Geometry Teacher Quotes

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

The waitress who handed her a menu was dressed in a black miniskirt and fishnet hose, but she had the face of a geometry teacher.

...there's something about telling other people what to ignore that just doesn't work for me. Especially things we shouldn't be ignoring. Kid bullying you at school? Ignore him. Girl passing rumors? Ignore her. Eighth-grade teacher pinch your friend's ass? Ignore it. Sexist geometry teacher says girls shouldn't go to college because they will only ever pop out babies and get fat? Ignore him. Hear that a girl in your class is being abused by her stepfather and had to go to the clinic? Hear she's bringing her mother's pills to school and selling them to pay for it? Ignore. Ignore. Ignore. Mind your own business. Don't make waves. Fly under the radar. It's just one of those things, Vera. I'm sorry, but I don't get it. If we're supposed to ignore everything that's wrong with our lives, then I can't see how we'll ever make things right.

Don’t forget to be specific…Details. Put in all the details. The boys appreciate all that detailed daily life sh*t they don’t get anymore. If you’ve got a teacher you’re hot for, tell ‘em what her hair looks like, what her legs look like, what she eats for lunch. If she’s teaching you geometry, tell ‘em how she draws a bloody triangle on the blackboard. If you went down the shop for a bag of sweets yesterday, did you ride your pushee? Did you go by foot? Did you see a rainbow along the way? Did you buy gobstoppers or clinkers or caramels? If you had a good meat pie last week was it steak and peas or curry or mushroom and beef? You catchin’ my drift? Details.

unsolicited advice to adolescent girls with crooked teeth and pink hair When your mother hits you, do not strike back. When the boys call asking your cup size, say A, hang up. When he says you gave him blue balls, say you’re welcome. When a girl with thick black curls who smells like bubble gum stops you in a stairwell to ask if you’re a boy, explain that you keep your hair short so she won’t have anything to grab when you head-butt her. Then head-butt her. When a guidance counselor teases you for handed-down jeans, do not turn red. When you have sex for the second time and there is no condom, do not convince yourself that screwing between layers of underwear will soak up the semen. When your geometry teacher posts a banner reading: “Learn math or go home and learn how to be a Momma,” do not take your first feminist stand by leaving the classroom. When the boy you have a crush on is sent to detention, go home. When your mother hits you, do not strike back. When the boy with the blue mohawk swallows your heart and opens his wrists, hide the knives, bleach the bathtub, pour out the vodka. Every time. When the skinhead girls jump you in a bathroom stall, swing, curse, kick, do not turn red. When a boy you think you love delivers the first black eye, use a screw driver, a beer bottle, your two good hands. When your father locks the door, break the window. When a college professor writes you poetry and whispers about your tight little ass, do not take it as a compliment, do not wait, call the Dean, call his wife. When a boy with good manners and a thirst for Budweiser proposes, say no. When your mother hits you, do not strike back. When the boys tell you how good you smell, do not doubt them, do not turn red. When your brother tells you he is gay, pretend you already know. When the girl on the subway curses you because your tee shirt reads: “I fucked your boyfriend,” assure her that it is not true. When your dog pees the rug, kiss her, apologize for being late. When he refuses to stay the night because you live in Jersey City, do not move. When he refuses to stay the night because you live in Harlem, do not move. When he refuses to stay the night because your air conditioner is broken, leave him. When he refuses to keep a toothbrush at your apartment, leave him. When you find the toothbrush you keep at his apartment hidden in the closet, leave him. Do not regret this. Do not turn red. When your mother hits you, do not strike back.

When I began my physical studies [in Munich in 1874] and sought advice from my venerable teacher Philipp von Jolly...he portrayed to me physics as a highly developed, almost fully matured science...Possibly in one or another nook there would perhaps be a dust particle or a small bubble to be examined and classified, but the system as a whole stood there fairly secured, and theoretical physics approached visibly that degree of perfection which, for example, geometry has had already for centuries.

You might ask why we cannot teach physics by just giving the basic laws on page one and then showing how they work in all possible circumstances, as we do in Euclidean geometry, where we state the axioms and then make all sorts of deductions. (So,

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.

