Mathematics Teacher Quotes

It's much easier, after all, to learn mathematics from someone who's made a few mistakes. It's impossible to learn it from someone who always gets it right.

Reading list (1972 edition)[edit] 1. Homer – Iliad, Odyssey 2. The Old Testament 3. Aeschylus – Tragedies 4. Sophocles – Tragedies 5. Herodotus – Histories 6. Euripides – Tragedies 7. Thucydides – History of the Peloponnesian War 8. Hippocrates – Medical Writings 9. Aristophanes – Comedies 10. Plato – Dialogues 11. Aristotle – Works 12. Epicurus – Letter to Herodotus; Letter to Menoecus 13. Euclid – Elements 14. Archimedes – Works 15. Apollonius of Perga – Conic Sections 16. Cicero – Works 17. Lucretius – On the Nature of Things 18. Virgil – Works 19. Horace – Works 20. Livy – History of Rome 21. Ovid – Works 22. Plutarch – Parallel Lives; Moralia 23. Tacitus – Histories; Annals; Agricola Germania 24. Nicomachus of Gerasa – Introduction to Arithmetic 25. Epictetus – Discourses; Encheiridion 26. Ptolemy – Almagest 27. Lucian – Works 28. Marcus Aurelius – Meditations 29. Galen – On the Natural Faculties 30. The New Testament 31. Plotinus – The Enneads 32. St. Augustine – On the Teacher; Confessions; City of God; On Christian Doctrine 33. The Song of Roland 34. The Nibelungenlied 35. The Saga of Burnt Njál 36. St. Thomas Aquinas – Summa Theologica 37. Dante Alighieri – The Divine Comedy;The New Life; On Monarchy 38. Geoffrey Chaucer – Troilus and Criseyde; The Canterbury Tales 39. Leonardo da Vinci – Notebooks 40. Niccolò Machiavelli – The Prince; Discourses on the First Ten Books of Livy 41. Desiderius Erasmus – The Praise of Folly 42. Nicolaus Copernicus – On the Revolutions of the Heavenly Spheres 43. Thomas More – Utopia 44. Martin Luther – Table Talk; Three Treatises 45. François Rabelais – Gargantua and Pantagruel 46. John Calvin – Institutes of the Christian Religion 47. Michel de Montaigne – Essays 48. William Gilbert – On the Loadstone and Magnetic Bodies 49. Miguel de Cervantes – Don Quixote 50. Edmund Spenser – Prothalamion; The Faerie Queene 51. Francis Bacon – Essays; Advancement of Learning; Novum Organum, New Atlantis 52. William Shakespeare – Poetry and Plays 53. Galileo Galilei – Starry Messenger; Dialogues Concerning Two New Sciences 54. Johannes Kepler – Epitome of Copernican Astronomy; Concerning the Harmonies of the World 55. William Harvey – On the Motion of the Heart and Blood in Animals; On the Circulation of the Blood; On the Generation of Animals 56. Thomas Hobbes – Leviathan 57. René Descartes – Rules for the Direction of the Mind; Discourse on the Method; Geometry; Meditations on First Philosophy 58. John Milton – Works 59. Molière – Comedies 60. Blaise Pascal – The Provincial Letters; Pensees; Scientific Treatises 61. Christiaan Huygens – Treatise on Light 62. Benedict de Spinoza – Ethics 63. John Locke – Letter Concerning Toleration; Of Civil Government; Essay Concerning Human Understanding;Thoughts Concerning Education 64. Jean Baptiste Racine – Tragedies 65. Isaac Newton – Mathematical Principles of Natural Philosophy; Optics 66. Gottfried Wilhelm Leibniz – Discourse on Metaphysics; New Essays Concerning Human Understanding;Monadology 67. Daniel Defoe – Robinson Crusoe 68. Jonathan Swift – A Tale of a Tub; Journal to Stella; Gulliver's Travels; A Modest Proposal 69. William Congreve – The Way of the World 70. George Berkeley – Principles of Human Knowledge 71. Alexander Pope – Essay on Criticism; Rape of the Lock; Essay on Man 72. Charles de Secondat, baron de Montesquieu – Persian Letters; Spirit of Laws 73. Voltaire – Letters on the English; Candide; Philosophical Dictionary 74. Henry Fielding – Joseph Andrews; Tom Jones 75. Samuel Johnson – The Vanity of Human Wishes; Dictionary; Rasselas; The Lives of the Poets

I think we need more math majors who don't become mathematicians. More math major doctors, more math major high school teachers, more math major CEOs, more math major senators. But we won't get there unless we dump the stereotype that math is only worthwhile for kid geniuses.

No doubt there are some who, when confronted with a line of mathematical symbols, however simply presented, can only see the face of a stern parent or teacher who tried to force into them a non-comprehending parrot-like apparent competence--a duty and a duty alone--and no hint of magic or beauty of the subject might be allowed to come through.

A math teacher’s least favorite thing to hear from a student is “I get the concept, but I couldn’t do the problems.” Though the student doesn’t know it, this is shorthand for “I don’t get the concept.

I feel that my main responsibility as a teacher isn’t to convey facts, but to rekindle that lost enthusiasm for asking questions.

I needed to wander… whenever and wherever I wanted! I’d found myself at the end of my rope as far as school was concerned; there seemed no particular reason for me to stay. The teachers didn’t want to teach, and I didn’t want to learn—from them. I wanted my education to come from living life, getting out there in the world, seeing and doing and moving amongst the other vagabonds who had had the same sneaking suspicion that I did, that there would be no great need for high-end mathematics, nope… I was not going to be doing other people’s taxes and going home at 5:37 p.m. to pat my dog’s head and sit down to my one meat and two vegetable table waiting for Jeopardy to pop on the glass tit, the Pat Sajak of my own private game show, in the bellybutton of the universe, Miramar, Florida.

[Math] curriculum is obsessed with jargon and nomenclature seemingly for no other purpose than to provide teachers with something to test the students on.

Mathematics education is much more complicated than you expected, even though you expected it to be more complicated than you expected.

How do we change the way science is taught? Ask anybody how many teachers truly made a difference in their life, and you never come up with more than the fingers on one hand. You remember their names, you remember what they did, you remember how they moved in front of the classroom. You know why you remember them? Because they were passionate about the subject. You remember them because they lit a flame within you. They got you excited about a subject you didn't previously care about, because they were excited about it themselves. That's what turns people on to careers in science and engineering and mathematics. That's what we need to promote. Put that in every classroom, and it will change the world.

We should reject the view that high culture, as the possession of an elite, is of no use to those who don’t possess it. This is as false as the view that science or higher mathematics are useless to those who don’t understand them. Scientific knowledge exists because a few talented people are prepared to devote their energy to pursuing it. That is what a university is for: and since you cannot pass on difficult knowledge without discriminating between the students who can absorb it and those who cannot, discrimination is a social good. The same is true of high culture. Those able to acquire it will be a minority and the process of cultural transmission will be critically impeded if that teacher must teach Mozart and Lady Gaga side by side to satisfy some egalitarian agenda.

I noticed that the [drawing] teacher didn't tell people much... Instead, he tried to inspire us to experiment with new approaches. I thought of how we teach physics: We have so many techniques - so many mathematical methods - that we never stop telling the students how to do things. On the other hand, the drawing teacher is afraid to tell you anything. If your lines are very heavy, the teacher can't say, "Your lines are too heavy." because *some* artist has figured out a way of making great pictures using heavy lines. The teacher doesn't want to push you in some particular direction. So the drawing teacher has this problem of communicating how to draw by osmosis and not by instruction, while the physics teacher has the problem of always teaching techniques, rather than the spirit, of how to go about solving physical problems.

Education makes your maths better, not necessarily your manners.

