn diagram provides students with a visual display of the similarities and differences between two items. The similarities between elements are listed in the intersection between the two circles. The differences are listed in the parts of each circle that do not intersect. Ideally, a new Venn diagram should be completed for each characteristic so that students can easily see how similar and different the elements are for each characteristic used in the comparison." (Robert J. Marzano et al, "Classroom Instruction that Works: Research-based strategies for increasing student achievement, 2001)

"Mathematics is often thought to be difficult and dull. Many people avoid it as much as they can and as a result much of the population is mathematically illiterate. This is in part due to the relative lack of importance given to numeracy in our culture, and to the way that the subject has been presented to students." (Julian Havil , "Gamma: Exploring Euler's Constant", 2003)

"Another aspect of representativeness that is misunderstood or ignored is the tendency of regression to the mean. Stochastic phenomena where the outcomes vary randomly around stable values (so-called stationary processes) exhibit the general tendency that extreme outcomes are more likely to be followed by an outcome closer to the mean or mode than by other extreme values in the same direction. For example, even a bright student will observe that her or his performance in a test following an especially outstanding outcome tends to be less brilliant. Similarly, extremely low or extremely high sales in a given period tend to be followed by sales that are closer to the stable mean or the stable trend." (Hans G Daellenbach & Donald C McNickle, "Management Science: Decision making through systems thinking", 2005)

"Many people believe that all of mathematics has already been discovered and codified. Mathematicians (they think) do nothing except rearrange the material in different ways for different types of students. This seems to be the result of the cut-and-dried method of teaching mathematics in many high schools and universities. The facts are laid out in the cleanest logical order. Little attempt is made to show how someone once had to invent it all, at first in a confused way, and that only later was it possible to give it this neat form." (Avner Ash & Robert Gross, "Fearless Symmetry: Exposing the hidden patterns of numbers", 2006)

"Math is actually very important, but because it genuinely is difficult, nearly all of the teaching slots are occupied with making sure that students learn how to solve certain types of problem and get the answers right. There isn't time to tell them about the history of the subject, about its connections with our culture and society, about the huge quantity of new mathematics that is created every year, or about the unsolved questions, big and little, that litter the mathematical landscape." (Ian Stewart, "Letters to a Young Mathematician", 2006)

"But in mathematics there is a kind of threshold effect, an intellectual tipping point. If a student can just get over the first few humps, negotiate the notational peculiarities of the subject, and grasp that the best way to make progress is to understand the ideas, not just learn them by rote, he or she can sail off merrily down the highway, heading for ever more abstruse and challenging ideas, while an only slightly duller student gets stuck at the geometry of isosceles triangles." (Ian Stewart, "Why Beauty is Truth: A history of symmetry", 2007)

"[…] mathematics is not only to teach the algorithms and skills of mathematics - which we will agree are very important - but also to teach for understanding, with an emphasis on reasoning." (Alfred S Posamentier et al, "Exemplary Practices for Secondary Math Teachers", 2007)

"As students, we learned mathematics from textbooks. In textbooks, mathematics is presented in a rigorous and logical way: definition, theorem, proof, example. But it is not discovered that way. It took many years for a mathematical subject to be understood well enough that a cohesive textbook could be written. Mathematics is created through slow, incremental progress, large leaps, missteps, corrections, and connections." (Richard S Richeson, "Eulers Gem: The Polyhedron Formula and the birth of Topology", 2008)

"Nonetheless, some hesitation persisted. After all, the very word imaginary betrays ambivalence, and suggests that in our heart of hearts we do not believe these numbers exist. On the other hand, by calling every number representable by a decimal expansion real, we are making the psychological distinction more stark. Indeed the adjective imaginary is a somewhat unfortunate one - although an intriguing name, some students’ perceptions are so colored by the word that they consequently fail to come to grips with the idea." (Peter M Higgins, "Number Story: From Counting to Cryptography", 2008)

"What could mathematics and poetry share, except that the mention of either one is sometimes enough to bring an uneasy chill into a conversation? [...] Both fields use analogies - comparisons of all sorts - to explain things, to express unknown or unknowable concepts, and to teach." (Marcia Birken & Anne C Coon, "Discovering Patterns in Mathematics and Poetry", 2008)

"Mathematics is the art of explanation. If you deny students the opportunity to engage in this activity - to pose their own problems, to make their own conjectures and discoveries, to be wrong, to be creatively frustrated, to have an inspiration, and to cobble together their own explanations and proofs - you deny them mathematics itself." (Paul Lockhart, "A Mathematician's Lament", 2009)

On Teaching (2010-2019)

"If we are to be effective mathematics teachers, we should endeavor to understand students’ values and students’ goals. Not to mention their motivations." (Steven G Krantz, "A Mathematician Comes of Age", 2012)

"Statistics is the scientific discipline that provides methods to help us make sense of data. […] The field of statistics teaches us how to make intelligent judgments and informed decisions in the presence of uncertainty and variation." (Roxy Peck & Jay L Devore, "Statistics: The Exploration and Analysis of Data" 7th Ed, 2012)

"Understanding chaos requires much less advanced mathematics than other current areas of physics research such as general relativity or particle physics. Observing chaos and fractals requires no specialized equipment; chaos is seen in scores of everyday phenomena - a boiling pot of water, a dripping faucet, shifting weather patterns. And fractals are almost ubiquitous in the natural world. Thus, it is possible to teach the central ideas and insights of chaos in a rigorous, genuine, and relevant way to students with relatively little mathematics background." (David P Feldman, "Chaos and Fractals: An Elementary Introduction", 2012)

"A mathematical entity is a concept, a shared thought. Once you have acquired it, you have it available, for inspection or manipulation. If you understand it correctly (as a student, or as a professional) your ‘mental model’ of that entity, your personal representative of it, matches those of others who understand it correctly. (As is verified by giving the same answers to test questions.) The concept, the cultural entity, is nothing other than the collection of the mutually congruent personal representatives, the ‘mental models’, possessed by those participating in the mathematical culture." (Reuben Hersh, "Experiencing Mathematics: What Do We Do, when We Do Mathematics?", 2014)

"Working an integral or performing a linear regression is something a computer can do quite effectively. Understanding whether the result makes sense - or deciding whether the method is the right one to use in the first place - requires a guiding human hand. When we teach mathematics we are supposed to be explaining how to be that guide. A math course that fails to do so is essentially training the student to be a very slow, buggy version of Microsoft Excel." (Jordan Ellenberg, "How Not to Be Wrong: The Power of Mathematical Thinking", 2014)

"Complex numbers do not fit readily into many people’s schema for ‘number’, and students often reject the concept when it is first presented. Modern mathematicians look at the situation with the aid of an enlarged schema in which the facts make sense." (Ian Stewart & David Tall, "The Foundations of Mathematics" 2nd Ed., 2015)