On the SB5 Stanford-Binet intelligence test Isaiah’s reasoning scores were near genius levels. His abilities came naturally but were honed in his math classes. He was formally introduced to inductive reasoning in geometry, a tenth-grade subject he took in the eighth. His teacher, Mrs. Washington, was a severe woman who looked to be all gristle underneath her brightly colored pantsuits. Lavender, Kelly green, peach. She talked to the class like somebody had tricked her into it. “All

It's one thing if he wants to ignore it. I guess that's fine. I mean, I ignore plenty of stuff, like school spirit days and the dirty looks I get from the Detentionheads while I try to slink through the halls unnoticed. But there's something about telling other people what to ignore that just doesn't work for me. Especially things we shouldn't be ignoring. Kid bullying you at school? Ignore him. Girl passing rumors? Ignore her. Eighth grade teacher pinch your friend's ass? Ignore it. Sexist geometry teacher says girls shouldn't go to college because they will only ever pop out babies and get fat? Ignore him. Hear that a girl in my class is being abused by her stepfather and had to go to the clinic? Hear she's bringing her mother's pills to school and selling them to pay for it? Ignore, ignore, ignore. Mind your own business. Don't make waves. Fly under the radar. It's just one of those things, Vera. I'm sorry, but I don't get it. If we're supposed to ignore everything that's wrong with our lives, then I can't see how we'll ever make things right.

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.

I try to tell the teacher, you know. I don't give a fuck about geometry or English. Like I'm probably going to drive a truck or something when I get out of school. Join the army or something simple. I'm sure in the army they're all going to be wondering what an acute angle is. I'm sure I'll make lots of friends driving my truck because I can diagram some lousy goddamn sentence. And then after school I'm free, right? What's that mean? I go down to the bowling alley or the shopping mall with my friends. We scope the grils, smoke a little doobidge, maybe a tab of acid every now and then. But that's not really living, is it? I mean, if that's living, then excuse me right now. I'll go out and put a bullet in the old brainpan. But if that's not all there is, right, well, maybe there's something I could do a little less radical, like, you know. I don't mind life or anything--I'm perfectly willing to give it a try. So what the hell, I figured. I'm sick of school, drugs, this goddamn oppressive house of Ethel's and all. Maybe it's time I experimented a little more with my life, took a few more chances. So that's when I decided to become a warlock. To master the satanic arts of black magic. Devil worshiping, for you laymen. I want to master what they call the black arts.

An Act for establishing religious Freedom. Section 1 Whereas, Almighty God hath created the mind free; That all attempts to influence it by temporal punishments or burthens, or by civil incapacitations tend only to beget habits of hypocrisy and meanness, and therefore are a departure from the plan of the holy author of our religion, who being Lord, both of body and mind yet chose not to propagate it by coercions on either, as was in his Almighty power to do, That the impious presumption of legislators and rulers, civil as well as ecclesiastical, who, being themselves but fallible and uninspired men have assumed dominion over the faith of others, setting up their own opinions and modes of thinking as the only true and infallible, and as such endeavouring to impose them on others, hath established and maintained false religions over the greatest part of the world and through all time; That to compel a man to furnish contributions of money for the propagation of opinions, which he disbelieves is sinful and tyrannical; That even the forcing him to support this or that teacher of his own religious persuasion is depriving him of the comfortable liberty of giving his contributions to the particular pastor, whose morals he would make his pattern, and whose powers he feels most persuasive to righteousness, and is withdrawing from the Ministry those temporary rewards, which, proceeding from an approbation of their personal conduct are an additional incitement to earnest and unremitting labours for the instruction of mankind; That our civil rights have no dependence on our religious opinions any more than our opinions in physics or geometry, That therefore the proscribing any citizen as unworthy the public confidence, by laying upon him an incapacity of being called to offices of trust and emolument, unless he profess or renounce this or that religious opinion, is depriving him injuriously of those privileges and advantages, to which, in common with his fellow citizens, he has a natural right, That it tends only to corrupt the principles of that very Religion it is meant to encourage, by bribing with a monopoly of worldly honours and emoluments those who will externally profess and conform to it; That though indeed, these are criminal who do not withstand such temptation, yet neither are those innocent who lay the bait in their way; That to suffer the civil magistrate to intrude his powers into the field of opinion and to restrain the profession or propagation of principles on supposition of their ill tendency is a dangerous fallacy which at once destroys all religious liberty because he being of course judge of that tendency will make his opinions the rule of judgment and approve or condemn the sentiments of others only as they shall square with or differ from his own; That it is time enough for the rightful purposes of civil government, for its officers to interfere when principles break out into overt acts against peace and good order; And finally, that Truth is great, and will prevail if left to herself, that she is the proper and sufficient antagonist to error, and has nothing to fear from the conflict, unless by human interposition disarmed of her natural weapons free argument and debate, errors ceasing to be dangerous when it is permitted freely to contradict them.