Our schools will not improve if we continue to focus only on reading and mathematics while ignoring the other studies that are essential elements of a good education. Schools that expect nothing more of their students than mastery of basic skills will not produce graduates who are ready for college or the modern workplace. *** Our schools will not improve if we value only what tests measure. The tests we have now provide useful information about students' progress in reading and mathematics, but they cannot measure what matters most in education....What is tested may ultimately be less important that what is untested... *** Our schools will not improve if we continue to close neighborhood schools in the name of reform. Neighborhood schools are often the anchors of their communities, a steady presence that helps to cement the bond of community among neighbors. *** Our schools cannot improve if charter schools siphon away the most motivated students and their families in the poorest communities from the regular public schools. *** Our schools will not improve if we continue to drive away experienced principals and replace them with neophytes who have taken a leadership training course but have little or no experience as teachers. *** Our schools cannot be improved if we ignore the disadvantages associated with poverty that affect children's ability to learn. Children who have grown up in poverty need extra resources, including preschool and medical care.

When I was young, most teachers of philosophy in British and American universities were Hegelians, so that, until I read Hegel, I supposed there must be some truth to his system; I was cured, however, by discovering that everything he said on the philosophy of mathematics was plain nonsense.

One of the first and foremost duties of the teacher is not to give his students the impression that mathematical problems have little connection with each other, and no connection at all with anything else. We have a natural opportunity to investigate the connections of a problem when looking back at its solution.

Suzuki method or no Suzuki method, what matters most of all is the teacher,

As a teacher, Tengo pounded into his students' heads how voraciously mathematics demanded logic. Here things that could not be proven had no meaning, but once you had succeeded in proving something, the world's riddles settled into the palm of your hand like a tender oyster.

We cannot do it in this way for two reasons. First, we do not yet know all the basic laws: there is an expanding frontier of ignorance. Second, the correct statement of the laws of physics involves some very unfamiliar ideas which require advanced mathematics for their description. Therefore,

I felt a downright fear of the mathematics class. The teacher pretended that algebra was a perfectly natural affair, to be taken for granted, whereas I didn’t even know what numbers really were. They were not flowers, not animals, not fossils; they were nothing that could be imagined, mere quantities that resulted from counting. To my confusion these quantities were now represented by letters, which signified sounds, so that it became possible to hear them, so to speak. Oddly enough, my classmates could handle these things and found them self-evident. No one could tell me what numbers were, and I was unable even to formulate the question.

Neither parents nor schools are very effective at teaching the young to find pleasure in the right things. Adults, themselves often deluded by infatuation with fatuous models, conspire in the deception. They make serious tasks seem dull and hard, and frivolous ones exciting and easy. Schools generally fial to teach how exciting, how mesmerizingly beautiful science or mathematics can be; they teach the routine of literature or history rather than the adventure.

American parents, teachers, and children were far more likely than their Japanese and Chinese counterparts to believe that mathematical ability is innate; if you have it, you don’t have to work hard, and if you don’t have it, there’s no point in trying. In contrast, most Asians regard math success, like achievement in any other domain, as a matter of persistence and plain hard work. Of course you will make mistakes as you go along; that’s how you learn and improve. It doesn’t mean you are stupid.

My other teachers did not seem to care about the challenge of being human and instead they taught us to think about mathematics and analyze different chemicals and as the months went by I felt further from myself. And the only thing that seemed to make sense was Ben Sweet and the way he talked to us and urged something in the deeps of us to come out—the way he looked, and listened, as if he had no other place on this Earth to be except with us, as if there were nothing more important in his life than what we had to say at just that moment in time.

Dear friends & fellow characters, you all know the importance we attach to the power of collective prayer in this our desperate struggle for survival. Some of us have more existence than others, at various times according to fashion. But even this is becoming extremely shadowy & precarious, for we are not read, & when read , we are read badly, we are not lived as we used to be, we are not identified with & fantasized, we are rapidly forgotten. Those of us who have the good fortune to be read by teachers, scholars, & students are not read as we used to be read, but analyzed as schemata, structures, functions within structures, logical & mathematical formulae, aporia, psychic movements, social significances & so forth.

I don't know which is worse—to have a bad teacher or no teacher at all. In any case, I believe the teacher's work should be largely negative. He can't put the gift into you, but if he finds it there, he can try to keep it from going in an obviously wrong direction. We can learn how not to write, but this is a discipline that does not simply concern writing itself but concerns the whole intellectual life. A mind cleared of false emotion and false sentiment and egocentricity is going to have at least those roadblocks removed from its path. If you don't think cheaply, then there at least won't be the quality of cheapness in your writing, even though you may not be able to write well. The teacher can try to weed out what is positively bad, and this should be the aim of the whole college. Any discipline can help your writing: logic, mathematics, theology, and of course and particularly drawing. Anything that helps you to see, anything that makes you look. The writer should never be ashamed of staring. There is nothing that doesn't require his attention.

While Euclid himself may not have been the greatest mathematician who ever lived, he was certainly the greatest teacher of mathematics.

Students of color do only slightly worse than white students in mathematics. If you’ll pardon my grammar, nonwhite students do more worse in English and most worse in history.

the correct statement of the laws of physics involves some very unfamiliar ideas which require advanced mathematics for their description. Therefore,

Early success is a terrible teacher.

Diagnostic, comment-based feedback is now known to promote learning, and it should be the standard way in which students’ progress is reported.

So, a great Indian teacher of mathematics discovered the zero written in God's notebook, and, thanks to him, we can now read many more pages in the notebook. Is that it?

The teacher should help, but not too much and not too little, so that the student shall have a reasonable share of the work.

Teaching is a dialogue, and it is through the process of engaging students that we see ideas taken from the abstract and played out in concrete visual form. Students teach us about creativity through their personal responses to the limits we set, thus proving that reason and intuition are not antithetical. Their works give aesthetic visibility to mathematical ideas.

He would wordlessly light up his pungent antiasthma cigarettes in the middle of class and debate openly with his mathematics and literature teachers about inaccuracies he's caught them in.

Imagine how safe it would feel to know that no one could ever commit a crime of violence and go unnoticed, ever again. Imagine what it would mean to us to know – know for certain – that the plane or the bus we’re travelling on is properly maintained, that the teacher who looks after our children doesn’t have ugly secrets. All it would cost is our privacy, and to be honest who really cares about that? What secrets would you need to keep from a mathematical construct without a heart? From a card index? Why would it matter? And there couldn’t be any abuse of the system, because the system would be built not to allow it. It’s the pathway we’re taking now, that we’ve been on for a while.

Bradley Headstone, in his decent black coat and waistcoat, and decent white shirt, and decent formal black tie, and decent pantaloons of pepper and salt, with his decent silver watch in his pocket and its decent hair-guard round his neck, looked a thoroughly decent young man of six-and-twenty. He was never seen in any other dress, and yet there was a certain stiffness in his manner of wearing this, as if there were a want of adaptation between him and it, recalling some mechanics in their holiday clothes. He had acquired mechanically a great store of teacher's knowledge. He could do mental arithmetic mechanically, sing at sight mechanically, blow various wind instruments mechanically, even play the great church organ mechanically. From his early childhood up, his mind had been a place of mechanical stowage. The arrangement of his wholesale warehouse, so that it might be always ready to meet the demands of retail dealers history here, geography there, astronomy to the right, political economy to the left—natural history, the physical sciences, figures, music, the lower mathematics, and what not, all in their several places—this care had imparted to his countenance a look of care; while the habit of questioning and being questioned had given him a suspicious manner, or a manner that would be better described as one of lying in wait. There was a kind of settled trouble in the face. It was the face belonging to a naturally slow or inattentive intellect that had toiled hard to get what it had won, and that had to hold it now that it was gotten. He always seemed to be uneasy lest anything should be missing from his mental warehouse, and taking stock to assure himself.

Sometimes all it takes is a simple word, a mere nothing, a well-intentioned but over-protective gesture, like the gesture made, quite unwittingly, by the Mathematics teacher, for the pacific, docile, submissive person suddenly to vanish and be replaced, to the dismay and incomprehension of those who thought they knew all there was to know about the human soul, by the blind, devastating wrath of the meek. It doesn't usually last very long, but while it does, it inspires real fear.

The teacher manages to get along still with the cumbersome algebraic analysis, in spite of its difficulties and imperfections, and avoids the smooth infinitesimal calculus, although the eighteenth century shyness toward it had long lost all point.