"Mathematics courses are hierarchical but every new course begins with the assumption that the student is at the level of conceptual development that would be implied by an optimal understanding of the previous course. Unfortunately many mathematical ideas are so subtle and logically complex that it may take students many years to develop an adequate conceptual understanding. As a result, in practice there is a lot of 'faking it' going on and not merely on the part of the students." (William Byers, "Deep Thinking: What Mathematics Can Teach Us About the Mind", 2015)

On Teaching (Unsourced)

"A good problem should be more than a mere exercise; it should be challenging and not too easily solved by the student, and it should require some ‘dreaming’ time." (Howard W Eves)

"A great teacher is not simply one who imparts knowledge to his students but is one who awakens their interest in the subject and makes them eager to pursue it for themselves. An outstanding teacher is a spark plug, not a fuel line." (Norman J Berrill)

"A good teacher protects his pupils from his own influence." (Bruce Lee)

"A good teacher will not just make people accept a form of life, he will also provide them with means of seeing it in perspective and perhaps of even rejecting it." (Paul K Feyerabend)

"A teacher is a compass that activates the magnets of curiosity, knowledge, and wisdom in the pupils." (Ever Garrison)

"[...] a truly popular lecture cannot teach, and a lecture that truly teaches cannot be popular" (Michael Faraday)

"All explanations should be given in a language that pupils understand." (John A Comenius)

"An extremely odd demand is often set forth but never met, even by those who make it; i.e., that empirical data should be presented without any theoretical context, leaving the reader, the student, to his own devices in judging it. This demand seems odd because it is useless simply to look at something. Every act of looking turns into observation, every act of observation into reflection, every act of reflection into the making of associations; thus it is evident that we theorize every time we look carefully at the world." (Johann Wolfgang von Goethe)

"[Education carries an impact] as long as the student has a need for it and applies it to some situation of his own. Every new idea should be worked out in application." (John A Comenius)

"[…] education is not something which the teacher does, but that it is a natural process which develops spontaneously in the human being." (Maria Montessori)

"Education is teaching our children to desire the right things." (Plato)

"Great teachers do not act important; they make their students feel important." (Todd Whitaker)

"Great teachers see challenging students as a reason to try that much harder." (Todd Whitaker)

"Great teachers have high expectations for their students, but higher expectations for themselves."  (Todd Whitaker)

"Great teachers treat their students the way their best teacher treated them." (Al Burr)

"Hypotheses are lullabies with which the teacher lulls his pupils to sleep. The thinking and faithful observer learns to know his limitation more and more; he sees that the further knowledge extends the more problems arise." (Johann Wolfgang von Goethe)

"I consider the teaching and study of the historical development of science as indispensable. [...] Our textbooks fail in this respect." (Richard Willstätter)

"It is not sufficient that the teacher should have a competent knowledge of the subject which he professes [...] he must (in addition) have considered his science from the point of view at which it appears as a human acquisition." (T F Nunn)

"It appears to me that if one wishes to make progress in mathematics, one should study the masters and not the pupils." (Niels H Abel)

"Life is good for only two things, discovering mathematics and teaching mathematics." (Simeon-Denis Poisson) [in Mathematical Magazine, Volume 64, Number 1, February 1991]

"Most students have to do some work to resuscitate their childlike curiosity. The best way to do that is to start asking questions again - lots of them." (Hal Gregersen)

"Only he who knows what mathematics is, and what its function in our present civilization, can give sound advice for the improvement of our mathematical teaching." (Hermann Weyl)

"Poor is the pupil who does not surpass his master." (Leonardo da Vinci)

"Science, as usually taught to liberal arts students, emphasizes results rather than method, and tries to teach technique rather than to give insight into and understanding of the scientific habit of thought. What is needed, however, is not a dose of metaphysics but a truly humanistic teaching of science." (Harry D Gideonse)

"Show all these fanatics a little geometry, and they learn it quite easily. But, strangely enough, their minds are not thereby rectified. They perceive the truths of geometry, but it does not teach them to weight probabilities. Their minds have set hard. They will reason in a topsy-turvy wall all their lives, and I am sorry for it." (Voltaire)

"Students shall themselves seek, discover, discuss, do, and repeat by their own efforts, examine everything themselves without abdicating to the teacher's authority. The teacher should be left with the task of seeing that the task is completed." (John A Comenius)

"Teaching is not about information. It's about having an honest intellectual relationship with your students." (Paul Lockhart)

"The best teachers make every decision based on what is best for their students." (Al Burr)

"The more abstract the truth you wish to teach, the more you need to seduce the senses to it." (Friedrich Nietzsche)

"The more the teacher 'teaches,' the less the student learns." (John A Comenius)

"The most important outcome of education is to help students to become independent of formal education." (Paul E Gray)

"The only instruction which a professor can give, in my opinion, is to think in front of his students." (Henry Lebesgue)

"The secret of education is respecting the pupil." (Ralph W Emerson)

"The teacher, if indeed wise, does not bid you to enter the house of their wisdom, but leads you to the threshold of your own mind." (Kahil Gibran)

"[...] the two functions of teaching and working in science should never be divorced." (James J Sylvester)

"Those who have had the good fortune to be students of the great mathematician cannot forget the almost religious accent of his teaching, the shudder of beauty or mystery that he sent through his audience, at some admirable discovery or before the unknown." (Charles Hermite [according to Paul Painlevé])

Out of Context: On Teaching (Definitions)

"Teaching is just pointing out what you cannot see without help." (Issai Chozan, "The Mysterious Skills of the Old Cat" ["Neko No Myoujutsu"], 1727)

"Teaching is not a lost art, but the regard for it is a lost tradition." (Jacques Barzun, "Teacher in America", 1945)

"Teaching is more difficult than learning because what teaching calls for is this: to let learn." (Martin Heidegger, "What is called thinking?", 1968)

"Learning is finding out what you already know. Doing is demonstrating that you know it. Teaching is reminding others that they know just as well as you. You are all learners, doers, teachers." (Richard Bach, "Illusions: The Adventures of a Reluctant Messiah", 1977)

"Teaching is an instinctual art, mindful of potential, craving of realizations, a pausing, seamless process." (A Bartlett Giamatti, "The University and the Public Interest", 1981)

"Teaching is only demonstrating that it is possible. Learning is making it possible for yourself." (Paulo Coelho, "The Pilgrimage", 1987)

"Teaching is mostly listening, and learning is mostly telling." (Deborah Meier, "The Power of Their Ideas", 1995)

"Teaching is not about how we see things, it is about how children see things." (Kavita B Ghosh)

"Teaching is not about information. It's about having an honest intellectual relationship with your students." (Paul Lockhart)

On Teaching (1875-1899)

"The most striking characteristic of the written language of algebra and of the higher forms of the calculus is the sharpness of definition, by which we are enabled to reason upon the symbols by the mere laws of verbal logic, discharging our minds entirely of the meaning of the symbols, until we have reached a stage of the process where we desire to interpret our results. The ability to attend to the symbols, and to perform the verbal, visible changes in the position of them permitted by the logical rules of the science, without allowing the mind to be perplexed with the meaning of the symbols until the result is reached which you wish to interpret, is a fundamental part of what is called analytical power. Many students find themselves perplexed by a perpetual attempt to interpret not only the result, but each step of the process. They thus lose much of the benefit of the labor-saving machinery of the calculus and are, indeed, frequently incapacitated for using it." (Thomas Hill, "Uses of Mathesis", Bibliotheca Sacra Vol. 32 (127), 1875)

