Education

"1. The human mind is so constructed that it must see every perception in a time-relation—in an order—and every perception of an object in a space-relation—as outside or beside our perceiving selves. 2. These necessary time-relations are reducible to Number, and they are studied in the theory of number, arithmetic and algebra. 3. These necessary space-relations are reducible to Position and Form, and they are studied in geometry. Mathematics, therefore, studies an aspect of all knowing, and reveals to us the universe as it presents itself, in one form, to mind. To apprehend this and to be conversant with the higher developments of mathematical reasoning, are to have at hand the means of vitalizing all teaching of elementary mathematics."

"More often than we are aware, it is the jargon which is the hurdle that a student cannot overcome rather than the mathematical concepts being introduced."

"By the beginning of the seventeenth century we may say that the fundamental principles of arithmetic, algebra, theory of equations, and trigonometry had been laid down, and the outlines of the subjects as we know them had been traced. It must be, however, remembered that there were no good elementary text-books on these subjects; and a knowledge of them was therefore confined to those who could extract it from the ponderous treatises in which it lay buried. Though much of the modern algebraical and trigonometrical notation had been inroduced, it was not familiar to mathematicians, nor was it even universally accepted; and it was not until the end of the seventeenth century that the language of the subjects was definitely fixed. Considering the absence of good text-books, I am inclined... to admire the rapidity with which it came into universal use, than to cavil at the hesitation to trust to it alone which many writers showed."

"Zoologists maintain that the embryonic development of an animal recapitulates in brief the whole history of its ancestors throughout geologic time. It seems it is the same in the development of minds. The teacher should make the child go over the path his fathers trod; more rapidly, but without skipping stations. For this reason the history of science should be our first guide."

"Because of white racism's ability to bludgeon us into believing that we are inferior beings and therefore incapable of learning math and the sciences, we must spend a significant amount of our learning and teaching time unlearning and unteaching. This is to say, for example, that when a brother or sister reaches the freshman college level, he or she has already been subjected to at least 17 years of conditioning that dictates: "You are too black and too ignorant to understand such lily-white and intelligent things as math, chemistry, physics, etc." Thus, a major part of the teacher's initial instructional time must be spent dealing with the psychological block against learning math—or any of the sciences."

"The study of mathematics is apt to commence in disappointment... We are told that by its aid the stars are weighed and the billions of molecules in a drop of water are counted. Yet, like the ghost of Hamlet's father, this great science eludes the efforts of our mental weapons to grasp it."

"Mathematics as an Element in the History of Thought."

"So far as civilisation is connected with the advance and diffusion of human knowledge, civilisation flourishes when the prevalent education is mathematical, and fades when philosophy is the subject most preferred."

"A liberal Education ought to include both Permanent Studies which connect men with the culture of past generations, and Progressive Studies which make them feel their community with the present generation, its businesses, interests and prospects. ...When the Student is once well disciplined in geometrical mathematics, he may pursue analysis safely and surely to any extent. But though modern mathematics may thus be very fitly studied as a sequel to the older forms of mathematical science... these modern methods cannot supply the place of the ancient subjects as the Permanent Studies in our Educational course."

"The use of mathematical study, with which we have to do, is not to produce a school of eminent mathematicians, but to contribute to a Liberal Education of the highest kind."

"The ancient Geometry had no symbols, nor any notation beyond ordinary language and the specific terms of the science. We may question the propriety of allowing a learner, at the commencement of his Geometrical studies, to exhibit Geometrical demonstrations in Algebraical symbols. Surely it is not too much to apprehend that such a practice may occasion serious confusion of thought."

"Since formal mathematics is no more than a culturally-dependent system of aesthetics, while it may continue to be taught like Western music, there is no need to impose its consequences on K-12 children. What we need to teach children is practical mathematics... In particular, it may be worth re-examining whether one might want to teach as entirely separate subjects, from the outset, the two mathematical streams: practical mathematics and formal mathematics, with their distinct notions of number and proof."