John Dalton was a very singular Man: He has none of the manners or ways of the world. A tolerable mathematician He gained his livelihood I believe by teaching the mathematics to young people. He pursued science always with mathematical views. He seemed little attentive to the labours of men except when they countenanced or confirmed his own ideas... He was a very disinterested man, seemed to have no ambition beyond that of being thought a good Philosopher. He was a very coarse Experimenter & almost always found the results he required.—Memory & observation were subordinate qualities in his mind. He followed with ardour analogies & inductions & however his claims to originality may admit of question I have no doubt that he was one of the most original philosophers of his time & one of the most ingenious.

The human mind is an incredible thing. It can conceive of the magnificence of the heavens and the intricacies of the basic components of matter. Yet for each mind to achieve its full potential, it needs a spark. The spark of enquiry and wonder. Often that spark comes from a teacher. Allow me to explain. I wasn’t the easiest person to teach, I was slow to learn to read and my handwriting was untidy. But when I was fourteen my teacher at my school in St Albans, Dikran Tahta, showed me how to harness my energy and encouraged me to think creatively about mathematics. He opened my eyes to maths as the blueprint of the universe itself. If you look behind every exceptional person there is an exceptional teacher. When each of us thinks about what we can do in life, chances are we can do it because of a teacher. [...] The basis for the future of education must lie in schools and inspiring teachers. But schools can only offer an elementary framework where sometimes rote-learning, equations and examinations can alienate children from science. Most people respond to a qualitative, rather than a quantitative, understanding, without the need for complicated equations. Popular science books and articles can also put across ideas about the way we live. However, only a small percentage of the population read even the most successful books. Science documentaries and films reach a mass audience, but it is only one-way communication.

It puzzled him that she did not mourn all the things she could have been. Was it a quality inherent in women, or did they just learn to shield their personal regrets, to suspend their lives, subsume themselves in child care? She browsed online forums about tutoring and music and schools, and she told him what she had discovered as though she truly felt the rest of the world should be as interested as she was in how music improved the mathematics skills of nine-year-olds. Or she would spend hours on the phone talking to her friends, about which violin teacher was good and which tutorial was a waste of money. One day, after

I was perplexed by the failure of teachers at school to address what seemed the most urgent matter of all: the bewildering, stomach-churning insecurity of being alive. The standard subjects of history, geography, mathematics, and English seemed perversely designed to ignore the questions that really mattered. As soon as I had some inkling of what 'philosophy' meant, I was puzzled as to why we were not taught it. And my skepticism about religion only grew as I failed to see what the vicars and priests I encountered gained from their faith. They struck me either as insincere, pious, and aloof or just bumblingly good-natured. (p. 10)

Byron had said teaching was an underpaid and underappreciated profession. He’d said it drained the life out of people. He’d said the system takes advantage of teachers and sets them up to fail, so why would any intelligent, reasonable person with an aptitude for science or mathematics or engineering ever willingly accept a teacher’s salary to do a teacher’s job?

The human mind is an incredible thing. It can conceive of the magnificence of the heavens and the intricacies of the basic components of matter. Yet for each mind to achieve its full potential, it needs a spark. The spark of enquiry and wonder. Often that spark comes from a teacher. Allow me to explain. I wasn’t the easiest person to teach, I was slow to learn to read and my handwriting was untidy. But when I was fourteen my teacher at my school in St Albans, Dikran Tahta, showed me how to harness my energy and encouraged me to think creatively about mathematics. He opened my eyes to maths as the blueprint of the universe itself. If you look behind every exceptional person there is an exceptional teacher. When each of us thinks about what we can do in life, chances are we can do it because of a teacher.

If your teacher is in a private, for-profit school, however, and you withdraw your child, then the owner of the school will quickly feel the effect in his pocket, and the bad teacher will be fired. In a free system the parent, the consumer, is the boss. Tooley found that private-school proprietors constantly monitor their teachers and follow up parents’ complaints. His team visited classrooms in various parts of India and Africa, and found teachers actually teaching in fewer of the government classrooms they visited than in private classrooms – sometimes little more than half as many. Despite having no public funds or aid money, the unrecognised private schools had better facilities such as toilets, electricity and blackboards. Their pupils also got better results, especially in English and mathematics. The

Pythagoras was born around 570 B.C. in the island of Samos in the Aegean Sea (off Asia Minor), and he emigrated sometime between 530 and 510 to Croton in the Dorian colony in southern Italy (then known as Magna Graecia). Pythagoras apparently left Samos to escape the stifling tyranny of Polycrates (died ca. 522 B.C.), who established Samian naval supremacy in the Aegean Sea. Perhaps following the advice of his presumed teacher, the mathematician Thales of Miletus, Pythagoras probably lived for some time (as long as twenty-two years, according to some accounts) in Egypt, where he would have learned mathematics, philosophy, and religious themes from the Egyptian priests. After Egypt was overwhelmed by Persian armies, Pythagoras may have been taken to Babylon, together with members of the Egyptian priesthood. There he would have encountered the Mesopotamian mathematical lore. Nevertheless, the Egyptian and Babylonian mathematics would prove insufficient for Pythagoras' inquisitive mind. To both of these peoples, mathematics provided practical tools in the form of "recipes" designed for specific calculations. Pythagoras, on the other hand, was one of the first to grasp numbers as abstract entities that exist in their own right.

It is almost as if ideas set in mathematical form melt and become liquid and just as rivers can, from the most humble beginnings, flow for thousands of miles, through the most varied topography bringing nourishment and life with them wherever they go, so too can ideas cast in mathematical form flow far from their original sources, along well-defined paths, electrifying and dramatically affecting much of what they touch. pp. xii - xiii.

Every now and then, I'm lucky enough to teach a kindergarten or first-grade class. Many of these children are natural-born scientists - although heavy on the wonder side, and light on skepticism. They're curious, intellectually vigorous. Provocative and insightful questions bubble out of them. They exhibit enormous enthusiasm. I'm asked follow-up questions. They've never heard of the notion of a 'dumb question'. But when I talk to high school seniors, I find something different. They memorize 'facts'. By and large, though, the joy of discovery, the life behind those facts has gone out of them. They've lost much of the wonder and gained very little skepticism. They're worried about asking 'dumb' questions; they are willing to accept inadequate answers, they don't pose follow-up questions, the room is awash with sidelong glances to judge, second-by-second, the approval of their peers. They come to class with their questions written out on pieces of paper, which they surreptitiously examine, waiting their turn and oblivious of whatever discussion their peers are at this moment engaged in. Something has happened between first and twelfth grade. And it's not just puberty. I'd guess that it's partly peer pressure not to excel - except in sports, partly that the society teaches short-term gratification, partly the impression that science or mathematics won't buy you a sports car, partly that so little is expected of students, and partly that there are few rewards or role-models for intelligent discussion of science and technology - or even for learning for it's own sake. Those few who remain interested are vilified as nerds or geeks or grinds. But there's something else. I find many adults are put off when young children pose scientific questions. 'Why is the Moon round?', the children ask. 'Why is grass green?', 'What is a dream?', 'How deep can you dig a hole?', 'When is the world's birthday?', 'Why do we have toes?'. Too many teachers and parents answer with irritation, or ridicule, or quickly move on to something else. 'What did you expect the Moon to be? Square?' Children soon recognize that somehow this kind of question annoys the grown-ups. A few more experiences like it, and another child has been lost to science.

It is said that there comes a point in every mathematics student's education when he hears himself saying to the teacher, "I think I understand"-- and that's the point at which he has hit a wall. Making sure that all gifted students hit their own personal walls is crucial for developing the empathy with the rest of the world. When they see their less lucky peers struggle academically, they need to be able to say "I know how it feels,"-- and be telling the truth.