"[…] not only a knowledge of the ideas that have been accepted and cultivated by subsequent teachers is necessary for the historical understanding of a science, but also that the rejected and transient thoughts of the inquirers, nay even apparently erroneous notions, may be very important and very instructive. The historical investigation of the development of a science is most needful, lest the principles treasured up in it become a system of half-understood prescripts, or worse, a system of prejudices." (Ernst Mach, "The Science of Mechanics", 1883)

"The theory most prevalent among teachers is that mathematics affords the best training for the reasoning powers; […] The modem, and to my mind true, theory is that mathematics is the abstract form of the natural sciences; and that it is valuable as a training of the reasoning powers, not because it is abstract, but because it is a representation of actual things." (Truman H Safford, "Mathematical Teaching and Its Modern Methods", 1886)

"Thoroughly to teach another is the best way to learn for yourself." (Tyron Edwards, "A Dictionary of Thoughts", 1891)

"And as the number of combinations that can be made on the chess-board, is so great that probably no two games exactly alike were ever played; so no two games which the student plays with nature to wrest from her hidden truths, which were worth playing at all, ever made use of quite the same methods in quite the same way." (Alfred Marshall, "Principles of Economics", 1890)

"One of the most baneful delusions by which the minds, not only of students, but even of many teachers of mathematics in our classical colleges, have been afflicted is that mathematics can be mastered by the favored few, but lies beyond the grasp and power of the ordinary mind." (Florian Cajori, "The Teaching and History of Mathematics in the United State", 1890)

"The true aim of the teacher must be to impart an appreciation of method and not a knowledge of facts." (Karl Pearson, "The Grammar of Science", 1892)

"The history of mathematics may be instructive as well as agreeable; it may not only remind us of what we have, but may also teach us to increase our store." (Florian Cajori, "A History of Mathematics", 1893)

On Teaching (1980-1989)

"Some people believe that a theorem is proved when a logically correct proof is given; but some people believe a theorem is proved only when the student sees why it is inevitably true." (Wesley R Hamming, "Coding and Information Theory", 1980)

“That is to say, intuition is not a direct perception of something existing externally and eternally. It is the effect in the mind of certain experiences of activity and manipulation of concrete objects (at a later stage, of marks on paper or even mental images). As a result of this experience, there is something (a trace, an effect) in the pupil's mind which is his representation of the integers. But his representation is equivalent to mine, in the sense that we both get die same answer to any question you ask - or if we get different answers, we can compare notes and figure out what's right. We do this, not because we have been taught a set of algebraic rules, but because our mental pictures match each other.” (Philip J Davis & Reuben Hersh, “The Mathematical Experience”, 1981)

"I am a firm believer that in studying mathematics one should never forget one’s common sense. Many years ago, I was teaching an elementary algebra course. On one exam, I had a standard-type question that involved finding the ages of the mother, father, and child. After the students read the question, I said, 'On this problem, I’ll give you one hint.' All eyes eagerly turned to me. I continued,: 'If the child should turn out to be older than either of the parents, then you’ve done something wrong.'" (Raymond Smullyan, "5000 B.C. and Other Philosophical Fantasies", 1983)

"It is the very strangeness of nature that makes science engrossing. That ought to be at the center of science teaching. There are more than seven-times-seven types of ambiguity in science, awaiting analysis." (Lewis Thomas, "Late Night Thoughts on Listening to Mahler’s Ninth Symphony", 1983

"A central problem in teaching mathematics is to communicate a reasonable sense of taste - meaning often when to, or not to, generalize, abstract, or extend something you have just done." (Richard W Hamming, "Methods of Mathematics Applied to Calculus, Probability, and Statistics", 1985)

"A teacher who is not always thinking about solving problems - ones he does not know the answer to - is psychologically simply not prepared to teach problem solving to his students." (Paul R Halmos, "I Want to Be A Mathematician", 1985)

"Contrary to the impression students acquire in school, mathematics is not just a series of techniques. Mathematics tells us what we have never known or even suspected about notable phenomena and in some instances even contradicts perception. It is the essence of our knowledge of the physical world. It not only transcends perception but outclasses it." (Morris Kline, "Mathematics and the Search for Knowledge", 1985)

"More fundamentally students should be taught that instead of asking ‘What techniques shall I use here?,’ they should ask ‘How can I summarize and understand the main features of this set of data?’" (Christopher Chatfield, "The initial examination of data", Journal of the Royal Statistical Society, Series A 14, 1985)

"Probability and statistics are now so obviously necessary tools for understanding many diverse things that we must not ignore them even for the average student." (Richard W Hamming, "Methods of Mathematics Applied to Calculus, Probability, and Statistics", 1985)

"The result is that non-statisticians tend to place undue reliance on single ‘cookbook’ techniques, and it has for example become impossible to get results published in some medical, psychological and biological journals without reporting significance values even if of doubtful validity. It is sad that students may actually be more confused and less numerate at the end of a ‘service course’ than they were at the beginning, and more likely to overlook a descriptive approach in favor of some inferential method which may be inappropriate or incorrectly executed." (Christopher Chatfield, "The initial examination of data", Journal of the Royal Statistical Society, Series A 14, 1985)

"Thus statistics should generally be taught more as a practical subject with analyses of real data. Of course some theory and an appropriate range of statistical tools need to be learnt, but students should be taught that Statistics is much more than a collection of standard prescriptions." (Christopher Chatfield, "The Initial Examination of Data", Journal of the Royal Statistical Society A Vol. 148, 1985)

"Experience without theory teaches nothing." (William E Deming, "Out of the Crisis", 1986)

"It is our responsibility as scientists, knowing the great progress which comes from a satisfactory philosophy of ignorance, the great progress which is the fruit of freedom of thought, to proclaim the value of this freedom; to teach how doubt is not to be feared but welcomed and discussed; and to demand this freedom as our duty to all coming generations." (Richard P (Feynman, "What Do You Care What Other People Think?", 1988)

"Science must be taught well, if a student is to understand the coming decades he must live through."(Isaac Asimov, "Isaac Asimov’s Book of Science and Nature Quotations", 1988)

On Teaching (1950-1959)

"It has been said, often enough and certainly with good reason, that teaching mathematics affords a unique opportunity to teach demonstrative reasoning. I wish to add that teaching mathematics also affords an excellent opportunity to teach plausible reasoning. A student of mathematics should learn, of course, demonstrative reasoning; it is his profession and the distinctive mark of his science. Yet he should also learn plausible reasoning; this is the kind of reasoning on which his creative work will mainly depend, The general student should get a taste of demonstrative reasoning; he may have little opportunity to use it directly, but he should acquire a standard with which he can compare alleged evidence of all sorts aimed at him in modern life. He needs, however, in all his endeavors plausible reasoning. At any rate, an ambitious teacher of mathematics should teach both kinds of reasoning to both kinds of students." (George Pólya, "On Plausible Reasoning", Proceedings of the International Congress of Mathematics, 1950)