"The education of the child must accord both in mode and arrangement with the education of mankind, considered historically. In other words, the genesis of knowledge in the individual, must follow the same course as the genesis of knowledge in the race. In strictness, this principle may be considered as already expressed by implication; since both being processes of evolution, must conform to those same general laws of evolution... and must therefore agree with each other. Nevertheless this particular parallelism is of value for the specific guidance it affords. To M. Comte we believe society owes the enunciation of it; and we may accept this item of his philosophy without at all committing ourselves to the rest."

"Why should the typical student be interested in those wretched triangles? ...Without trigonometry we put back the clock millennia to Standard Darkness Time and antedate the Greeks."

"I shall often discuss mathematical discoveries... I shall try to make up a likely story how the discovery could have happened. I shall try to emphasize the motives underlying the discovery, the plausible inferences that led to it... everything that deserves imitation. ...I... present also examples of historic interest, examples of real mathematical beauty, and examples illustrating the parallelism of the procedures in other sciences, or in everyday life."

""Groping" and "muddling through" is usually described as a solution by trial and error. ...a series of trials, each of which attempts to correct the error committed by the preceding and, on the whole, the errors diminished as we proceed and the successive trials come closer and closer to the desired final result. ...we may wish a better characterization ..."successive trials" or "successive corrections" or "successive approximations." ...You use successive approximations when ...looking for a word in the dictionary ...A mathematician may apply the term ...to a highly sophisticated procedure ...to treat some very advanced problem ...that he cannot treat otherwise. The term even applies to science as a whole; the scientific theories which succeed each other, each claiming a better explanation ...may appear as successive approximations to the truth. Therefore, the teacher should not discourage his students from using trial and error—on the contrary, he should encourage the intelligent use of the fundamental method of successive approximations. Yet he should convincingly show that for ...many ... situations, straightforward algebra is more efficient than successive approximations."

"Nowhere in all ancient mathematics do we find any attempt at what we call demonstration. No argumentation was presented, but only the prescription of certain rules: "Do such and so." We are ignorant of the way the theorems were found... To those who have been educated in Euclid's strict argumentation, the entire Oriental way of reasoning seems at first strange and highly unsatisfactory. But this strangeness wears off when we realize that most of the mathematics we teach our present-day engineers and technicians is still of the "do such, do so" type, without much attempt at rigorous demonstration. Algebra is still being taught in many high schools as a set of rules rather than as a science of deduction. Oriental mathematics never seems to have been emancipated from the millenial influence of the problems of technology and administration, for the use of which it had been invented."

"At each stage of its development the human race has had a certain climate of opinion, a way of looking, conceptually, at the world. The next glimmer of fresh understanding had to grow out of what was already understood. The next move forward, halting shuffle, faltering step, or stride with some confidence, was developed upon how well the [human] race could then walk. As for the human race, so for the human child. But this is not to say that to teach science we must repeat the thousand and one errors of the past, each ill-directed shuffle. It is to say that the sequence in which the major strides forward were made is a good sequence in which to teach them. The genetic method is a guide to, not a substitute for, judgement."

"Euclid's manner of exposition, progressing relentlessly from the data to the unknown and from the hypothesis to the conclusion, is perfect for checking the argument in detail but far from being perfect for making understandable the main line of the argument."

"Pedantry and mastery are opposite attitudes toward rules. To apply a rule to the letter, rigidly, unquestioningly, in cases where it fits and in cases where it does not fit, is pedantry. … To apply a rule with natural ease, with judgment, noticing the cases where it fits, and without ever letting the words of the rule obscure the purpose of the action or the opportunities of the situation, is mastery."

"The cookbook gives a detailed description of ingredients and procedures but no proofs for its prescriptions or reasons for its recipes; the proof of the pudding is in the eating. … Mathematics cannot be tested in exactly the same manner as a pudding; if all sorts of reasoning are debarred, a course of calculus may easily become an incoherent inventory of indigestible information."

"Histories make men wise; poets, witty; the mathematics, subtile; natural philosophy, deep; morals, grave; logic and rhetoric, able to contend."