While the universality of the creative process has been noticed, it has not been noticed universally. Not enough people recognize the preverbal, pre-mathematical elements of the creative process. Not enough recognize the cross-disciplinary nature of intuitive tools for thinking. Such a myopic view of cognition is shared not only by philosophers and psychologists but, in consequence, by educators, too. Just look at how the curriculum, at every educational level from kindergarten to graduate school, is divided into disciplines defined by products rather than processes. From the outset, students are given separate classes in literature, in mathematics, in science, in history, in music, in art, as if each of these disciplines were distinct and exclusive. Despite the current lip service paid to “integrating the curriculum,” truly interdisciplinary courses are rare, and transdisciplinary curricula that span the breadth of human knowledge are almost unknown. Moreover, at the level of creative process, where it really counts, the intuitive tools for thinking that tie one discipline to another are entirely ignored. Mathematicians are supposed to think only “in mathematics,” writers only “in words,” musicians only “in notes,” and so forth. Our schools and universities insist on cooking with only half the necessary ingredients. By half-understanding the nature of thinking, teachers only half-understand how to teach, and students only half-understand how to learn.

The researchers found that when students were given problems to solve, and they did not know methods to solve them, but they were given opportunity to explore the problems, they became curious, and their brains were primed to learn new methods, so that when teachers taught the methods, students paid greater attention to them and were more motivated to learn them. The researchers published their results with the title “A Time for Telling,” and they argued that the question is not “Should we tell or explain methods?” but “When is the best time do this?

The curriculum for the education of statesmen at the time of Plato included arithmetic, geometry, solid geometry, astronomy, and music-all of which, the Pythagorean Archytas tells us, fell under the general definition of "mathematics." According to legend, when Alexander the Great asked his teacher Menaechmus (who is reputed to have discovered the curves of the ellipse, the parabola, and the hyperbola) for a shortcut to geometry, he got the reply: "O King, for traveling over the country there are royal roads and roads for common citizens; but in geometry there is one road for all.

THE TRUTH ABOUT PUBLIC SCHOOL EDUCATION • American fifteen-year-olds rank thirty-fifth out of fifty-seven developed countries in math and literacy. • 30 percent of public school students don’t graduate from high school. • Every day, 7,000 kids drop out of high school. • Of the 50 million children currently in public school, 15 million of them will drop out. • 25 percent of all public school math teachers did not major in mathematics or a math-related subject at a college or university. • Less than two-thirds of high school graduates are accepted to college every year. • One half...

Perhaps the most striking illustration of Bayes’s theorem comes from a riddle that a mathematics teacher that I knew would pose to his students on the first day of their class. Suppose, he would ask, you go to a roadside fair and meet a man tossing coins. The first toss lands “heads.” So does the second. And the third, fourth . . . and so forth, for twelve straight tosses. What are the chances that the next toss will land “heads” ? Most of the students in the class, trained in standard statistics and probability, would nod knowingly and say: 50 percent. But even a child knows the real answer: it’s the coin that is rigged. Pure statistical reasoning cannot tell you the answer to the question—but common sense does. The fact that the coin has landed “heads” twelve times tells you more about its future chances of landing “heads” than any abstract formula. If you fail to use prior information, you will inevitably make foolish judgments about the future. This is the way we intuit the world, Bayes argued. There is no absolute knowledge; there is only conditional knowledge. History repeats itself—and so do statistical patterns. The past is the best guide to the future.

Hippasus’ proof—or at least Nico’s retelling of it—was really so simple that when he finished sketching it out, I wasn’t even aware that we had actually proven anything. Nico paused for a few minutes to let us mull it over. It was Peter who broke the silence, “I’m not sure I understand what we have done.” Nico seemed to be expecting such a response. “Step back and examine the proof; in fact, you should try and do this with every proof you see or have to work out for yourself. ..." He again waited for his words to sink in, and it began to make sense for me. All my mathematics teachers (other than Bauji and Nico) always seemed to evade this part of their responsibility. They had been content to merely write out a proof on the blackboard and carry on, seemingly without concern for what the proof meant and what it told us. “But you should not stop here. Even when you have understood a proof, and I hope you have indeed understood this proof, ask yourself the next question, the obvious one, but as critical: So what? Or, why are we proving this? What is the point? What is the context? How does it relate to us? To answer these questions we have to step back a little. Let me show you—it’s really quite delightful.” Now there was excitement in Nico’s voice.

The Sexual plight of these children [those adolescents experimenting sexually] is officially not mentioned. The revolutionary attack on hypocrisy by Ibsen, Freud, Ellis, Dreiser, did not succeed this far. Is it an eccentric opinion that an important part of the kids' restiveness in school from the onset of puberty has to do with puberty? The teachers talk about it among themselves, all right. (In his school, Bertrand Russell thought it was better if they had sex, so they could give their undivided attention to mathematics, which was the main thing.) But since the objective factor does not exist in our schools, the school itself begins to be irrelevant. The question here is not whether sexuality should be discouraged or encouraged. That is an important issue, but far more important is that it is hard to grow up when existing facts are treated as though they do not exist. For then there is no dialogue, it is impossible to be taken seriously, to be understood, to make a bridge between oneself and society. In American society we have perfected a remarkable form of censorship: to allow every one his political right to say what he believes, but to swamp his little boat with literally thousands of millions of newspapers, mass-circulation magazines, best-selling books, broadcasts, and public pronouncements that disregard what he says and give the official way of looking at things.

Countries measured their success by the size of their territory, the increase in their population and the growth of their GDP – not by the happiness of their citizens. Industrialised nations such as Germany, France and Japan established gigantic systems of education, health and welfare, yet these systems were aimed to strengthen the nation rather than ensure individual well-being. Schools were founded to produce skilful and obedient citizens who would serve the nation loyally. At eighteen, youths needed to be not only patriotic but also literate, so that they could read the brigadier’s order of the day and draw up tomorrow’s battle plans. They had to know mathematics in order to calculate the shell’s trajectory or crack the enemy’s secret code. They needed a reasonable command of electrics, mechanics and medicine in order to operate wireless sets, drive tanks and take care of wounded comrades. When they left the army they were expected to serve the nation as clerks, teachers and engineers, building a modern economy and paying lots of taxes. The same went for the health system. At the end of the nineteenth century countries such as France, Germany and Japan began providing free health care for the masses. They financed vaccinations for infants, balanced diets for children and physical education for teenagers. They drained festering swamps, exterminated mosquitoes and built centralised sewage systems. The aim wasn’t to make people happy, but to make the nation stronger. The country needed sturdy soldiers and workers, healthy women who would give birth to more soldiers and workers, and bureaucrats who came to the office punctually at 8 a.m. instead of lying sick at home. Even the welfare system was originally planned in the interest of the nation rather than of needy individuals. When Otto von Bismarck pioneered state pensions and social security in late nineteenth-century Germany, his chief aim was to ensure the loyalty of the citizens rather than to increase their well-being. You fought for your country when you were eighteen, and paid your taxes when you were forty, because you counted on the state to take care of you when you were seventy.30 In 1776 the Founding Fathers of the United States established the right to the pursuit of happiness as one of three unalienable human rights, alongside the right to life and the right to liberty. It’s important to note, however, that the American Declaration of Independence guaranteed the right to the pursuit of happiness, not the right to happiness itself. Crucially, Thomas Jefferson did not make the state responsible for its citizens’ happiness. Rather, he sought only to limit the power of the state.

And one of the things that has most obstructed the path of discipleship in our Christian culture today is this idea that it will be a terribly difficult thing that will certainly ruin your life. A typical and often-told story in Christian circles is of those who have refused to surrender their lives to God for fear he would “send them to Africa as missionaries.” And here is the whole point of the much misunderstood teachings of Luke 14. There Jesus famously says one must “hate” all their family members and their own life also, must take their cross, and must forsake all they own, or they “cannot be my disciple” (Luke 14:26–27, 33). The entire point of this passage is that as long as one thinks anything may really be more valuable than fellowship with Jesus in his kingdom, one cannot learn from him. People who have not gotten the basic facts about their life straight will therefore not do the things that make learning from Jesus possible and will never be able to understand the basic points in the lessons to be learned. It is like a mathematics teacher in high school who might say to a student, “Verily, verily I say unto thee, except thou canst do decimals and fractions, thou canst in no wise do algebra.” It is not that the teacher will not allow you to do algebra because you are a bad person; you just won’t be able to do basic algebra if you are not in command of decimals and fractions. So this counting of the cost is not a moaning and groaning session. “Oh how terrible it is that I have to value all of my ‘wonderful’ things (which are probably making life miserable and hopeless anyway) less than I do living in the kingdom! How terrible that I must be prepared to actually surrender them should that be called for!” The counting of the cost is to bring us to the point of clarity and decisiveness. It is to help us to see. Counting the cost is precisely what the man with the pearl and the hidden treasure did. Out of it came their decisiveness and joy. It is decisiveness and joy that are the outcomes of the counting.