"Statistics is not the easiest subject to teach, and there are those to whom anything savoring of mathematics is regarded as for ever anathema." (Michael J Moroney, "Facts from Figures", 1951)

"[...] science taught [...] without a sense of history is robbed of those very qualities that make it worth teaching to the student of the humanities and the social sciences." (I. Bernard Cohen, 1952)

"Finally, students must learn to realize that mathematics is a science with a long history behind it, and that no true insight into the mathematics of the present day can be obtained without some acquaintance with its historical background. In the first-place time gives an additional dimension to one's mental picture both of mathematics as a whole, and of each individual branch." (André Weil, "The Mathematical Curriculum", 1954)

"For that theory [mathematical theory of statistics] is solely concerned with working out the properties of the theoretical models, whereas what matters - and what in one sense is most difficult - is to decide what theoretical model best corresponds to the real world-situation to which statistical methods must be applied. There is a great danger that mathematical pupils will imagine that a knowledge of mathematical statistics alone makes a statistician." (David G Champemowne, "A Discussion on the Teaching of Mathematical Statistics at the University Level", Journal of the Royal Statistical Society Vol. 118, 1955)

"It is proper to the role of the scientist that he not merely find new truth and communicate it to his fellows, but that he teach, that he try to bring the most honest and intelligible account of new knowledge to all who will try to learn." (J Robert Oppenheimer, "The Open Mind", 1955) 

"The true responsibility of a scientist, as we all know, is to the integrity and vigor of his science. And because most scientists, like all men of learning, tend in part also to be teachers, they have a responsibility for the communication of the truths they have found. This is at least a collective, if not an individual responsibility. That we should see in this any insurance that the fruits of science will be used for man’s benefit, or denied to man when they make for his distress or destruction, would be a tragic naiveté."  (J Robert Oppenheimer, "The Open Mind", 1955) 

"At bottom, the society of scientists is more important than their discoveries. What science has to teach us here is not its techniques but its spirit: the irresistible need to explore." (Jacob Bronowski, "Science and Human Values", 1956)

"There are two ways to teach mathematics. One is to take real pains toward creating understanding - visual aids, that sort of thing. The other is the old British style of teaching until you’re blue in the face." (James R Newman, New York Times, 1956)

"Just as there is an applied mathematics of games, genetics, and mechanics, so there should be an applied mathematics (at least in terms of concepts, perhaps with techniques and operations) of the applications of mathematics. When there is, mathematicians will be able to teach the applications of mathematics." (John W Tukey, "The Teaching of Concrete Mathematics", The American Mathematical Monthly Vol. 65 (1), 1958)

"Mathematical examination problems are usually considered unfair if insoluble or improperly described: whereas the mathematical problems of real life are almost invariably insoluble and badly stated, at least in the first balance. In real life, the mathematician's main task is to formulate problems by building an abstract mathematical model consisting of equations, which will be simple enough to solve without being so crude that they fail to mirror reality. Solving equations is a minor technical matter compared with this fascinating and sophisticated craft of model-building, which calls for both clear, keen common-sense and the highest qualities of artistic and creative imagination." (John Hammersley & Mina Rees, "Mathematics in the Market Place", The American Mathematical Monthly 65, 1958) 

"The diagrams and circles aid the understanding by making it easy to visualize the elements of a given argument. They have considerable mnemonic value […] They have rhetorical value, not only arousing interest by their picturesque, cabalistic character, but also aiding in the demonstration of proofs and the teaching of doctrines. It is an investigative and inventive art. When ideas are combined in all possible ways, the new combinations start the mind thinking along novel channels and one is led to discover fresh truths and arguments, or to make new inventions. Finally, the Art possesses a kind of deductive power." (Martin Gardner, "Logic Machines and Diagrams", 1958)

"The world of today demands more mathematical knowledge on the part of more people than the world of yesterday and the world of tomorrow will demand even more. It is therefore important that mathematics be taught in a vital and imaginative way which will make students aware that it is a living, growing subject which plays an increasingly important part in the contemporary world." (Edward G Begle, "The School Mathematics Study Group," The Mathematics Teacher 51, 1958)

"We believe that student will come to understand mathematics when his textbook and teacher use unambiguous language and when he is enabled to discover generalizations by himself." (Max Beberman, "An Emerging Program of Secondary School Mathematics", 1958)

On Teaching (1925-1949)

"Science is a magnificent force, but it is not a teacher of morals. It can perfect machinery, but it adds no moral restraints to protect society from the misuse of the machine. It can also build gigantic intellectual ships, but it constructs no moral rudders for the control of storm tossed human vessel. It not only fails to supply the spiritual element needed but some of its unproven hypotheses rob the ship of its compass and thus endangers its cargo." (William J Bryan, "Undelivered Trial Summation Scopes Trial", 1925)

"It seems to be the impression among students that mathematical physics consists in deriving a large number of partial differential equations and then solving them, individually, by an assortment of special mutually unrelated devices. It has not been made clear that there is any underlying unity of method and one has often been left entirely in the dark as to what first suggested a particular device to the mind of its inventor." (Arthur G Webster, "Partial Differential Equations of Mathematical Physics", 1927)

"The primary purposes of the teaching of mathematics should be to develop those powers of understanding und analyzing relations of quantity and of space which are necessary to an insight into and a control over our environment and to an appreciation of the progress of civilization its various aspects, and to develop those habits of thought and of action which will make those powers in effective in the life of the individual." (J W Young [Ed] The Reorganization of Mathematics in Secondary Education, 1927)

"What had already been done for music by the end of the eighteenth century has at last been begun for the pictorial arts. Mathematics and physics furnished the means in the form of rules to be followed and to be broken. In the beginning it is wholesome to be concerned with the functions and to disregard the finished form. Studies in algebra, in geometry, in mechanics characterize teaching directed towards the essential and the functional, in contrast to apparent. One learns to look behind the façade, to grasp the root of things. One learns to recognize the undercurrents, the antecedents of the visible. One learns to dig down, to uncover, to find the cause, to analyze." (Paul Klee, "Bauhaus prospectus", 1929)

"Before teachers can properly correlate mathematics with other fields, they ought to learn how to correlate the various parts of mathematics. They should first learn how and where arithmetic and informal geometry can be correlated, how and where algebra can be best correlated with arithmetic and informational geometry, and so on. Unless we can do this, there is small chance that we can successfully correlate mathematics with science, music, the arts, and other applied fields." (W D Reeve "Mathematics and the Integrated Program in Secondary Schools", Teachers College Record 36, 1935)

"In scientific subjects, the natural remedy for dogmatism has been found in research. By temperament and training, the research worker is the antithesis of the pundit. What he is actively and constantly aware of is his ignorance, not his knowledge; the insufficiency of his concepts, of the terms and phrases in which he tries to excogitate his problems: not their final and exhaustive sufficiency. He is, therefore, usually only a good teacher for the few who wish to use their mind as a workshop, rather than to store it as a warehouse." (Ronald A Fisher, "Eugenics, Academic and Practical Eugenics" Review Vol. 27, 1935)

