Math Professor Quotes

The Professor never really seemed to care whether we figured out the right answer to a problem. He preferred our wild, desperate guesses to silence, and he was even more delighted when those guesses led to new problems that took us beyond the original one. He had a special feeling for what he called the "correct miscalculation," for he believed that mistakes were often as revealing as the right answers.

Math has proven the existence of God, because it is absolute and without contradiction; but the devil must exist as well, because we cannot prove it

He walked straight out of college into the waiting arms of the Navy. They gave him an intelligence test. The first question on the math part had to do with boats on a river: Port Smith is 100 miles upstream of Port Jones. The river flows at 5 miles per hour. The boat goes through water at 10 miles per hour. How long does it take to go from Port Smith to Port Jones? How long to come back? Lawrence immediately saw that it was a trick question. You would have to be some kind of idiot to make the facile assumption that the current would add or subtract 5 miles per hour to or from the speed of the boat. Clearly, 5 miles per hour was nothing more than the average speed. The current would be faster in the middle of the river and slower at the banks. More complicated variations could be expected at bends in the river. Basically it was a question of hydrodynamics, which could be tackled using certain well-known systems of differential equations. Lawrence dove into the problem, rapidly (or so he thought) covering both sides of ten sheets of paper with calculations. Along the way, he realized that one of his assumptions, in combination with the simplified Navier Stokes equations, had led him into an exploration of a particularly interesting family of partial differential equations. Before he knew it, he had proved a new theorem. If that didn't prove his intelligence, what would? Then the time bell rang and the papers were collected. Lawrence managed to hang onto his scratch paper. He took it back to his dorm, typed it up, and mailed it to one of the more approachable math professors at Princeton, who promptly arranged for it to be published in a Parisian mathematics journal. Lawrence received two free, freshly printed copies of the journal a few months later, in San Diego, California, during mail call on board a large ship called the U.S.S. Nevada. The ship had a band, and the Navy had given Lawrence the job of playing the glockenspiel in it, because their testing procedures had proven that he was not intelligent enough to do anything else.

Befuddlement is a healthy part of the learning process. When students approach a problem and don’t know how to do it, they’ll often decide they’re no good at the subject. Brighter students, in particular, can have difficulty in this way—their breezing through high school leaves them no reason to think that being confused is normal and necessary. But the learning process is all about working your way out of confusion. Articulating your question is 80 percent of the battle. By the time you’ve figured out what’s confusing, you’re likely to have answered the question yourself!” —Kenneth R. Leopold, Distinguished Teaching Professor, Department of Chemistry, University of Minnesota

Something, somewhere, somewhen, must have happened differently... PETUNIA EVANS married Michael Verres, a Professor of Biochemistry at Oxford. HARRY JAMES POTTER-EVANS-VERRES grew up in a house filled to the brim with books. He once bit a math teacher who didn't know what a logarithm was. He's read Godel, Escher, Bach and Judgment Under Uncertainty: Heuristics and Biases and volume one of The Feynman Lectures on Physics. And despite what everyone who's met him seems to fear, he doesn't want to become the next Dark Lord. He was raised better than that. He wants to discover the laws of magic and become a god. HERMIONE GRANGER is doing better than him in every class except broomstick riding. DRACO MALFOY is exactly what you would expect an eleven-year-old boy to be like if Darth Vader were his doting father. PROFESSOR QUIRRELL is living his lifelong dream of teaching Defense Against the Dark Arts, or as he prefers to call his class, Battle Magic. His students are all wondering what's going to go wrong with the Defense Professor this time. DUMBLEDORE is either insane, or playing some vastly deeper game which involved setting fire to a chicken. DEPUTY HEADMISTRESS MINERVA MCGONAGALL needs to go off somewhere private and scream for a while. Presenting: HARRY POTTER AND THE METHODS OF RATIONALITY You ain't guessin' where this one's going.

I have an English degree, which has as much value as The Papiermark in The Weimar Republic in 1923. It's worthless, and why I'm now a duck farmer. Plus, what could I do with English but be a classroom colonialist? If I wanted to teach something racist, I'd be a math professor.

