Education

"A good many times I have been present at gatherings of people who, by the standards of the traditional culture, are thought highly educated and who have with considerable gusto been expressing their incredulity at the illiteracy of scientists. Once or twice I have been provoked and have asked the company how many of them could describe the Second Law of Thermodynamics. The response was cold: it was also negative. Yet I was asking something which is about the scientific equivalent of: Have you read a work of Shakespeare's?"

"Increasingly, our leaders must deal with dangers that threaten the entire world, where an understanding of those dangers and the possible solutions depends on a good grasp of science. The ozone layer, the greenhouse effect, acid rain, questions of diet and heredity. All require scientific literacy. Can Americans choose the proper leaders and support the proper programs if they themselves are scientifically illiterate? The whole premise of democracy is that it is safe to leave important questions to the court of public opinion—but is it safe to leave them to the court of public ignorance?"

"Let me suggest to you a simple test one can apply to scientific activities to determine whether or not they can constitute the practice of physics. Is what you are doing beautiful? Many beautiful things are created without the use of physical knowledge, but I know of no really worthwhile physics that isn’t beautiful. Indeed, one of the most distressing symptoms of scientific illiteracy is the impression so often given to school children that science is a mechanistic activity subject to algorithmic description."

"Rampant scientific illiteracy in the general public is, in my opinion, one major cause of the current lack of opportunities for scientists. … A public that is ignorant of science, and of how science is done, is not going to support scientific research enthusiastically."

"It was a clever saying of Bion, the philosopher, that, just as the suitors, not being able to approach Penelope, consorted with her maid-servants, so also do those who are not able to attain to philosophy wear themselves to a shadow over the other kinds of education which have no value."

"Specialization develops only part of a man; a man partially developed is deformed."

"Men go to sea before they know the unhappiness of that way of life; and when they have come to know it, they cannot escape from it, because it is then too late to choose another profession; as indeed is generally the case with men, when they have once engaged in any particular way of life."

"Nothing is more certain than that whatever has to court public favor for its support will sooner or later be prostituted to utilitarian ends. The educational institutions of the United States afford a striking demonstration of this truth. Virtually without exception, liberal education, that is to say, education centered about ideas and ideals, has fared best in those institutions which draw their income from private sources. They have been able … to insist that education be not entirely a means for breadwinning. This means that they have been relatively free to promote pure knowledge and the training of the mind. … In state institutions, always at the mercy of elected bodies and of the public generally, and under obligation to show practical fruits for their expenditure of money, the movement toward specialism and vocationalism has been irresistible. They have never been able to say that they will do what they will with their own because their own is not private. It seems fair to say that the opposite of the private is the prostitute."

"Of what use is it, ask these slaves of the ledger, to spend the greater part of a lifetime in acquiring a competency only to find after it has been acquired that its acquisition has taken all the savour of enjoyment out of life?"

"The Confucian aspirant to office, stemming from the old tradition, could hardly help viewing a specialized, professional training of European stamp as anything but a conditioning in the dirtiest Philistinism. ... The fundamental assertion, ‘a cultured man is not a tool’ meant that he was an end in himself and not just a means for a specified useful purpose."

"The United States ... celebrates rote vocational training and the singular, amoral skill of making money. It churns out stunted human products, lacking the capacity and vocabulary to challenge the assumptions and structures of the corporate state. It funnels them into a caste system of drones and systems managers. It transforms a democratic state into a feudal system of corporate masters and serfs."

"When I speak of the purpose of self-culture, I mean that it should be sincere. In other words, we must make self-culture really and truly our end, or choose it for its own sake, and not merely as a means or instrument of something else. And here I touch a common and very pernicious error. Not a few persons desire to improve themselves only to get property and to rise in the world; but such do not properly choose improvement, but something outward and foreign to themselves; and so low an impulse can produce only a stinted, partial, uncertain growth. A man, as I have said, is to cultivate himself because he is a man. He is to start with the conviction that there is something greater within him than in the whole material creation, than in all the worlds which press on the eye and ear; and that inward improvements have a worth and dignity in themselves quite distinct from the power they give over outward things. Undoubtedly a man is to labor to better his condition, but first to better himself. If he knows no higher use of his mind than to invent and drudge for his body, his case is desperate as far as culture is concerned."