"An effective way to teach the methods of science is to show how our great scientists reached their goals and how their minds worked in the process." (Bernard Jaffe, Journal of Chemical Education Vol. 15, 1938)

"Science teaching has long concerned itself chiefly with the mastery of laws, facts, and principles to the neglect of certain of the less tangible, but non the less desirable outcomes, such as attitude of mind." (Elwood D Heiss et al, "Modern Methods and Materials for Teaching Science", 1940)

"Of all the fantastic ideas that belong to science fiction, the most remarkable - and, perhaps, the most fascinating - is that of time travel [...] Indeed, so fantastic a notion does it seem, and so many apparently obvious absurdities and bewildering paradoxes does it present, that some of the most imaginative students of science refuse to consider it as a practical proposition." (Idrisyn O Evans, "Can We Conquer Time?", Tales of Wonder, 1940)

"Many teachers and textbook writers have never recognized the power of sheer intellectual curiosity as a motive for the highest type of work in mathematics. and as a consequence they have failed to organize and present the work in a manner designed to stimulate the student's interest through a challenge to his curiosity." (Charles H Butler & E Lynwood Wren, "The Teaching of Secondary Mathematics, 1941)

"The scientist collects crude facts, but he stores only what he has converted them into: laws. Laws are the body of science. Laws are what it is a scientist’s business to come at. Laws are what a master-scientist has to teach. Laws are what a pupil-scientist has to learn." (Robin G Collingwood, "The New Leviathan: Or Man, Society, Civilization and Barbarism", 1942)

"[…] there are terms which cannot be defined, such as number and quantity. Any attempt at a definition would only throw difficulty in the student’s way, which is already done in geometry by the attempts at an explanation of the terms point, straight line, and others, which are to be found in treatise on that subject." (Augustus de Morgan, "On the Study and Difficulties of Mathematics", 1943)

"A good teacher should understand and impress on his students the view that no problem whatever is completely exhausted. There remains always something to do; with sufficient study and penetration, we could improve any solution, and, in any case, we can always improve our understanding of the solution." (George Pólya, "How to Solve It", 1945)

"One of the most important tasks of the teacher is to help his students. This task is not quite easy; it demands time, practice, devotion, and sound principles." (George Pólya, "How to Solve It", 1945)

"Teaching to solve problems is education of the will. Solving problems which are not too easy for him, the student learns to persevere through unsuccess, to appreciate small advances, to wait fo rthe essential idea, to concentrate with all his might when it appears." (George Pólya, "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." (George Pólya, "How to solve it", 1945)

"The mathematical experience of the student is incomplete if he never had an opportunity to solve a problem invented by himself." (George Pólya, "How to Solve It", 1945) 

"When a student makes really silly blunders or is exasperatingly slow, the trouble is almost always the same; he has no desire at all to solve the problem, even no desire to understand it properly, and so he has not understood it. Therefore, a teacher wishing seriously to help the student should. first of all, stir up his curiosity, give him some desire to solve the problem. The teacher should also allow some time to the student to make up his mind to settle down to his task. Teaching to solve problems is education of the will. Solving problems which are not too easy for him, the student learns to persevere through success, to appreciate small advance, to wait for the essential idea, to concentrate with all his might when it appears, If the student had no opportunity in school to familiarize himself with the varying emotions of the struggle for the solution his mathematical education failed in the most vital point." (George Pólya, "How to Solve It", 1945)

"To some people, statistics is ‘quartered pies, cute little battleships and tapering rows of sturdy soldiers in diversified uniforms’. To others, it is columns and columns of numerical facts. Many regard it as a branch of economics. The beginning student of the subject considers it to be largely mathematics." (The Editors, "Statistics, The Physical Sciences and Engineering", The American Statistician, Vol. 2, No. 4, 1948)

"Unfortunately, the mechanical way in which calculus sometimes is taught fails to present the subject as the outcome of a dramatic intellectual struggle which has lasted for twenty-five hundred years or more, which is deeply rooted in many phases of human endeavors and which will continue as long as man strives to understand himself as well as nature. Teachers, students, and scholars who really want to comprehend the forces and appearances of science must have some understanding of the present aspect of knowledge as a result of historical evolution." (Richard Curand [forward to Carl B Boyer’s "The History of the Calculus and Its Conceptual Development", 1949])

On Teaching (1900-1924)

"The chief end of mathematical instruction is to develop certain powers of the mind, and among these the intuition is not the least precious. By it the mathematical world comes in contact with the real world, and even if pure mathematics could do without it, it would always be necessary to turn to it to bridge the gulf between symbol and reality. The practician will always need it, and for one mathematician there are a hundred practicians. However, for the mathematician himself the power is necessary, for while we demonstrate by logic, we create by intuition; and we have more to do than to criticize others’ theorems, we must invent new ones, this art, intuition teaches us." (Henri Poincaré, "The Value of Science", 1905)

"The aims in teaching geometry should he, according to my views: (1) That the pupil should acquire an accurate thorough knowledge of geometrical truths. (2) That he should develop the power o! original, logical, geometrical reasoning. (3) That he should acquire a habit of thought which will give him a practical sagacity; which will develop his judgment, increase his resourcefulness, and fit him to cope more successfully with the many and varied problems of his after life; which will teach him to rake a many-sided view of things, that if the avenue of attack is blocked, he should able to promptly, cheerfully and successfully attack from another." (W E Bond, "The Aims in Teaching Geometry and HOW to Attain Them", The Mathematics Teacher, 1908) [source]

"The study of mathematics - from ordinary reckoning up to the higher processes - must be connected with knowledge of nature, and at the same time with experience, that it may enter the pupil’s circle of thought." (Johann F Herbart, "Letters and Lectures on Education", 1908)

"There is no science which teaches the harmonies of nature more clearly than mathematics […]." , (William Andrews, "Magic Squares and Cubes", 1908)

"It is my opinion that in teaching it is not only admissible, but absolutely necessary, to be less abstract at the start, to have constant regard to the applications, and to refer to the refinements only gradually as the student becomes able to understand them. This is, of course, nothing but a universal pedagogical principle to be observed in all mathematical instruction." (Felix Klein, "Lectures on Mathematics", 1911)

"It is my opinion that in teaching it is not only admissible, but absolutely necessary, to be less abstract at the start, to have constant regard to the applications, and to refer to the refinements only gradually as the student becomes able to understand them. This is, of course, nothing but a universal pedagogical principle to be observed in all mathematical instruction." (Felix Klein, "Lectures on Mathematics", 1911)

"The student of arithmetic who has mastered the first four rules of his art, and successfully striven with money sums and fractions, finds himself confronted by an unbroken expanse of questions known as problems." (Stephen Leacock, "Literary Lapses", 1911)