Kids aren’t learning science right these days! The teachers teach it like it’s just supposed to be useful, like, here, learn this geometry so you can design a building, here, learn this chemistry so you can make a plastic bag. Of course kids don’t like it! No kid comes home from school and says, ‘I want to make plastic bags when I grow up!’ We already have plastic bags, and comfy chairs, and flying cars, we’ve had them for centuries, and they aren’t getting better because they work already so no one’s interested in replacing them, just making them cheaper, or with more games. That isn’t science! Science is figuring out where the universe is going! Science is noticing that the ants crawling up the picnic table like your sandwich better than your ba’sib’s and asking, ‘Why?’ Not ‘How is this useful?’ not ‘Can I make this into a plastic bag?’ but ‘Why?

When they enter the classroom, however, they may well discover that the algorithms have got there first. Companies such as Mindojo are developing interactive algorithms that will not only teach me maths, physics and history, but will also simultaneously study me and get to know exactly who I am. Digital teachers will closely monitor every answer I give, and how long it took me to give it. Over time, they will discern my unique weaknesses as well as my strengths and will identify what gets me excited, and what makes my eyelids droop. They could teach me thermodynamics or geometry in a way that suits my personality type, even if that particular method doesn’t suit 99 per cent of the other pupils. And these digital teachers will never lose their patience, never shout at me, and never go on strike. It remains unclear, however, why on earth I would need to know thermodynamics or geometry in a world containing such intelligent computer programs.

When they enter the classroom, however, they may well discover that the algorithms have got there first. Companies such as Mindojo are developing interactive algorithms that will not only teach me maths, physics and history, but will also simultaneously study me and get to know exactly who I am. Digital teachers will closely monitor every answer I give, and how long it took me to give it. Over time, they will discern my unique weaknesses as well as my strengths and will identify what gets me excited, and what makes my eyelids droop. They could teach me thermodynamics or geometry in a way that suits my personality type, even if that particular method doesn’t suit 99 per cent of the other pupils. And these digital teachers will never lose their patience, never shout at me, and never go on strike. It remains unclear, however, why on earth I would need to know thermodynamics or geometry in a world containing such intelligent computer programs.7

Look around on your next plane trip. The iPad is the new pacifier for babies and toddlers… Parents and other passengers read on Kindles… Unbeknownst to most of us, an invisible, game-changing transformation links everyone in this picture: the neuronal circuit that underlies the brain’s ability to read is subtly, rapidly changing… As work in neurosciences indicates, the acquisition of literacy necessitated a new circuit in our species’ brain more than 6,000 years ago… My research depicts how the present reading brain enables the development of some of our most important intellectual and affective processes: internalized knowledge, analogical reasoning, and inference; perspective-taking and empathy; critical analysis and the generation of insight. Research surfacing in many parts of the world now cautions that each of these essential “deep reading” processes may be under threat as we move into digital-based modes of reading… Increasing reports from educators and from researchers in psychology and the humanities bear this out. English literature scholar and teacher Mark Edmundson describes how many college students actively avoid the classic literature of the 19thand 20th centuries because they no longer have the patience to read longer, denser, more difficult texts. We should be less concerned with students’ “cognitive impatience,” however, than by what may underlie it: the potential inability of large numbers of students to read with a level of critical analysis sufficient to comprehend the complexity of thought and argument found in more demanding texts… Karin Littau and Andrew Piper have noted another dimension: physicality. Piper, Littau and Anne Mangen’s group emphasize that the sense of touch in print reading adds an important redundancy to information – a kind of “geometry” to words, and a spatial “thereness” for text. As Piper notes, human beings need a knowledge of where they are in time and space that allows them to return to things and learn from re-examination – what he calls the “technology of recurrence”. The importance of recurrence for both young and older readers involves the ability to go back, to check and evaluate one’s understanding of a text. The question, then, is what happens to comprehension when our youth skim on a screen whose lack of spatial thereness discourages “looking back.