"A man has within him capacities of growth which deserve and will reward intense, unrelaxing toil. I do not look on a human being as a machine, made to be kept in action by a foreign force, to accomplish an unvarying succession of motions, to do a fixed amount of work, and then to fall to pieces at death, but as a being of free spiritual powers; and I place little value on any culture but that which aims to bring out these, and to give them perpetual impulse and expansion. I am aware that this view is far from being universal. The common notion has been, that the mass of the people need no other culture than is necessary to fit them for their various trades; and, though this error is passing away, it is far from being exploded. But the ground of a man’s culture lies in his nature, not in his calling. His powers are to be unfolded on account of their inherent dignity, not their outward direction. He is to be educated, because he is a man, not because he is to make shoes, nails, or pins. A trade is plainly not the great end of his being, for his mind cannot be shut up in it; his force of thought cannot be exhausted on it. He has faculties to which it gives no action, and deep wants it cannot answer. Poems, and systems of theology and philosophy, which have made some noise in the world, have been wrought at the work-bench and amidst the toils of the field. How often, when the arms are mechanically plying a trade, does the mind, lost in reverie or day-dreams, escape to the ends of the earth! How often does the pious heart of woman mingle the greatest of all thoughts, that of God, with household drudgery! Undoubtedly a man is to perfect himself in his trade, for by it he is to earn his bread and to serve the community. But bread or subsistence is not his highest good; for, if it were, his lot would be harder than that of the inferior animals, for whom nature spreads a table and weaves a wardrobe, without a care of their own. Nor was he made chiefly to minister to the wants of the community. A rational, moral being cannot, without infinite wrong, be converted into a mere instrument of others’ gratification. He is necessarily an end, not a means. A mind, in which are sown the seeds of wisdom, disinterestedness, firmness of purpose, and piety, is worth more than all the outward material interests of a world. It exists for itself, for its own perfection, and must not be enslaved to its own or others’ animal wants."

"It is in the forms and under the forms of ideological subjection that provision is made for the reproduction of the skills of labour power."

"The cultured man is not a tool."

"For the Confucian, the specialistic expert could not be raised to truly positive dignity, no mater what his social usefulness. The decisive factor was that the "cultured man" (gentleman) was "not a tool"; that is, in his adjustment to the world and in his self-perfection he was an end unto himself, not a means for any functional end. This core of Confucian ethics rejected professional specialization, modern expert bureaucracy, and special training; above all, it rejected training in economics for the pursuit of profit."

"When we call for education we mean real education. We believe in work. We ourselves are workers, but work is not necessarily education. Education is the development of power and ideal. We want our children trained as intelligent human beings should be, and we will fight for all time against any proposal to educate black boys and girls simply as servants and underlings, or simply for the use of other people. They have a right to know, to think, to aspire."

"Everyone who achieves strives for totality, and the value of his achievement lies in that totality—that is, in the fact that the whole, undivided nature of a human being should be expressed in his achievement. But when determined by our society, as we see it today, achievement does not express a totality; it is completely fragmented and derivative. It is not uncommon for the community to be the site where a joint and covert struggle is waged against higher ambitions and more personal goals. ... The socially relevant achievement of the average person serves in the vast majority of cases to repress the original and nonderivative, inner aspirations of the human being."

"The science, which teaches arts and handicrafts Is merely science for the gaining of a living; But the science which teaches deliverance from worldly existence, Is not that the true science?"

"At the end of a life spent in the pursuit of knowledge Faust has to confess: "I now see that we can nothing know." That is the answer to a sum, it is the outcome of a long experience. But as Kierkegaard observed, it is quite a different thing when a freshman comes up to the university and uses the same sentiment to justify his indolence. As the answer to a sum it is perfectly true, but as the initial data it is a piece of self-deception."

"Alexander W. Astin’s research tells us that in the mid-1960s, more than 80 percent of entering college freshmen reported that nothing was more important than “developing a meaningful philosophy of life.” Astin, director of the Higher Education Research Institute at UCLA, reports that “being very well off financially” was only an afterthought, one that fewer than 45 percent of those freshmen thought to be an essential goal. As the years went on, however, and as tuition shot up, the two traded places; by 1977, financial goals had surged past philosophical ones, and by the year 2001 more than 70 percent of undergraduate students had their eyes trained on financial realities, while only 40 percent were still wrestling with meaningful philosophies."

"Multis ne magistri veri essent magisterii falsum nomen obstitit: dum de se plus omnibus quam sibi dumque quod dicebantur, sed non erant, esse crediderunt, quod esse poterant non fuerunt."

"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."

"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."

"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."

"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."

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

"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."

"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."

"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."

""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."

"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."

"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."

"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."

"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 principal aim of mathematical education is to develop certain faculties of the mind, and among these intuition is not the least precious. It is through it that the mathematical world remains in touch with the real world, and even if pure mathematics could do without it, we should still have to have recourse to it to fill up the gulf that separates the symbol from reality."

"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."

"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."

"I am well aware of the negative attitude of so many students toward the subject. There are many reasons for this, one of them no doubt being the esoteric, dry way in which we teach the subject. We tend to overwhelm our students with formulas, definitions, theorems, and proofs, but we seldom mention the historical evolution of these facts, leaving the impression that these facts were handed to us, like the Ten Commandments, by some divine authority. The history of mathematics is a good way to correct these impressions."