"Science is the most intimate school of resignation and humility, for it teaches us to bow for the seemingly most insignificant of facts." (Miguel de Unamuno, "The Tragic Sense of Life in Men and in Peoples", 1912)

"The ends to be attained [in mathematical teaching] are the knowledge of a body of geometrical truths to be used. In the discovery of new truths, the power to draw correct inferences from given premises, the power to use algebraic processes as a means of finding results in practical problems, and the awakening of interest In the science of mathematics." (J Craig, "A Course of Study for the Preparation of Rural School Teachers", 1912)

"The theoretical side of physical chemistry is and will probably remain the dominant one; it is by this peculiarity that it has exerted such a great influence upon the neighboring sciences, pure and applied, and on this ground physical chemistry may be regarded as an excellent school of exact reasoning for all students of the natural sciences." (Svante Arrhenius, "Theories of Solutions", 1912)

"[…] mathematical verities flow from a small number of self-evident propositions by a chain of impeccable reasonings; they impose themselves not only on us, but on nature itself. They fetter, so to speak, the Creator and only permit him to choose between some relatively few solutions. A few experiments then will suffice to let us know what choice he has made. From each experiment a number of consequences will follow by a series of mathematical deductions, and in this way each of them will reveal to us a corner of the universe. This, to the minds of most people, and to students who are getting their first ideas of physics, is the origin of certainty in science." (Henri Poincaré, "The Foundations of Science", 1913)

"The way to enable a student to apprehend the instrumental value of arithmetic is not to lecture him on the benefit it will be to him in some remote and uncertain future, but to let him discover that success in something he is interested in doing depends on ability to use numbers." (John Dewey, "Democracy and Education: An Introduction to the Philosophy of Education", 1916)

"To humanize the teaching of mathematics means so to present the subject, so to interpret its ideas and doctrines, that they shall appeal, not merely to the computatory faculty or to the logical faculty but to all the great powers and interests of the human mind." (Cassius J Keyser, "The Human Worth of Rigorous Thinking: Essays and Addresses", 1916)

"To come very near to a true theory, and to grasp its precise application, are two very different things, as the history of a science teaches us. Everything of importance has been said before by somebody who did not discover it." (Alfred N Whitehead, "The Organization of Thought", 1917)

"Abstract as it is, science is but an outgrowth of life. That is what the teacher must continually keep in mind. […] Let him explain […] science is not a dead system - the excretion of a monstrous pedantism - but really one of the most vigorous and exuberant phases of human life." (George A L Sarton, "The Teaching of the History of Science", The Scientific Monthly, 1918)

"No student ought to complete a course in mathematics without the feeling that there must be something in it, without catching a glimpse, however fleeting, of its possibilities, without at least a few moments of pleasure in achievement and insight." (Helen A Merrill, 'Why Students Fail in Mathematics", The Mathematics Teacher, 1918) [source]

"Most teachers waste their time by asking questions which are intended to discover what a pupil does not know whereas the true art of questioning has for its purpose to discover what the pupil knows or is capable of knowing." (Albert Einstein, 1920)

"[…] teachers are simply your guides. You yourselves must do the travelling." (William J Gies, "Research in Destiny", 1921)

"Our work is great in the classroom it we feel the nobility of that work, if we love the human souls we work with more than the division of fractions, if we love our subject so much that we make our pupils love it, and if we remember that our duty to the world is to help fix in the minds of our pupils the facts of number that they must have in after life." (David E Smith, "The Progress of Arithmetic", 1923) 

"We have come to believe that a pupil in school should feel that he is living his own life naturally. with a minimum of restraint and without tasks that are unduly irksome; that he should find his way through arithmetic largely hoy his own spirit of curiosity; and that he should be directed in arithmetic as he would he directed in any other game, - not harshly driven, hardly even led, but proceeding with the feeling that he is being accompanied and that he is doing his share in finding the way." (David E Smith, "The Progress of Arithmetic", 1923)

Mathematics through Students' Eyes II

"[…] mathematical verities flow from a small number of self-evident propositions by a chain of impeccable reasonings; they impose themselves not only on us, but on nature itself. They fetter, so to speak, the Creator and only permit him to choose between some relatively few solutions. A few experiments then will suffice to let us know what choice he has made. From each experiment a number of consequences will follow by a series of mathematical deductions, and in this way each of them will reveal to us a corner of the universe. This, to the minds of most people, and to students who are getting their first ideas of physics, is the origin of certainty in science." (Henri Poincaré, "The Foundations of Science", 1913)

"The way to enable a student to apprehend the instrumental value of arithmetic is not to lecture him on the benefit it will be to him in some remote and uncertain future, but to let him discover that success in something he is interested in doing depends on ability to use numbers." (John Dewey, "Democracy and Education: An Introduction to the Philosophy of Education", 1916)

"It seems to be the impression among students that mathematical physics consists in deriving a large number of partial differential equations and then solving them, individually, by an assortment of special mutually unrelated devices. It has not been made clear that there is any underlying unity of method and one has often been left entirely in the dark as to what first suggested a particular device to the mind of its inventor." (Arthur G Webster, "Partial Differential Equations of Mathematical Physics", 1927)

"[…] there are terms which cannot be defined, such as number and quantity. Any attempt at a definition would only throw difficulty in the student’s way, which is already done in geometry by the attempts at an explanation of the terms point, straight line, and others, which are to be found in treatise on that subject." (Augustus de Morgan, "On the Study and Difficulties of Mathematics", 1943)

"To some people, statistics is ‘quartered pies, cute little battleships and tapering rows of sturdy soldiers in diversified uniforms’. To others, it is columns and columns of numerical facts. Many regard it as a branch of economics. The beginning student of the subject considers it to be largely mathematics." (The Editors, "Statistics, The Physical Sciences and Engineering", The American Statistician, Vol. 2, No. 4, 1948)

"Unfortunately, the mechanical way in which calculus sometimes is taught fails to present the subject as the outcome of a dramatic intellectual struggle which has lasted for twenty-five hundred years or more, which is deeply rooted in many phases of human endeavors and which will continue as long as man strives to understand himself as well as nature. Teachers, students, and scholars who really want to comprehend the forces and appearances of science must have some understanding of the present aspect of knowledge as a result of historical evolution." (Richard Curand [forward to Carl B Boyer’s "The History of the Calculus and Its Conceptual Development", 1949])

Mathematics through Students' Eyes I

"Often I have considered the fact that most of the difficulties which block the progress of students trying to learn analysis stem from this: that although they understand little of ordinary algebra, still they attempt this more subtle art. From this it follows not only that they remain on the fringes, but in addition they entertain strange ideas about the concept of the infinite, which they must try to use." (Leonhard Euler, "Introduction to Analysis of the Infinite", 1748)

"The first thing to be attended to in reading any algebraical treatise, is the gaining a perfect understanding of the different processes there exhibited, and of their connection with one another. This cannot be attained by a mere reading of the book, however great the attention which may be given. It is impossible, in a mathematical work, to fill up every process in the manner in which it must be filled up in the mind of the student before he can be said to have completely mastered it. Many results must be given of which the details are suppressed, such are the additions, multiplications, extractions of the square root, etc., with which the investigations abound. These must not be taken on trust by the student, but must be worked by his own pen, which must never be out of his hand, while engaged in any algebraical process." (Augustus de Morgan, "On the Study and Difficulties of Mathematics", 1830)