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

I was somewhat surprised, therefore, to find that my college teachers—famous academics and composers—inhabited an entirely different musical universe. They knew nothing about, and cared little for, the music I had grown up with. Instead, their world revolved around the dissonant, cerebral music of Arnold Schoenberg and his followers. As I quickly learned, in this environment not everything was possible: tonality was considered passé and “unserious”; electric guitars and saxophones were not to be mixed with violins and pianos; and success was judged by criteria I could not immediately fathom. Music, it seemed, was not so much to be composed as constructed—assembled painstakingly, note by note, according to complicated artificial systems. Questions like “does this chord sound good?” or “does this compositional system produce likeable music?” were frowned upon as naive or

The most innovative schools, from the established Montessori to the new-age MUSE, know this and have modeled themselves accordingly. Students in these schools don’t learn geometry from a textbook; they learn it by building real structures and observing real phenomena. They don’t study biology by listening to a teacher drone on; they learn it by cultivating gardens that feed the entire school.

If only my high school geometry teacher had used a pool table as a visual aide, he might've had a class of Einsteins instead of bored-to-death teenagers.

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.

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.

Adam: Adam was a young man whose anxiety turned into a monster. Where Shelly had a very mild case of social anxiety, Adam’s case could only be called severe. Over a period of several years, his underlying social fears developed into a full-blown school phobia. A quiet, unassuming person, Adam had never stood out in the classroom. Through elementary school and on into high school, he neither excelled nor failed his subjects. By no means a discipline problem, the “shy” Adam kept to himself and seldom talked in class, whether to answer a teacher’s question or chat with his buddies. In fact, he really had no friends, and the only peers he socialized with were his cousins, whom he saw at weekly family gatherings. Though he watched the other kids working together on projects or playing sports together, Adam never approached them to join in. Maybe they wouldn’t let him, he thought. Maybe he wasn’t good enough. Being rejected was not a chance he was willing to take. Adam never tried hard in school either. If he didn’t understand something, he kept quiet, fearful that raising his hand would bring ridicule. When he did poorly on an exam or paper, it only confirmed to him what he was sure was true: He didn’t measure up. He became so apprehensive about his tests that he began to feel physically ill at the thought of each approaching reminder of his inadequacy. Even though he had studied hard for a math test, for example, he could barely bring himself to get out of bed on the morning it was to take place. His parents, who thought of their child as a reserved but obedient boy who would eventually grow out of this awkward adolescent stage, did not pressure him. Adam was defensive and withdrawn, overwrought by the looming possibility that he would fail. For the two class periods preceding the math test, Adam’s mind was awash with geometry theorems, and his stomach churning. As waves of nausea washed over him, he began to salivate and swallowed hard. His eyes burned and he closed them, wishing he could block the test from his mind. When his head started to feel heavy and he became short of breath, he asked for a hall pass and headed for the bathroom. Alone, he let his anxiety overtake him as he stared into the mirror, letting the cool water flow from the faucet and onto his sweaty palms. He would feel better, he thought, if he could just throw up. But even when he forced his finger down his throat, there was no relief. His dry heaves made him feel even weaker. He slumped to the cold tile and began to cry. Adam never went back to math class that day; instead, he got a pass from the nurse and went straight home. Of course, the pressure Adam was feeling was not just related to the math test. The roots of his anxiety went much deeper. Still, the physical symptoms of anxiety became so debilitating that he eventually quit going to school altogether. Naturally, his parents were extremely concerned but also uncertain what to do. It took almost a year before Adam was sufficiently in control of his symptoms to return to school.