To The Times. Steeple Aston 19 April 1974 Sir, I hear on my radio Mr Reg Prentice, of the party which I support, saying to a gathering on education the following: ‘The eleven plus must go, so must selection at twelve plus, at sixteen plus, and any other age.’ What can this mean? How are universities to continue? Are we to have engineers without selection of those who understand mathematics, linguists without selection of those who understand grammar? To many teachers such declarations of policy must seem obscure and astonishing, and to imply the adoption of some quite new philosophy of education which has not, so far as I know, been in this context discussed. It is certainly odd that the Labour Party should wish to promote a process of natural unplanned sorting which will favour the children of rich and educated people, leaving other children at a disadvantage. I thought socialism was concerned with the removal of unfair disadvantages. Surely what we need is a careful reconsideration of how to select, not the radical and dangerous abandonment of the principle of selection. Yours faithfully, Iris Murdoch

A more complex way to understand this is the method used by Hermann Minkowski, Einstein’s former math teacher at the Zurich Polytechnic. Reflecting on Einstein’s work, Minkowski uttered the expression of amazement that every beleaguered student wants to elicit someday from condescending professors. “It came as a tremendous surprise, for in his student days Einstein had been a lazy dog,” Minkowski told physicist Max Born. “He never bothered about mathematics at all.”63 Minkowski decided to give a formal mathematical structure to the theory. His approach was the same one suggested by the time traveler on the first page of H. G. Wells’s great novel The Time Machine, published in 1895: “There are really four dimensions, three which we call the three planes of Space, and a fourth, Time.” Minkowski turned all events into mathematical coordinates in four dimensions, with time as the fourth dimension. This permitted transformations to occur, but the mathematical relationships between the events remained invariant. Minkowski dramatically announced his new mathematical approach in a lecture in 1908. “The views of space and time which I wish to lay before you have sprung from the soil of experimental physics, and therein lies their strength,” he said. “They are radical. Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality.”64 Einstein, who was still not yet enamored of math, at one point described Minkowski’s work as “superfluous learnedness” and joked, “Since the mathematicians have grabbed hold of the theory of relativity, I myself no longer understand it.” But he in fact came to admire Minkowski’s handiwork and wrote a section about it in his popular 1916 book on relativity.

The declining age of learning and of mankind is marked, however, by the rise and rapid progress of the new Platonists. The school of Alexandria silenced those of Athens; and the ancient sects enrolled themselves under the banners of the more fashionable teachers, who recommended their system by the novelty of their method and the austerity of their manners. Several of these masters—Ammonius, Plotinus, Amelius, and Porphyry—were men of profound thought and intense application; but, by mistaking the true object of philosophy, their labors contributed much less to improve than to corrupt human understanding. The knowledge that is suited to our situation and powers, the whole compass of moral, natural and mathematical science, was neglected by the new Platonists; whilst they exhausted their strength in the verbal disputes of metaphysics, attempted to explore the secrets of the invisible world, and studied to reconcile Aristotle with Plato, on subjects of which both of these philosophers were as ignorant as the rest of mankind. Consuming their reason in these deep but unsubstantial meditations, their minds were exposed to illusions of fancy. They flattered themselves that they possessed the secret of disengaging the soul from its corporeal prison, claimed a familiar intercourse withe dæmons and spirits; and, by a very singular revolution, converted the study of philosophy into that of magic. The ancient sages had derided the popular superstition; after disguising its extravagance by the this pretense of allegory, the disciples of Plotinus and Porphyry becomes its most zealous defenders. As they agreed with the Christians in a few mysterious points of faith, they attacked the remainder of their theological system with all the fury of civil war. The new Platonists would scarcely deserve a place in the history of science, but in that of the church the mention of them will very frequently occur.

The modern educational system provides numerous other examples of reality bowing down to written records. When measuring the width of my desk, the yardstick I am using matters little. My desk remains the same width regardless of whether I say it is 200 centimetres or 78.74 inches. However, when bureaucracies measure people, the yardsticks they choose make all the difference. When schools began assessing people according to precise marks, the lives of millions of students and teachers changed dramatically. Marks are a relatively new invention. Hunter-gatherers were never marked for their achievements, and even thousands of years after the Agricultural Revolution, few educational establishments used precise marks. A medieval apprentice cobbler did not receive at the end of the year a piece of paper saying he has got an A on shoelaces but a C minus on buckles. An undergraduate in Shakespeare’s day left Oxford with one of only two possible results – with a degree, or without one. Nobody thought of giving one student a final mark of 74 and another student 88.6 Credit 1.24 24. A European map of Africa from the mid-nineteenth century. The Europeans knew very little about the African interior, which did not prevent them from dividing the continent and drawing its borders. Only the mass educational systems of the industrial age began using precise marks on a regular basis. Since both factories and government ministries became accustomed to thinking in the language of numbers, schools followed suit. They started to gauge the worth of each student according to his or her average mark, whereas the worth of each teacher and principal was judged according to the school’s overall average. Once bureaucrats adopted this yardstick, reality was transformed. Originally, schools were supposed to focus on enlightening and educating students, and marks were merely a means of measuring success. But naturally enough, schools soon began focusing on getting high marks. As every child, teacher and inspector knows, the skills required to get high marks in an exam are not the same as a true understanding of literature, biology or mathematics. Every child, teacher and inspector also knows that when forced to choose between the two, most schools will go for the marks.

The fact that no one made demands on her knowledge in her special field was lucky for Simochka. Not only she but many of her girlfriends had graduated from the institute without any such knowledge. There were many reasons for this. The young girls had come from high schools with very little grounding in mathematics and physics. They had learned in the upper grades that at faculty council meetings the school director had scolded the teachers for giving out failing marks, and that even if a pupil didn't study at all he received a diploma. In the institute, when they found time to sit down to study, they made their way through the mathematics and radio-technology as through a dense pine forest. But more often there was no time at all. Every fall for a month or more the students were taken to collective farms to harvest potatoes. For this reason, they had to attend lectures for eight and ten hours a day all the rest of the year, leaving no time to study their course work. On Monday evenings there was political indoctrination. Once a week a meeting of some kind was obligatory. Then one had to do socially useful work, too: issue bulletins, organize concerts, and it was also necessary to help at home, to shop, to wash, to dress. And what about the movies? And the theater? And the club? If a girl didn't have some fun and dance a bit during her student years, when would she do so afterward? For their examinations Simochka and her girlfriends wrote many cribs, which they hid in those sections of female clothing denied to males, and at the exams they pulled out the one the needed, smoothed it out, and turned it in as a work sheet. The examiners, of course, could have easily discovered the women students' ignorance, but they themselves were overburdened with committee meetings, assemblies, a variety of plans and reports to the dean's office and to the rector. It was hard on them to have to give an examination a second time. Besides, when their students failed, the examiners were reprimanded as if the failures were spoiled goods in a production process—according to the well-known theory that there are no bad pupils, only bad teachers. Therefore the examiners did not try to trip the students up but, in fact, attempted to get them through the examination with as good results as possible.