"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 usual courses in mathematics are... deceptive in a basic respect. They give an organized logical presentation which leaves the impression that mathematicians go from theorem to theorem almost naturally, that mathematicians can master almost any difficulty, and that subjects are completely thrashed out and settled. The succession of theorems overwhelms the student... The history, by contrast, teaches us the development of a subject is made bit by bit with results coming from various directions. We learn, too, that often decades and even hundreds of years of effort were required before significant steps could be made. In place of the impression that subjects are completely thrashed out one finds that what is attained is often but a start, that many gaps have to be filled, or that the really important extensions remain to be created."

"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."

"There is a danger to the humanities in the present educational crash programs designed to produce a large number of mathematicians, physical scientists, engineers, and technical workers. ...Part of the... objective of the present work is to supply material which can serve as a cultural background or supplement for all those who are receiving rapid, concentrated exposure to recent advanced mathematical concepts, without any opportunity to examine the origins or gradual historical development of such ideas. Hence, although designed for the layman, this book would be helpful in courses in the history, philosophy, or fundamental concepts of mathematics."

"Geometry is grasping space. And since it is about the education of children, it is grasping that space in which the child lives, breathes and moves. The space that the child must learn to know, explore, conquer, in order to live, breathe and move better in it."

"The history of Greek mathematics is, for the most part, only the history of such mathematics as are learnt daily in all our public schools. ...If it was not wanted, as it ought to have been, by our classical professors and our mathematicians, it would have served at any rate to quicken, with some human interest, the melancholy labours of our schoolboys."

"Using the history of algebra, teachers of the subject, either at the school or at the college level, can increase students' overall understanding of the material. The "logical" development so prevalent in our textbooks is often sterile because it explains neither why people were interested in a particular algebraic topic in the first place nor why our students should be interested in that topic today. History, on the other hand, often demonstrates the reasons for both. With the understanding of the historical development of algebra, moreover, teachers can better impart to their students an appreciation that algebra is not arbitrary, that it is not created "full-blown" by fiat. Rather, it develops at the hands of people who need to solve vital problems, problems the solutions of which merit understanding. Algebra has been and is being created in many areas of the world, with the same solution often appearing in disparate times and places. ...professors can stimulate their students to master often complex notions by motivating the material through the historical questions that prompted its development. In absorbing the idea, moreover, that people struggled with many important mathematical ideas before finding their solutions, that they frequently could not solve problems entirely, and that they consciously left them for their successors to explore, students can better appreciate the mathematical endeavor and its shared purpose."

"It is a student's understanding of mathematics, above all other subjects, which suffers most from unenlightened teaching methods. ...the troubles may well stem mainly from the first year or two of the child's encounter with numbers... if children come to fear them or to be bored with them, they will eventually join the ranks of the present majority... If, on the other hand, numbers are made a... source of adventure and exploration from the beginning, there is a good chance that the level of numeracy in society can be raised... There is a real role here for the history of mathematics—and the history of number in particular—for history emphasizes the diversity of approaches and methods which are possible and frees us from the straitjacket of contemporary fashions in mathematics education. It is, at the same time, interesting and stimulating in its own right."

"Reflect on how many real problems of life require solutions which are meaningless unless they are whole numbers. The absence of Diophantine analysis from school curricula is difficult to justify. The methods... are not beyond the capabilities of schoolchildren."

"We cannot expect to present at pre-university level the more abstract approaches to the definition[s]... of the late nineteenth and early twentieth century. We... accept... that the natural numbers are 'given'... we do not need to define them. Our pupils will discover ways in which numbers relate to the real world for themselves—all we have to do is to provide the environment in which this can happen easily and effectively."

"Algebra is a machine, or more accurately, a collection of machines. There is a machinery to factor expressions, to decompose complicated fractions into simpler ones, and so on. The object of the conversion in every case is to obtain a form more useful for the problem in hand. ...Elementary algebra as a whole is a huge machine to mechanize thinking. ...The mechanization of processes that have to be used repeatedly is... a great gain, since one does not have to think about them. They become habitual like washing and dressing. ...in itself elementary algebra is of no great interest. ...generally speaking, a machine is valuable because it turns out a useful product. ... The individual techniques of algebra are like single notes selected at random from large and magnificent musical compositions. ...these notes ...employed in the investigation of more significant undertakings, help to form beautiful patterns of reasoning. ...Unfortunately, the usefulness of the techniques of algebra has caused many people to mistake the means for the end and to emphasize these menial techniques to the exclusion of the larger ideas and goals of mathematics. The students who are bored by the process of algebra are more perceptive than those who have mistakenly identified algebraic processes with mathematics."