As I headed down the library steps, I turned back to look. The mathematics stacks were as silent and empty as ever. Apparently no one suspected the riches hidden there.

To paraphrase one of my favorite math professors: The difference between great artists, and artists who are not so great, is that great artists think deeply about simple things.

According to a 1995 study, a sample of Japanese eighth graders spent 44 percent of their class time inventing, thinking, and actively struggling with underlying concepts. The study's sample of American students, on the other hand, spent less than 1 percent of their time in that state. “The Japanese want their kids to struggle,” said Jim Stigler, the UCLA professor who oversaw the study and who cowrote The Teaching Gap with James Hiebert. “Sometimes the [Japanese] teacher will purposely give the wrong answer so the kids can grapple with the theory. American teachers, though, worked like waiters. Whenever there was a struggle, they wanted to move past it, make sure the class kept gliding along. But you don't learn by gliding.

The easiest explanations are often the right ones," his math professor, Dr. Li, always said, and maybe the sam principle applied here. Except he knew it didn't. Math was one thing. Nothing else was that reductive.

The community certainly included black English professors, like my mother, as well as black doctors and dentists, black mechanics, janitors, and contractors, black cobblers, wedding planners, real estate agents, and undertakers, several black lawyers, and a handful of black Mary Kay salespeople. As a child, however, I knew so many African Americans working in science, math, and engineering that I thought that’s just what black folks did.

Basically, having a gift for happiness was a bit like being good at maths or games: it depended partly on the development of the brain after you were born, ad even before, but also on how your parents or other adults had brought you up when you were small. And of course on your own efforts and subsequent encounters. 'Nature or nurture,' said the professor. 'Whichever way, the parents are to blame!

Monty Jones: Dad, is there a word to describe answers that are completely correct but entirely useless under the circumstances? Professor Jones: Yes, yes there is.

In the midst of a vast field of numbers, a straight path opened before my eyes.

Math has proven the existence of God, because it is absolute and without contradiction; but the devil must exist as well, because we cannot prove it.

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.

Going through old papers I came across the transcript of a university debate on Rublyov. God, what a level. Abysmal, pathetic. But there is one remarkable contribution by a maths professor called Manin, Lenin Prize winner, who can hardly be more than thirty. I share his views. Not that one should say that about oneself. But it's exactly what I felt when I was making Andrey. And I'm grateful to Manin for that. "Almost every speaker has asked why they have to be made to suffer all through the three hours of the film. I'll try to reply to that question. It is because the twentieth century has seen the rise of a kind of emotional inflation. When we read in a newspaper that two million people have been butchered in Indonesia, it makes as much impression on us as an account of our hockey team winning a match. The same degree of impression! We fail to notice the monstrous discrpancy between these two events. The channels of our perception have been smoothed out to the point where we are no longer aware. However, I don't want to preach about this. It may be that without it life would be impossible. Only the point is that there are some artists who do make us feel the true measure of things. It is a burden which they carry throughout their lives, and we must be thankful to them.

Violetta: If I had a Belgian chocolate mimmoth for every hour I've had to listen to Tarvek blather on about those stupid muses, I'd weigh a thousand kilos. Professor Mezzasalma: ... really? Violetta: Oh yeah. I worked it out once.