Ibn Sina was born in a tiny settlement called Afshanah, outside the village of Kharmaythan, and soon after his birth his family moved to the nearby city of Bukhara. While he was still a small boy his father, a tax collector, arranged for him to study with a teacher of Qu’ran and a teacher of literature, and by the time he was ten he had memorized the entire Qu’ran and absorbed much of Muslim culture. His father met a learned vegetable peddler named Mahmud the Mathematician, who taught the child Indian calculation and algebra. Before the gifted youth grew his first facial hairs he had qualified in law and delved into Euclid and geometry, and his teachers begged his father to allow him to devote his life to scholarship. He began the study of medicine at eleven and by the time he was sixteen he was lecturing to older physicians and spending much of his time in the practice of law. All his life he would be both jurist and philosopher, but he noted that although these learned pursuits were given deference and respect by the Persian world in which he lived, nothing mattered more to an individual than his well-being and whether he would live or die. At an early age, fate made Ibn Sina the servant of a series of rulers who used his genius to guard their health, and though he wrote dozens of volumes on law and philosophy—enough to win him the affectionate sobriquet of Second Teacher (First Teacher being Mohammed)—it was as the Prince of Physicians that he gained the fame and adulation that followed him wherever he traveled. In Ispahan, where he had gone at

Twelve is the number which fits around one in three dimensions in the same way that six fits around one in two dimensions. The New Testament is a story of a teacher surrounded by twelve disciples.

Don’t forget to be specific,’ Slim always says. ‘Details. Put in all the details. The boys appreciate all that detailed daily life shit they don’t get any more. If you’ve got a teacher you’re hot for, tell ’em what her hair looks like, what her legs look like, what she eats for lunch. If she’s teaching you geometry, tell ’em how she draws a bloody triangle on the blackboard. If you went down the shop for a bag of sweets yesterday, did you ride your pushy, did you go by foot, did you see a rainbow along the way? Did you buy gobstoppers or clinkers or caramels? If you ate a good meat pie last week, was it steak and peas or curry or mushroom beef? You catchin’ my drift? Details.

Early education,” says President Thwing, “occupies itself with description (geometry, space, arithmetic, time, science, the world of nature). Later education with comparison and relations.” If one asks, “Why not both together? Why learn facts at one time and their relations at another? Is it not the most vital possible way to learn facts to learn them in their relations?”—the answer that would be generally made reveals that most teachers are pessimists, that they have very small faith in what can be expected of the youngest pupils. The theory is that interpretative minds must not be expected of them. Some of us find it very hard to believe as little as this, in any child. Most children have such an incorrigible tendency for putting things together that they even put them together wrong rather than not put them together at all. Under existing educational conditions a child is more of a philosopher at six than he is at twenty-six.

Early education,” says President Thwing, “occupies itself with description (geometry, space, arithmetic, time, science, the world of nature). Later education with comparison and relations.” If one asks, “Why not both together? Why learn facts at one time and their relations at another? Is it not the most vital possible way to learn facts to learn them in their relations?”—the answer that would be generally made reveals that most teachers are pessimists, that they have very small faith in what can be expected of the youngest pupils. The theory is that interpretative minds must not be expected of them. Some of us find it very hard to believe as little as this, in any child. Most children have such an incorrigible tendency for putting things together that they even put them together wrong rather than not put them together at all. Under existing educational conditions a child is more of a philosopher at six than he is at twenty-six. The third stage of education for which Dr. Thwing partitions off the human mind is the “stage in which a pupil becomes capable of original research, a discoverer of facts and relations” himself. In theory this means that when a man is thirty years old and all possible habits of originality have been trained out of him, he should be allowed to be original. In practice it means removing a man’s brain for thirty years and then telling him he can think.

What he found was the geometry of the universe. Looking at the bubbles made by the Wego’s propellers, he recalled his boarding school math teachers, who had taught him to measure a sphere’s volume in terms of pi. He also remembered that pi was an irrational number, a decimal that never ended. He asked himself how nature could ever make bubbles in such circumstances. Did nature approximate? The rules his teachers had taught him must be mistaken. Spheres ought to be understood in terms of the forces that made them. At the age of twenty-one, Bucky determined that the universe had no objects. Geometry described forces. It was an insight bound to shape Bucky’s entire worldview—informing every future invention—but