"This science, Geometry, is one of indispensable use and constant reference, for every student of the laws of nature; for the relations of space and number are the alphabet in which those laws are written. But besides the interest and importance of this kind which geometry possesses, it has a great and peculiar value for all who wish to understand the foundations of human knowledge, and the methods by which it is acquired. For the student of geometry acquires, with a degree of insight and clearness which the unmathematical reader can but feebly imagine, a conviction that there are necessary truths, many of them of a very complex and striking character; and that a few of the most simple and self-evident truths which it is possible for the mind of man to apprehend, may, by systematic deduction, lead to the most remote and unexpected results." (William Whewell, "The Philosophy of the Inductive Sciences", 1858)

"Besides accustoming the student to demand complete proof, and to know when he has not obtained it, mathematical studies are of immense benefit to his education by habituating him to precision. It is one of the peculiar excellencies of mathematical discipline, that the mathematician is never satisfied with à peu près. He requires the exact truth." (John S Mill, "An Examination of Sir William Hamilton's Philosophy", 1865)

"Therefore, the great business of the scientific teacher is, to imprint the fundamental, irrefragable facts of his science, not only by words upon the mind, but by sensible impressions upon the eye, and ear, and touch of the student, in so complete a manner, that every term used, or law enunciated, should afterwards call up vivid images of the particular structural, or other, facts which furnished the demonstration of the law, or the illustration of the term." (Thomas H Huxley, "Lay Sermons, Addresses and Reviews", 1870)

"Thought is symbolical of Sensation as Algebra is of Arithmetic, and because it is symbolical, is very unlike what it symbolises. For one thing, sensations are always positive; in this resembling arithmetical quantities. A negative sensation is no more possible than a negative number. But ideas, like algebraic quantities, may be either positive or negative. However paradoxical the square of a negative quantity, the square root of an unknown quantity, nay, even in imaginary quantity, the student of Algebra finds these paradoxes to be valid operations. And the student of Philosophy finds analogous paradoxes in operations impossible in the sphere of Sense. Thus although it is impossible to feel non-existence, it is possible to think it; although it is impossible to frame an image of Infinity, we can, and do, form the idea, and reason on it with precision." (George H Lewes "Problems of Life and Mind", 1873)

"The most striking characteristic of the written language of algebra and of the higher forms of the calculus is the sharpness of definition, by which we are enabled to reason upon the symbols by the mere laws of verbal logic, discharging our minds entirely of the meaning of the symbols, until we have reached a stage of the process where we desire to interpret our results. The ability to attend to the symbols, and to perform the verbal, visible changes in the position of them permitted by the logical rules of the science, without allowing the mind to be perplexed with the meaning of the symbols until the result is reached which you wish to interpret, is a fundamental part of what is called analytical power. Many students find themselves perplexed by a perpetual attempt to interpret not only the result, but each step of the process. They thus lose much of the benefit of the labor-saving machinery of the calculus and are, indeed, frequently incapacitated for using it." (Thomas Hill, "Uses of Mathesis", Bibliotheca Sacra Vol. 32 (127), 1875)

Jacob W A Young - Collected Quotes

"All thinking and all actions are influenced by conclusions that have been consciously or unconsciously drawn." (Jacob W A Young, "The Teaching of Mathematics", 1907)

"Little can be understood of even the simplest phenomena of nature without some knowledge of mathematics, and the attempt to penetrate deeper into the mysteries of nature compels simultaneous development of the mathematical processes." (Jacob W A Young, "The Teaching of Mathematics", 1907)

"Mathematics has beauties of its own - a symmetry and proportion in its results, a lack of superfluity, an exact adaptation of means to ends, which is exceedingly remarkable and to be found elsewhere only in the works of the greatest beauty." (Jacob W A Young, "The Teaching of Mathematics", 1907)

"Mathematics makes constant demands upon the imagination, calls for picturing in space (of one, two, three dimensions), and no considerable success can be attained without a growing ability to imagine all the various possibilities of a given case, and to make them defile before the mind's eye." (Jacob W A Young, "The Teaching of Mathematics", 1907)

"Mathematics is a type of thought which seems ingrained in the human mind, which manifests itself to some extent with even the primitive races, and which is developed to a high degree with the growth of civilization. […] A type of thought, a body of results, so essentially characteristic of the human mind, so little influenced by environment, so uniformly present in every civilization, is one of which no well-informed mind today can be ignorant." (Jacob W A Young, "The Teaching of Mathematics", 1907)

"One of these modes of thought is the ability to grasp a situation, to seize the facts, and to perceive correctly the state of affairs. […] The web of facts and relationships in which every one moves is moreover extremely tangled. Much practice is requisite to even fair success in grasping situations, and we look to the school to furnish and direct such practice. Mathematics is specially adapted to the beginning of the practice because its facts are few and uncomplicated." (Jacob W A Young, "The Teaching of Mathematics", 1907)

"So completely is nature mathematical that some of the more exact natural sciences, in particular astronomy and physics, are in their theoretic phases largely mathematical in character, while other sciences which have hitherto been compelled by the complexity of their phenomena and the inexactitude of their data to remain descriptive and empirical, are developing towards the mathematical ideal, proceeding upon the fundamental assumption that mathematical relations exist between the forces and the phenomena, and that nothing short, of the discovery and formulations of these relations would constitute definitive knowledge of the subject. Progress is measured by the closeness of the approximation to this ideal formulation." (Jacob W A Young, "The Teaching of Mathematics", 1907)

"Still more important than x the subject matter of mathematics is the fact that it exemplifies 'most typically, clearly and simply certain modes of thought which are of the utmost importance to every one." (Jacob W A Young, "The Teaching of Mathematics", 1907)

"The training which mathematics gives in working with symbols is an excellent preparation for other sciences; […] the world's work requires constant mastery of symbols." (Jacob W A Young, "The Teaching of Mathematics", 1907)

"The type of reasoning found in mathematics seems thus not only available but essentially interwoven with every inference in non-mathematical reasoning, being always used in one of its two steps ; facility in making the other step, the more difficult one, must be attained through other than purely mathematical training." (Jacob W A Young, "The Teaching of Mathematics", 1907)

"There is one, but only one branch which may claim certainty; in which experts have not seriously disagreed, - Mathematics. […] Mathematics possesses also the second desideratum, - Simplicity. It begins, as is well known, with few and uncomplicated definitions and postulates, and proceeds, step by step, to quite elaborate cases. […] Mathematics possesses also the third desideratum, - Applicability of the skill acquired." (Jacob W A Young, "The Teaching of Mathematics", 1907)

Florian Cajori - Collected Quotes

"The best review of arithmetic consists in the study of algebra." (Florian Cajori, "Teaching and History of Mathematics in U. S.", 1896)