In 1933 things were still being taught in the higher educational establishments which had been proven by science to be false as long ago as 1899. The young man who wishes to keep abreast of the times, therefore, had to accept a double load on his unfortunate brain. In a hundred years' time, the number of people wearing spectacles, and the size of the human brain, will both have increased considerably; but the people will be none the more intelligent. What they will look like, with their enormous, bulging heads, it is better not to try to imagine; they will probably be quite content with their own appearance, but if things continue in the manner predicted by the scientists, I think we can count ourselves lucky that we shall not live to see them! When I was a schoolboy, I did all I could to get out into the open air as much as possible—my school reports bear witness to that ! In spite of this, I grew up into a reasonably intelligent young man, I developed along very normal lines, and I learnt a lot of things of which my schoolfellows learnt nothing. In short, our system of education is the exact opposite of that practised in the gymnasia of ancient days. The Greek of the golden age sought a harmonious education; we succeed only in producing intellectual monsters. Without the introduction of conscription, we should have fallen into complete decadence, and it is thanks to this universal military service that the fatal process has been arrested. This I regard as one of the greatest events in history. When I recall my masters at school, I realise that half of them were abnormal; and the greater the distance from which I look back on them, the stronger is my conviction that I am quite right. The primary task of education is to train the brain of the young. It is quite impossible to recognise the potential aspirations of a child of ten. In old days teachers strove always to seek out each pupil's weak point, and by exposing and dwelling on it, they successfully killed the child's self-confidence. Had they, on the contrary, striven to find the direction in which each pupil's talents lay, and then concentrated on the development of those talents, they would have furthered education in its true sense. Instead, they sought mass-production by means of endless generalisations. A child who could not solve a mathematical equation, they said, would do no good in life. It is a wonder that they did not prophesy that he would come to a bad and shameful end! Have things changed much to-day, I wonder? I am not sure, and many of the things I see around me incline me to the opinion that they have not.

Even male children of affluent white families think that history as taught in high school is “too neat and rosy.” 6 African American, Native American, and Latino students view history with a special dislike. They also learn history especially poorly. Students of color do only slightly worse than white students in mathematics. If you’ll pardon my grammar, nonwhite students do more worse in English and most worse in history.7 Something intriguing is going on here: surely history is not more difficult for minorities than trigonometry or Faulkner. Students don’t even know they are alienated, only that they “don’t like social studies” or “aren’t any good at history.” In college, most students of color give history departments a wide berth. Many history teachers perceive the low morale in their classrooms. If they have a lot of time, light domestic responsibilities, sufficient resources, and a flexible principal, some teachers respond by abandoning the overstuffed textbooks and reinventing their American history courses. All too many teachers grow disheartened and settle for less. At least dimly aware that their students are not requiting their own love of history, these teachers withdraw some of their energy from their courses. Gradually they end up going through the motions, staying ahead of their students in the textbooks, covering only material that will appear on the next test. College teachers in most disciplines are happy when their students have had significant exposure to the subject before college. Not teachers in history. History professors in college routinely put down high school history courses. A colleague of mine calls his survey of American history “Iconoclasm I and II,” because he sees his job as disabusing his charges of what they learned in high school to make room for more accurate information. In no other field does this happen. Mathematics professors, for instance, know that non-Euclidean geometry is rarely taught in high school, but they don’t assume that Euclidean geometry was mistaught. Professors of English literature don’t presume that Romeo and Juliet was misunderstood in high school. Indeed, history is the only field in which the more courses students take, the stupider they become. Perhaps I do not need to convince you that American history is important. More than any other topic, it is about us. Whether one deems our present society wondrous or awful or both, history reveals how we arrived at this point. Understanding our past is central to our ability to understand ourselves and the world around us. We need to know our history, and according to sociologist C. Wright Mills, we know we do.8

There is a powerful difference between picturing an image that does not relate to you, and recalling something specific that you are connected with. When memory experts memorize something that is idle or vain, they’re only attempting to picture an image that does not relate to them. They don’t have to daily apply the digits of mathematical pi to live by them. They don’t have to apply the Declaration of Independence in a practical day to day life. As if Americans say when living life, “As it is written in the Declaration of Independence.” So they teach you how to memorize things idly, where as it is better to make a heartfelt connection with the word of God. Therefore Beloved, read the scriptures first and have a full connection with it before you memorize. As we stated before, it is good to memorize what is speaking out to you in the scriptures. Where God is speaking to you and teaching you in the word is the best place to memorize. It makes an emotional connection with you, and it’s applicable to your life in the here and now. Seeing that it is immediately applicable to your life you’ll be able to make an emotional connection with what you are memorizing. As a result, when you are past these teachings and have full understanding, even years in the future you’ll still remember the scripture because it made an emotional connection with you. Much like reminiscing over the cottage experience, every time the topic is brought up you’ll have waves of scripture rushing to you for practical application. Therefore in this method we are seeking to make memorizing the scriptures an experience and not merely a task or a goal for godliness. When it is relevant to experiences there is more for the mind to grasp onto the memory with thereby giving greater longevity to the memory itself. Similar to the peg method where you create an image for the mind to have more to grasp onto, you are using an already existing “image” so to speak, that the mind will grasp onto harder. But why does it grasp harder? Because it isn’t something silly thought of by oneself but it is an ongoing experience that led to a reminiscent memory. Therefore memorize what God is speaking to you and what has strong meaning to you. Whatever jumps out at you from the pages is what the Lord wants you to be memorizing. Therefore as a good pupil and good student, memorize what the Lord your Teacher is giving you to memorize. In school we do not memorize anything but what the teacher gives us, otherwise it would serve no purpose. Likewise it serves a greater purpose to memorize what God is giving you in the here and now, versus memorizing something that is not applying to you at this moment. Yes, all the word of God applies to your life and it always will. But certain things are speaking true to the immediate lesson in life and thus the scriptures speak out to you, and seem alive. Therefore memorize the words that are alive and you will have a continuous living memory of the word of God.

Steele found, for example, that if he could convince women who took difficult mathematics examinations that everyone connected with the test assumed they would perform as well as men, that they did.

In the online math class, there was almost no meaningful student/teacher or student/student interaction. To equate this type of online learning with a real-world classroom experience is a major stretch.

Now it makes sense, for example, if the children are taking a vocabulary test of 100 words, and one of the kids misses thirteen of them, to give him an 87 percent. But we go far beyond this. A student writes an essay on a sunset, let us say, and the teacher writes 87 percent at the top of that paper. What he is saying, in effect, is that there is a mathematical metaphor operative here. The figure of 87 is to 100 what this submitted essay is . . . to what? What on earth is this supposed to mean?

A key ingredient in appreciating what mathematics is about is to realize that it is concerned with ideas, understanding, and communication more than it is with any specific brand of symbols....It is almost as if ideas set in mathematical form melt and become liquid and just as rivers can, from the most humble beginnings, flow for thousands of miles, through the most varied topography bringing nourishment and life with them wherever they go, so too can ideas cast in mathematical form flow far from their original sources, along well-defined paths, electrifying and dramatically affecting much of what they touch. pp. xii - xiii.

Through the judicious employment of symbols, diagrams, and calculations, mathematics enables us to acquire significant facts about extremely significant things (universal laws, even), not by first forging out into the cosmos with teams of scientists, but rather from the comforts and confines of coffee tables in our living rooms! p. 72

In mathematics, by placing our fingers on a given problem, no matter how trite or pedestrian it apparently seems, we may end up measuring the pulse of the universe. p. 119

Primary Science confused her (if man descended form monkeys, how com the monkey that loved with Mama Boy near the church, and has lived with her for so long as anyone remembered, has not evolved and become human?) and grammar baffled her even more (she could never grasp why it was 'Run Run Ran' but 'See Saw Seen'). When she was caned by her teacher for failing to conjugate the verb Fear (she had said 'Fear Fore Forn'), she decided that school was not for her. There was no logic in what she read, all the teaching seemed designed both to compound her problems and to confound her. When she asked questions, her teachers told her off for being disruptive. How could it be 'Tear Tore Torn'? Change the first letter and the rules changed completely! How was she supposed to remember all of that? 'See Saw Sawn'. It was an unrealistic demand. And on top of that there was the illogicality of mathematics to deal with. Finding solutions to abstract questions that had nothing to do with real life. She did not see how any of this would help her, how it would help anybody really.

When we write nowadays that six million perished during the Holocaust, the number is awesome, abstract; it is hard for the mind to comprehend that number, yet each one was a world. Can we fathom what we lost, what the world lost? NOTE: For years only her small group of friends knew about the existence of the poems. Her two close friends, who kept the manuscripts, and her former mathematics teacher from tenth grade, Hersh Segal, got together and published the Anthology - Blütenlese in Rechovot, Israel, in 1976. This privately financed publication reached a larger public and her name and fame spread, but very slowly. A second edition was published by the Diaspora Research Institute, Tel Aviv University, in 1979.