An extreme representative of this view is Ted Kaczynski, infamously known as the Unabomber. Kaczynski was a child prodigy who enrolled at Harvard at 16. He went on to get a PhD in math and become a professor at UC Berkeley. But you’ve only ever heard of him because of the 17-year terror campaign he waged with pipe bombs against professors, technologists, and businesspeople. In late 1995, the authorities didn’t know who or where the Unabomber was. The biggest clue was a 35,000-word manifesto that Kaczynski had written and anonymously mailed to the press. The FBI asked some prominent newspapers to publish it, hoping for a break in the case. It worked: Kaczynski’s brother recognized his writing style and turned him in. You might expect that writing style to have shown obvious signs of insanity, but the manifesto is eerily cogent. Kaczynski claimed that in order to be happy, every individual “needs to have goals whose attainment requires effort, and needs to succeed in attaining at least some of his goals.” He divided human goals into three groups: 1. Goals that can be satisfied with minimal effort; 2. Goals that can be satisfied with serious effort; and 3. Goals that cannot be satisfied, no matter how much effort one makes. This is the classic trichotomy of the easy, the hard, and the impossible. Kaczynski argued that modern people are depressed because all the world’s hard problems have already been solved. What’s left to do is either easy or impossible, and pursuing those tasks is deeply unsatisfying. What you can do, even a child can do; what you can’t do, even Einstein couldn’t have done. So Kaczynski’s idea was to destroy existing institutions, get rid of all technology, and let people start over and work on hard problems anew. Kaczynski’s methods were crazy, but his loss of faith in the technological frontier is all around us. Consider the trivial but revealing hallmarks of urban hipsterdom: faux vintage photography, the handlebar mustache, and vinyl record players all hark back to an earlier time when people were still optimistic about the future. If everything worth doing has already been done, you may as well feign an allergy to achievement and become a barista.

About 4,400 years ago 8 people stepped off Noah’s ark. According to the United Nations Population Growth Statistics, the world’s population grows at about .47% per year. That is the growth rate for all civilizations who kept records. Suppose you put $8.00 in the bank 4,400 years ago and received .47% a year. How much money would you have? What a coincidence! It would be about $7,000,000,000. That’s kind of odd, because 4,400 years ago 8 people stepped off the ark and now we have about 7,000,000,000 people on planet earth. God’s math works! Compound interest is something we teach to seventh-graders. You don’t have to be a professor to figure this out. A twelve-year-old can do the calculation. Ask any seventh-grader, the algebraic equation looks like this: A=P (1+r/n)t . . . where "A " is the ending amount (about 7,000,000,000 in this case), "P " is the beginning amount (8 in this case), "r " is the interest rate (.47% in this case), "n " is the number of compoundings a year (1 in this case), and "t " is the total number of years (4,400 in this case).

The appearance of Professor Benjamin Peirce, whose long gray hair, straggling grizzled beard and unusually bright eyes sparkling under a soft felt hat, as he walked briskly but rather ungracefully across the college yard, fitted very well with the opinion current among us that we were looking upon a real live genius, who had a touch of the prophet in his make-up.

It's an old story," Julia says, leaning back in her chair. "Only for me, it's new. I went to school for industrial design. All my life I've been fascinated by chairs - I know it sounds silly, but it's true. Form meets purpose in a chair. My parents thought I was crazy, but somehow I convinced them to pay my way to California. To study furniture design. I was all excited at first. It was totally unlike me to go so far away from home. But I was sick of the cold and sick of the snow. I figured a little sun might change my life. So I headed down to L.A. and roomed with a friend of an ex-girlfriend of my brother's. She was an aspiring radio actress, which meant she was home a lot. At first, I loved it. I didn't even let the summer go by. I dove right into my classes. Soon enough, I learned I couldn't just focus on chairs. I had to design spoons and toilet-bowl cleaners and thermostats. The math never bothered me, but the professors did. They could demolish you in a second without giving you a clue if how to rebuild. I spent more and more time in the studio, with other crazed students who guarded their projects like toy-jealous kids. I started to go for walks. Long walks. I couldn't go home because my roommate was always there. The sun was too much for me, so I'd stay indoors. I spent hours in supermarkets, walking aisle to aisle, picking up groceries and then putting them back. I went to bowling alleys and pharmacies. I rode buses that kept their lights on all night. I sat in Laundromats because once upon a time Laundromats made me happy. But now the hum of the machines sounded like life going past. Finally, one night I sat too long in the laundry. The woman who folded in the back - Alma - walked over to me and said, 'What are you doing here, girl?' And I knew that there wasn't any answer. There couldn't be any answer. And that's when I knew it was time to go.

Soft power is not limited to moral exemplars like Mahatma Gandhi. Consider, for example, the much-ballyhooed excellence of Asians in fields like math and science. Professor Ni defines soft power as “quiet persistence,” and this trait lies at the heart of academic excellence as surely as it does in Gandhi’s political triumphs. Quiet persistence requires sustained attention—in effect restraining one’s reactions to external stimuli.