"The contemplation of the various steps by which mankind has come into possession of the vast stock of mathematical knowledge can hardly fail to interest the mathematician. He takes pride in the fact that his science, more than any other, is an exact science, and that hardly anything ever done in mathematics has proved to be useless." (Florian Cajori, "A History of Mathematics", 1893)

"The history of mathematics is important […] as a valuable contribution to the history of civilisation. Human progress is closely identified with scientific thought. Mathematical and physical researches are a reliable record of intellectual progress. The history of mathematics is one of the large windows through which the philosophic eye looks into past ages and traces the line of intellectual development." (Florian Cajori, "A History of Mathematics", 1893)

"The history of mathematics may be instructive as well as agreeable; it may not only remind us of what we have, but may also teach us to increase our store." (Florian Cajori, "A History of Mathematics", 1893)

"This history constitutes a mirror of past and present conditions in mathematics which can be made to bear on the notational problems now confronting mathematics. The successes and failures of the past will contribute to a more speedy solution of notational problems of the present time." (Florian Cajori, "A History of Mathematical Notations", 1928)

"One of the most baneful delusions by which the minds, not only of students, but even of many teachers of mathematics in our classical colleges, have been afflicted is that mathematics can be mastered by the favored few, but lies beyond the grasp and power of the ordinary mind." (Florian Cajori, "The Teaching and History of Mathematics in the United States", 1890)

André Weil - Collected Quotes

"Finally, students must learn to realize that mathematics is a science with a long history behind it, and that no true insight into the mathematics of the present day can be obtained without some acquaintance with its historical background. In the first-place time gives an additional dimension to one's mental picture both of mathematics as a whole, and of each individual branch." (André Weil, "The Mathematical Curriculum", 1954)

"Rigor is to the mathematician what morality is to man. It does not consist in proving everything, but in maintaining a sharp distinction between what is assumed and what is proved, and in endeavoring to assume as little as possible at every stage." (André Weil, Mathematical Teaching in Universities", The American Mathematical Monthly Vol. 61 (1), 1954)

"As every mathematician knows, nothing is more fruitful than these obscure analogies, these indistinct reflections of one theory into another, these furtive caresses, these inexplicable disagreements; also nothing gives the researcher greater pleasure." (André Weil, "De la Métaphysique aux Mathématiques", 1960)

"Nothing is more fruitful - all mathematicians know it - than those obscure analogies, those disturbing reflections of one theory on another; those furtive caresses, those inexplicable discords; nothing also gives more pleasure to the researcher. The day comes when this illusion dissolves: the presentiment turns into certainty; the yoked theories reveal their common source before disappearing. As the Gita teaches, one achieves knowledge and indifference at the same time. Metaphysics has become Mathematics, ready to form the material of some treatise whose cold beauty has lost the power to move us." (André Weil, "De la métaphysique aux mathématiques", 1960)

"Alternative models are neither right nor wrong, just more or less useful in allowing us to operate in the world and discover more and better options for solving problems." (Andrew Weil," The Natural Mind: A Revolutionary Approach to the Drug Problem", 2004)

"A tediously laborious proof may be a sign that the writer has been less than felicitous in expressing himself; but more often than not, as we know, it indicates that he has been laboring under limitations which prevented him from translating directly into words or formulas some very simple ideas." (André Weil)

"In the future, as in the past, the great ideas [of mathematics] must be simplifying ideas, the creator must always be one who clarifies, for himself, and for others, the most complicated issues of formulas and concepts." (André Weil)

"Strategy means the art of recognizing the main problems, attacking them at their weak points, setting up future lines of advance. Mathematical strategy is concerned with long-range objectives; it requires a deep understanding of broad trends and of the evolution of ideas over long periods." (André Weil)

Mental Models XXII

"The impossibility of separating the nomenclature of a science from the science itself is owing to this, that every branch of physical science must consist of three things: the series of facts which are the objects of the science, the ideas which represent these facts, and the words by which these ideas are expressed. Like three impressions of the same seal, the word ought to produce the idea, and the idea to be a picture of the fact." (Antoine L Lavoisier, "Elements of Chemistry in a New Systematic Order", 1790)

"Therefore, the great business of the scientific teacher is, to imprint the fundamental, irrefragable facts of his science, not only by words upon the mind, but by sensible impressions upon the eye, and ear, and touch of the student, in so complete a manner, that every term used, or law enunciated, should afterwards call up vivid images of the particular structural, or other, facts which furnished the demonstration of the law, or the illustration of the term." (Thomas H Huxley, "Lay Sermons, Addresses and Reviews", 1870) 

"The sciences bring into play the imagination, the building of images in which the reality, of the past is blended with the ideals for the future, and from the picture there springs the prescience of genius." (William J Mayo, "Contributions of Pure Science to Progressive Medicine", The Journal of the American Medical Association Vol. 84 (20), 1925)

"It is not surprising that our language should be incapable of describing the processes occurring within the atoms, for, as has been remarked, it was invented to describe the experiences of daily life, and these consist only of processes involving exceedingly large numbers of atoms. Furthermore, it is very difficult to modify our language so that it will be able to describe these atomic processes, for words can only describe things of which we can form mental pictures, and this ability, too, is a result of daily experience." (Werner K Heisenberg, "The Physical Principles of the Quantum Theory", 1930)

"A picture of reality drawn in a few sharp lines cannot be expected to be adequate to the variety of all its shades. Yet even so the draftsman must have the courage to draw the lines firm." (Hermann Weyl, "Philosophy of Mathematics and Natural Science", 1949)

"Man tries to make for himself in the fashion that suits him best a simplified and intelligible picture of the world; he then tries to some extent to substitute this cosmos of his for the world of experience, and thus to overcome it. This is what the painter, the poet, the speculative philosopher, and the natural scientist do, each in his own way." (Albert Einstein, "The World as I See It", 1949)

"Thus a word or an image is symbolic when it implies something more than its obvious and immediate meaning. It has a wider ‘unconscious’ aspect that is never precisely defined or fully explained. […] As the mind explores the symbols it is led to ideas that lie beyond the grasp of reason." (Carl G Jung, "Man and His Symbols", 1964)

"It is hard for us today to assimilate all the new ideas that are being suggested in response to the new information we have. We must remember that our picture of the universe is based not only on our scientific knowledge but also on our culture and our philosophy. What new discoveries lie ahead no one can say. There may well be civilizations in other parts of our galaxy or in other galaxies that have already accomplished much of what lies ahead for mankind. Others may just be beginning. The universe clearly presents an unending challenge." (Necia H Apfel & J Allen Hynek, "Architecture of the Universe", 1979)

“Simple observation generally gets us nowhere. It is the creative imagination that increases our understanding by finding connections between apparently unrelated phenomena, and forming logical, consistent theories to explain them. And if a theory turns out to be wrong, as many do, all is not lost. The struggle to create an imaginative, correct picture of reality frequently tells us where to go next, even when science has temporarily followed the wrong path.” (Richard Morris, “The Universe, the Eleventh Dimension, and Everything: What We Know and How We Know It”, 1999)