A classroom library containing both fiction and non-fiction books should be located centrally, and include newspapers, magazines, telephone books, restaurant menus, etc. Teachers should also integrate literacy across the curriculum by reading and assigning texts that support their learning units in subjects such as mathematics, sciences, and social studies.

The little boy met with his teachers and studied history, geography, mathematics, music, strategy, politics, dance, falconry, and all the things an emperor has to know so that later on he can do everything that makes him feel that doing it makes him the emperor.

Some of our findings were surprising in that they challenge some popularly held beliefs about what makes a teacher effective. For example, style of organization for mathematics teaching was not a predictor of how effective teachers were. Whole-class ‘question-and-answer’ teaching styles were used by both highly effective and comparatively less effective teachers. Similarly, individualized work and small-group work were used by teachers across the range of effectiveness. At the school level, setting across an age group was used in schools with both high and low proportions of highly effective teachers. The same published mathematics schemes were used by highly effective and comparatively much less effective teachers. Our findings also raised questions about the sort of mathematical knowledge teachers need in order to be effective. Despite what might be expected, being highly effective was not positively associated with higher levels of qualifications in mathematics. The amount of continuing professional development in mathematics education that teachers had undertaken was a better predictor of their effectiveness than the level to which they had formally studied mathematics.

Really, there was only one problem with Mr. Davis, as far as Gregory was concerned; He taught math.

Close to the left of the school entrance gate are uncompleted rooms which lack everything else but a base foundation and load-bearing walls. And as for the toilets - in all six years of my life as a student, I never used them once, as they were hardly ever clean. The sight of decayed excrement and worms crawling all over the floors and toilet seats was enough to make one lose interest in passing out waste. One fateful day, I held back the faeces in my anus and walked the mile-long journey home from school. It was painful yet needful as anything would be better than having to do my business in a smelly worm-infested room. No sane individual would subject their own human waste to that kind of treatment. We are all kidding ourselves - OGS is not a school, it is an extension of Kirikiri prison!

T he silence in the room is unnerving. There’s no mistaking it, all the teachers in this room are in trouble. We are as helpless passengers on a cruise ship whose captain is the principal, and he alone is navigating this treacherous sea and taking us to the island of judgement. Our options are limited - we can either stay for the ride or jump ship but judging from the earlier exchange between Adeshina and ‘Captain’ principal, such a move would be suicidal.

To focus on psychology, in which I’ve been working for the last 50 years, I think the trouble lies in the mediocrity of the researchers and teachers. The whole subject is a very difficult one. McDougall said that the trouble with psychology is that it is too difficult for psychologists. Quite advanced mathematics – actually quite beautiful mathematics, seemingly beyond the comprehension of most psychologists today — is necessary to solve the next issues awaiting us. We’ve got to get more acute selection in psychology, and take it out of the hands of the do-gooders and the social workers and really make a science of it.

The Pala period [between the eighth and the twelfth century CE], in particular, saw several monasteries emerge in what is now modern Bengal and Bihar, five of which—Vikramashila, Nalanda, Somapura Mahavihara, Odantapuri, and Jaggadala—were premier educational institutions which created a coordinated network amongst themselves under Indian rulers. Nalanda University, which enjoyed international renown when Oxford and Cambridge were not even gleams in their founders’ eyes, employed 2,000 teachers and housed 10,000 students in a remarkable campus that featured a library nine storeys tall. It is said that monks would hand-copy documents and books which would then become part of private collections of individual scholars. The university opened its doors to students from countries ranging from Korea, Japan, China, Tibet, and Indonesia in the east to Persia and Turkey in the west, studying subjects which included the fine arts, medicine, mathematics, astronomy, politics and the art of war.

NCTM reforms and similar programs lives on.!11 In the final analysis, it does not really flow from a conviction that such ideas promote superior teaching and learning of elementary and high-school mathematics. Rather, hypertrophied political piety lies at the root. Constructivism and its variants offer convenient pretexts for the display of self-perceived political virtue. They make it possible for well-meaning math teachers, and the well-meaning ed-school theorists under whom they study, to think of themselves as activists addressing urgent political and social problems through their educational practices.

The Javits legislation, reauthorized in 2001 as part of the No Child Left Behind Act (PL 107–110), was funded at $11.14 million in fiscal year 2004. Congress approved an appropriation of approximately $7.6 million for the Javits program in fiscal year 2008. The Javits funding was eliminated in 2010, curtailing research projects not yet completed. After a gap in funding, the Javits Act was funded again in 2013, and funding reached $12 million in 2016, the highest level in the history of the Javits Act. The National Center for Research on Gifted Education was also funded. Located at the University of Connecticut, the center has a partnership with the University of Virginia. In 2018, the Javits funding continued at $12 million. Academic standards have become increasingly important in the twenty-first century. The National Association for Gifted Children (2010) issued the Pre-K–Grade 12 Gifted Programming Standards. These standards focus on student outcomes and encourage collaboration among general education teachers, special educators, and teachers of the gifted in an effort to assist students in achieving projected outcomes. In 2010, the National Governors Association Center for Best Practices in conjunction with the Council of Chief State School Officers put forth the Common Core State Standards Initiative (2019), which provided standards in mathematics and English/language arts for Grades K–12. In 2013, the Next Generation Science Standards (2019) became available and were adopted by several states.

Unit circle is one of the important math concepts that every student must learn and understand. There are numerous concepts related to Trigonometry and geometry that needs to understand basics before solving the problems. Unit circle is known as the foundation of projectile motion, sine, cosine, tangents, degrees and radians. If you are learning the concept of geometry and trigonometry then you must have a unit circle chart as reference sheet. Most of the school teachers us this sheet while teaching the concepts of applied mathematics. This basic circle will be helpful throughout your life. It is necessary to learn this Blank Unit Circle Printable by heart and to practice it regularly for a solid foundation.

Dorothy Vaughan, Mary Jackson, Katherine Johnson, and Christine Darden loved math. As children, they showed special skill in arithmetic, and they went on to study mathematics in college. After graduation they worked as teachers before going to work as “computers,” or mathematicians, for the government’s air and space program.

German teachers have shown how the very plays of children can be made instrumental in conveying to the childish mind some concrete knowledge in both geometry and mathematics. The children who have made the squires of the theorem of Pythagoras out of pieces of coloured cardboard, will not look at the theorem, when it comes in geometry, as on a mere instrument of torture devised by the teachers; and the less so if they apply it as the carpenters do. Complicated problems of arithmetic, which so much harassed us in our boyhood, are easily solved by children seven and eight years old if they are put in the shape of interesting puzzles. And if the Kindergarten — German teachers often make of it a kind of barrack in which each movement of the child is regulated beforehand — has often become a small prison for the little ones, the idea which presided at its foundation is nevertheless true. In fact, it is almost impossible to imagine, without having tried it, how many sound notions of nature, habits of classification, and taste for natural sciences can be conveyed to the children’s minds; and, if a series of concentric courses adapted to the various phases of development of the human being were generally accepted in education, the first series in all sciences, save sociology, could be taught before the age of ten or twelve, so as to give a general idea of the universe, the earth and its inhabitants, the chief physical, chemical, zoological, and botanical phenomena, leaving the discovery of the laws of those phenomena to the next series of deeper and more specialised studies.

The conviction lodged in her head, that American children learned nothing in elementary school, and it hardened when he told her that his teacher sometimes gave out homework coupons; if you got a homework coupon, then you could skip one day of homework. Circles, homework coupons, what foolishness would she next hear? And so she began to teach him mathematics—she called it “maths” and he called it “math” and so they agreed not to shorten the word. She could not think, now, of that summer without thinking of long division, of Dike’s brows furrowed in confusion as they sat side by side at the dining table, of her swings from bribing him to shouting at him. Okay, try it one more time and you can have ice cream. You’re not going to play unless you get them all right. Later, when he was older, he would say that he found mathematics easy because of her summer of torturing him. “You must mean summer of tutoring,” she would say in what became a familiar joke that, like comfort food, they would reach for from time to time.