"One makes sense of , whether fictional or factual, by a mental construction that is sometimes called the world of story ...[T]he imaginative effort is a standard way of understanding what people say... In order to understand connected speech about concrete things, one imagines them. ...The capacity to do this... encourages empathy, but it also allows one to do mathematics. ...This is often fun, and it is a form of playing with ideas. ...This ludic aspect of mathematics is emphasized by in his semiotic analysis of mathematics and acknowledged by David Wells in his comparison of mathematics and games. ...The ludic aspect is something that undergraduates, many of whom have decided that mathematics is either a guessing game... or the execution of rigidly defined procedures, need to be encouraged to do when they are learning new ideas. They need to fool around with them to become familiar... Changing the s and seeing what a function looks like... to learn how the function behaves. ...Mathematical research involves a good deal of fooling around, which is part of why it is a pleasurable activity... This is not competitive..."

"Everyone knows that mathematics offers an excellent opportunity to learn demonstrative reasoning, but I contend also that there is no other subject in the usual curricula of the schools that affords a comparable opportunity to learn plausible reasoning. ...let us learn proving, but also let us learn guessing."

"Knowledge is indivisible. When people grow wise in one direction, they are sure to make it easier for themselves to grow wise in other directions as well. On the other hand, when they split up knowledge, concentrate on their own field, and scorn and ignore other fields, they grow less wise — even in their own field."

"The problem of déformation professionnelle is real but mostly American. It would be suicide in the American academy to show too early an interest beyond your doctoral specialization: charges of everything from charlatanry to ambition would be levied and tenure denied. I’ve seen this first-hand. This is because the American graduate school universe was created by Germans (refugees) and echoes many of the worst as well as the best features of its model: deep academic research, carefully limited range of materials, engagement in internally-referential debates and utter unconcern for the market. These are not all bad qualities—without them we would not have had some of the world’s greatest historical monographs. But they inhibit people for decades from putting their nose above a parapet. I have always loved sticking my nose about a parapet, so long as I had a decent weapon to hand. That’s partly an English trait, partly an Oxbridge trait and probably mostly a recessive Judt trait. John Dunn, my favorite King’s supervisor, once described me as “the silver-tongued orator”: a barbed compliment, since it suggested that I spoke before I thought and seduced rather than convinced, but I like it all the same."

"The subject of industrial design is one of three important practical co-related subjects which should be taught in public schools, and to which practice and skill in drawing should be applied. Satisfactory results in this subject, however, depend entirely upon the manner in which it is taught. Instruction in industrial design means a clear presentation of the principles which obtain in the construction and harmonious arrangement of geometric form for decorative purposes, the proper use of plant forms in ornamental arrangements, and the principles of good taste to be found in the great history styles of art."

"Fine Art then, records by idealised imitation the glorious works of good men, whilst it holds those of bad men up to our abhorrence — it gives to posterity their images, either on the tinted canvass or the sculptured marble — it imitates the beautiful effects of nature as seen in the glowing landscape or the rising storm, and perpetuates the appearance of those beauteous gems of the seasons — flowers and fruits, which, though fading whilst the painter catches their tints, yet live after decay by and through his genius. Industrial Art, on the contrary, aims at the embellishment of the works of man, by and through that power which is given to the artist for the investigation of the beautiful in nature; and in transferring it to the loom, the printing machine, the potter's wheel, or the metal worker's mould, he reproduces nature in a new form, adapting it to his purpose by an intelligence arising out of his knowledge as an artist and as a workman. In short, the adaptation of the natural type to a new material compels him to reproduce, almost create, as well as imitate — invent as well as copy — design as well as draw!"

"To be a teacher is my greatest work of art. The rest is the waste product, a demonstration. If you want to express yourself you must present something tangible. But after a while this has only the function of a historic document. Objects aren't very important any more. I want to get to the origin of matter, to the thought behind it."

"I discovered soon that teaching has the handicap of retrospection. And that I don't believe in. So I started instead a method of handling material with the material itself. So that was my main change. Whereas Itten before (Itten left the Bauhaus in 1923 and Albers followed him as art teacher, ed.) had only spoken about the appearance, "matiere" - (the French word) and I said I would turn from "matiere" - the outside - to the inside, to the capacity of the material, before the appearance. And that changed the attitude basically I think."

"If the passion for truthfulness is merely controlled and stilled without being satisfied, it will kill the activities it is supposed to support. This may be one of the reasons why, at the present time, the study of the humanities runs a risk of sliding from professional seriousness, through professionalization, to a finally disenchanted careerism."

"When we see a woman bartering beauty for gold, we look upon such a one as no other than a common prostitute. ... It is the very same with philosophy: he who sets it forth for public sale, to be disposed of to the highest bidder, is a sophist, a public prostitute."

"Nothing but ambition, nothing but the desire to get on, that's all there is in his soul," she thought; "as for these lofty ideals, love of culture, religion, they are only so many tools for getting on."

"The argument that will be made here can be simply stated. It is that the vast majority of the so-called research turned out in the modern university is essentially worthless. It does not result in any measurable benefit to anything or anybody. It does not push back those omnipresent 'frontiers of knowledge' so confidently evoked; it does not in the main result in greater health or happiness among the general populace or any particular segment of it. It is busywork on a vast, almost incomprehensible scale."

"Those who are ignorant of all good, careless, or enemies to it, take a more compendious way; their slavish, vicious, and base natures inclining them to seek only private and present advantages, they easily slide into a blind dependence upon one who has wealth and power; and desiring only to know his will, care not what injustice they do, if they may be rewarded. They worship what they find in the temple, though it be the vilest of idols; and always like that best which is worst, because it agrees with their inclinations and principles."

"Even though philosophers should be in a position to discover the truth, which of them would take any interest in it? Each one knows well that his system is not better founded than the others, but he supports it because it is his. There is not a single one of them who, if he came to know the true and the false, would not prefer the falsehood that he had found to the truth discovered by another. Where is the philosopher who would not willingly deceive mankind for his own glory? Where is he who in the secret of his heart does not propose to himself any other object than to distinguish himself? Provided that he lifts himself above the vulgar, provided that he outshines the brilliance of his competitors, what does he demand more? The essential thing is to think differently from others. With believers he is an atheist; with atheists he would be a believer."

"The intellectual origins of literary theory in Europe were, I think it is accurate to say, insurrectionary. The traditional university, the hegemony of determinism and positivism, the reification of ideological bourgeois "humanism," the rigid barriers between academic specialties: it was powerful responses to all these that linked together such influential progenitors of today's literary theorist as Saussure, Lukács, Bataille, Lévi-Strauss, Freud, Nietzsche, and Marx. Theory proposed itself as a synthesis overriding the petty fiefdoms within the world of intellectual production, and it was manifestly to be hoped as a result that all the domains of human activity could be seen, and lived, as a unity. ..."

"Each of these private teachers who work for pay … inculcates nothing else than these opinions of the multitude which they opine when they are assembled and calls this knowledge wisdom."

"If from the wilderness the righteous and honest John were actually to come who, clothed in skins and living on locusts and untouched by all the terrible mischief, were meanwhile to apply himself with a pure heart and in all seriousness to the investigation of truth and to offer the fruits thereof, what kind of reception would he have to expect from those businessmen of the chair, who are hired for State purposes and with wife and family have to live on philosophy, and whose watchword is, therefore, Primum vivere, deinde philosophari [first live and then philosophize]? These men have accordingly taken possession of the market and have already seen to it that here nothing is of value except what they allow; consequently merit exists only in so far as they and their mediocrity are pleased to acknowledge it. They thus have on a leading rein the attention of that small public, such as it is, that is concerned with philosophy. For on matters that do not promise, like the productions of poetry, amusement and entertainment but only instruction, and financially unprofitable instruction at that, that public will certainly not waste its time, effort, and energy, without first being thoroughly assured that such efforts will be richly rewarded. Now by virtue of its inherited belief that whoever lives by a business knows all about it, this public expects an assurance from the professional men who from professor's chairs and in compendiums, journals, and literary periodicals, confidently behave as if they were the real masters of the subject. Accordingly, the public allows them to sample and select whatever is worth noting and what can be ignored."

"The two greatest blinders of the intellectual who today might fight against the main drift are new and fascinating career chances, which often involve opportunities to practice his skill rather freely, and the ideology of liberalism, which tends to expropriate his chance to think straight. The two go together, for the liberal ideology, as now used by intellectuals, acts as a device whereby he can take advantage of the new career chances but retain the illusion that his soul remains his own."