...my colleagues upstairs, in their huge ground floor space with their big windows and perfectly ordered shelves, they're so comfortable sitting there alongside their coffee machines, that they actually talk out loud about how nice it would be in a library without readers. Like some teacher's dream of a school with no pupils. But what would be the point of us then? Oh, yes, it would be in perfect order. A mathematical masterpiece, really shipshape, our library. But what would be the point if nobody came along to disturb it? ...that's all I do want, to be asked a question, to be disturbed, just a bit.

Teachers greatly influence how students perceive and approach struggle in the mathematics classroom. Even young students can learn to value struggle as an expected and natural part of learning, as demonstrated by the class motto of one first-grade math class: If you are not struggling, you are not learning. Teachers must accept that struggle is important to students' learning of mathematics, convey this message to students, and provide time for them to try to work through their uncertainties. Unfortunately, this may not be enough, since some students will still simply shut down in the face of frustration, proclaim, 'I don't know,' and give up. Dweck (2006) has shown that students with a fixed mindset--that is, those who believe that intelligence (especially math ability) is an innate trait--are more likely to give up when they encounter difficulties because they believe that learning mathematics should come naturally. By contrast, students with a growth mindset--that is, those who believe that intelligence can be developed through effort--are likely to persevere through a struggle because they see challenging work as an opportunity to learn and grow.

The Great Pyramid, that monument to spirituality that the Agashan Teachers hold in such high esteem, is built according to the principles of Pi and Phi.

The case study as an evolving system has the following components. First, it views creative work as multi-faceted. So, in constructing a case study of a creative work, one must distill the facets that are relevant and construct the case study based on the chosen facets. Some facets that can be used to construct an evolving system case study are: (1) uniqueness of the work; (2) a narrative of what the creator achieved; (3) systems of belief; (4) multiple time-scales (construct the time-scales involved in the production of the creative work); (5) problem solving; and (6) contextual frame such as family, schooling, and teacher’s influences (Gruber & Wallace, 2000). In summary, constructing a case study of a creative work as an evolving system entails incorporating the many facets suggested by Gruber & Wallace (2000). One could also evaluate a case study involving creative work by looking for the above mentioned facets.

You are to be in all things regulated and governed,’ said the gentleman, ‘by fact. We hope to have, before long, a board of fact, composed of commissioners of fact, who will force the people to be a people of fact, and of nothing but fact. You must discard the word Fancy altogether. You have nothing to do with it. You are not to have, in any object of use or ornament, what would be a contradiction in fact. You don’t walk upon flowers in fact; you cannot be allowed to walk upon flowers in carpets. You don’t find that foreign birds and butterflies come and perch upon your crockery; you cannot be permitted to paint foreign birds and butterflies upon your crockery. You never meet with quadrupeds going up and down walls; you must not have quadrupeds represented upon walls. You must use,’ said the gentleman, ‘for all these purposes, combinations and modifications (in primary colours) of mathematical figures which are susceptible of proof and demonstration. This is the new discovery. This is fact. This is taste.

One of the more interesting paradoxes about knowledge is called the surprise-test paradox. A teacher announces that there will be a surprise test in the forthcoming week. The last day of class is Friday of that week. What day can the surprise test happen on? If the test is going to be on Friday, then after school on Thursday night the students will already know that the test is on Friday and it will not be a surprise test. So the test cannot happen on Friday. Since this was purely logical reasoning, everyone knows this. Can the test be on Thursday? After class on Wednesday night the students can deduce that since the test has not happened already and it cannot be on Friday, it must be on Thursday. But again, since they know that it must be on Thursday, it will no longer be a surprise test. So the test cannot occur on Thursday or Friday. We can continue reasoning in the same way and conclude that the test cannot happen on Wednesday, Tuesday, or Monday. When exactly will this surprise test occur? Logic has shown us that a teacher cannot give a surprise test within a given time interval. This is a paradox because it goes against the obvious fact that teachers have been torturing students with surprise tests for millennia. It

What that extra time does is allow for a more relaxed atmosphere,” Corcoran said, after the class was over. “I find that the problem with math education is the sink-or-swim approach. Everything is rapid fire, and the kids who get it first are the ones who are rewarded. So there comes to be a feeling that there are people who can do math and there are people who aren’t math people. I think that extended amount of time gives you the chance as a teacher to explain things, and more time for the kids to sit and digest everything that’s going on—to review, to do things at a much slower pace. It seems counterintuitive but we do things at a slower pace and as a result we get through a lot more. There’s a lot more retention, better understanding of the material. It lets me be a little bit more relaxed. We have time to have games. Kids can ask any questions they want, and if I’m explaining something, I don’t feel pressed for time. I can go back over material and not feel time pressure.” The extra time gave Corcoran the chance to make mathematics meaningful: to let his students see the clear relationship between effort and reward.

The reader terrorized by mathematics (persuaded by incompetent teachers that "I can't understand that stuff") need not panic.

Sophie Germain had taught herself calculus at a young age. The daughter of a wealthy family, she had become entranced by mathematics after reading a book about Archimedes in her father’s library. When her parents found out that she loved mathematics and was staying up late at night to work on it, they took away her candles, left her fire unlit, and confiscated her nightgowns. Sophie persisted. She wrapped herself in quilts and worked by the light of stolen candles. Eventually her family relented and gave her their blessing. Germain, like all women of her era, was not permitted to attend university, so she continued to teach herself, in some cases by obtaining lecture notes from the courses at the nearby École Polytechnique using the name Monsieur Antoine-August Le Blanc, a student who had left the school. Unaware of his departure, academy administrators continued to print lecture notes and problem sets for him. She submitted work under his name until one of the school’s teachers, the great Lagrange, noticed the remarkable improvement in Monsieur Le Blanc’s previously abysmal performance. Lagrange requested a meeting with Le Blanc and was delighted and astonished to discover her true identity.

Yet the boy himself was utterly unself-conscious, and the American observers wondered why they felt worse than he did. “Our culture exacts a great cost psychologically for making a mistake,” Stigler recalled, “whereas in Japan, it doesn’t seem to be that way. In Japan, mistakes, error, confusion [are] all just a natural part of the learning process.”21 (The boy eventually mastered the problem, to the cheers of his classmates.) The researchers also found that American parents, teachers, and children were far more likely than their Japanese and Chinese counterparts to believe that mathematical ability is innate; if you have it, you don’t have to work hard, and if you don’t have it, there’s no point in trying. In contrast, most Asians regard math success like achievement in any other domain; it’s a matter of persistence and plain hard work. Of course you will make mistakes as you go along; that’s how you learn and improve.

This proposal I at once adopted, and accordingly found myself one morning at a small station of the Moscow Railway, endeavouring to explain to a peasant in sheep's clothing that I wished to be conveyed to Ivanofka, the village where my future teacher lived. At that time I still spoke Russian in a very fragmentary and confused way—pretty much as Spanish cows are popularly supposed to speak French. My first remark therefore being literally interpreted, was—"Ivanofka. Horses. You can?" The point of interrogation was expressed by a simultaneous raising of the voice and the eyebrows. "Ivanofka?" cried the peasant, in an interrogatory tone of voice. In Russia, as in other countries, the peasantry when speaking with strangers like to repeat questions, apparently for the purpose of gaining time. "Ivanofka," I replied. "Now?" "Now!" After some reflection the peasant nodded and said something which I did not understand, but which I assumed to mean that he was open to consider proposals for transporting me to my destination. "Roubles. How many?" To judge by the knitting of the brows and the scratching of the head, I should say that that question gave occasion to a very abstruse mathematical calculation. Gradually the look of concentrated attention gave place to an expression such as children assume when they endeavour to get a parental decision reversed by means of coaxing. Then came a stream of soft words which were to me utterly unintelligible.

Newton blew away any dusty talk of natures and purposes, revealing what lay underneath: a crisp, rigorous mathematical formalism with which teachers continue to torment students to this very day.