"I now address the graduate students. This is a time of enormous opportunity for you. There is an ossified political establishment of invested self-interest. Conformism and empty pieties dominate academe. Rebel. Do not read Lacan, Derrida, and Foucault, and treat as insignificant nothings those that still prate of them. You need no contemporaries to interpret the present for you. Born here, alive now, you are modernity. You are the living link between the past and future. Charge yourself with the high ideal of scholarship, connecting you to Alexandria and to the devoted, distinguished scholars who came before you. When you build on learning, you build on rock. You become greater by a humility toward great things. Let your work follow its own organic rhythm. Seek no material return from it, and it will reward you with spiritual gold. Hate dogma. Shun careerists. If you keep the faith, the gods may give you, at midlife, the sweet pleasure of seeing the hotshots who were so fast out of the gate begin to flag and sink, just as your studies are reaching their point of maturation."

"Jesus ... combines all duties (1) in one universal rule (which includes within itself both the inner and the outer moral relations of men), namely: Perform your duty for no motive other than unconditioned esteem for duty itself, i.e., love God (the Legislator of all duties) above all else; and (2) in a particular rule, that, namely, which concerns man's external relation to other men as universal duty: Love every one as yourself, i.e., further his welfare from good-will that is immediate and not derived from motives of self-advantage. These commands are not mere laws of virtue but precepts of holiness which we ought to pursue, and the very pursuit of them is called virtue."

"The place-seeker will resort to methods from which self-respecting men would shrink with as much aversion as the ancient Jew shrank from contact with the leper. The true purpose of education is not office. "The true purpose of education," says one, "is to cherish and unfold the seed of immortality already sown within us; to develop to their fullest extent the capacities of every kind with which the God who made us has endowed us." He, therefore, who fixes a limit of any kind to his intellectual attainments dwarfs himself, and cramps the growth of that mind given to us by the Creator, and capable of indefinite expansion."

"Everyone knows how compromised the idea of bureaucracy as a meritocratic system is. The first criterion of loyalty to any organization is therefore complicity. Career advancement is not based on merit but on a willingness to play along with the fiction that career advancement is based on merit, or with the fiction that rules and regulations apply to everyone equally, when in fact they are often deployed as an instrument of arbitrary personal power. ... As whole societies have come to represent themselves as giant credentialized meritocracies, rather than as systems of predatory extraction, we bustle about, trying to curry favor by pretending we actually believe it to be true."

"To noble minds profit should not appear to be a dignified reward for studies. It fits a craftsman to seek profit; generous arts know a nobler goal."

"Some misguided men learn the Dhamma ... only for the sake of criticising others and for winning in debates, and they do not experience the good for the sake of which they learned the Dhamma."

"Neither the black boy nor the white will ever be educated in the best and broadest sense of the term who seeks an education merely to reach an office, for, as in nature a stream never rises higher than its source, so in life men never rise higher than their ideals. The education that merely seeks an office must of necessity be limited to the dimensions of that office."

"The true clerc is Vauvenargues, Lamarck, Fresnel, Spinoza, Schiller, Baudelaire, César Franck, who were never diverted from single-hearted adoration of the beautiful and the divine by the necessity of earning their daily bread. But such clercs are inevitably rare. ... The rule is that the living creature condemned to struggle for life turns to practical passions, and thence to the sanctifying of those passions."

"We have reached the point, it is painful to recognize, where the only persons accounted wise are those who can reduce the pursuit of wisdom to a profitable traffic."

"This book argues that we are currently witnessing not merely a decline in the quality of social science research, but the proliferation of meaningless research, of no value to society, and modest value to its authors—apart from securing employment and promotion. The explosion of published outputs, at least in social science, creates a noisy, cluttered environment which makes meaningful research difficult, as different voices compete to capture the limelight even briefly. Older, more significant contributions are easily neglected, as the premium is to write and publish, not read and learn. The result is a widespread cynicism among academics on the value of academic research, sometimes including their own. Publishing comes to be seen as a game of hits and misses, devoid of intrinsic meaning and value, and of no wider social uses whatsoever. Academics do research in order to get published, not to say something socially meaningful."