Yes, in Math City: 2 + 2 + him (her) = 5 not 4. And again yes: A stupid man + 5 stupid women + him (her) = 7 stupid or A stupid woman + 5 stupid men + him (her) = 7 stupid. Professor Six discovered this formula when he was studying about businessman. He saw that a businessman buys 3 apple, but the businessman sales 3 apple + his own benefit which is = 4 apple. Yes, Professor Six announced: "Businessman means "Business" + "Man" = Business + Him = 2

Olo, Remi, Kwuga, Nur, Anajama, Rhoden. Only Olo and Remi were in my group. Everyone else I met in the dining area or the learning room where various lectures were held by professors onboard the ship. They were all girls who grew up in sprawling houses, who’d never walked through the desert, who’d never stepped on a snake in the dry grass. They were girls who could not stand the rays of Earth’s sun unless it was shining through a tinted window. Yet they were girls who knew what I meant when I spoke of “treeing.” We sat in my room (because, having so few travel items, mine was the emptiest) and challenged each other to look out at the stars and imagine the most complex equation and then split it in half and then in half again and again. When you do math fractals long enough, you kick yourself into treeing just enough to get lost in the shallows of the mathematical sea. None of us would have made it into the university if we couldn’t tree, but it’s not easy. We were the best and we pushed each other to get closer to “God.

Something, somewhere, somewhen, must have happened differently… PETUNIA EVANS married Michael Verres, a Professor of Biochemistry at Oxford. HARRY JAMES POTTER-EVANS-VERRES grew up in a house filled to the brim with books. He once bit a math teacher who didn’t know what a logarithm was. He’s read Godel, Escher, Bach and Judgment Under Uncertainty: Heuristics and Biases and volume one of The Feynman Lectures on Physics. And despite what everyone who’s met him seems to fear, he doesn’t want to become the next Dark Lord. He was raised better than that. He wants to discover the laws of magic and become a god. HERMIONE GRANGER is doing better than him in every class except broomstick riding. DRACO MALFOY is exactly what you would expect an eleven-year-old boy to be like if Darth Vader were his doting father. PROFESSOR QUIRRELL is living his lifelong dream of teaching Defense Against the Dark Arts, or as he prefers to call his class, Battle Magic. His students are all wondering what’s going to go wrong with the Defense Professor this time. DUMBLEDORE is either insane, or playing some vastly deeper game which involved setting fire to a chicken. DEPUTY HEADMISTRESS MINERVA MCGONAGALL needs to go off somewhere private and scream for a while. Presenting: HARRY POTTER AND THE METHODS OF RATIONALITY You ain’t guessin’ where this one’s going.

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 students from the literary section had three written tests: Romanian, French and Latin; the scientific section needed math instead of Latin. Most of us had private tutoring in French and Latin for a few months before the matura. About a third of the candidates failed the written exam. The ones who succeeded took the orals. They were held in a large auditorium, at a high school and were open to the public. Five candidates faced a committee of eight teachers; the head of the committee was a university professor. None of the teachers were from our town. The orals took four to five hours, one group in the morning, another in the afternoon. The results were announced daily, late in the evening.

Memorizing multiplication tables may be a seminal school experience, among the few that kids today share with their grandparents. But a Stanford University professor says rapid-fire math drills are also the reason so many children fear and despise the subject.

IN THE SCHOOLS Memorizing multiplication tables may be a seminal school experience, among the few that kids today share with their grandparents. But a Stanford University professor says rapid-fire math drills are also the reason so many children fear and despise the subject. Moreover, the traditional approach to math instruction — memorization, timed testing and the pressure to speedily arrive at answers — may actually damage advanced-level skills by undermining the development of a deeper understanding about the ways numbers work. “There is a common and damaging misconception in mathematics — the idea that strong math students are fast math students,” says Jo Boaler, who teaches math education at the California university and has authored a new paper, “Fluency Without Fear.” In fact, many mathematicians are not speedy calculators, Boaler says. Laurent Schwartz, the French mathematician whose work is considered key to the theory of partial differential equations, wrote that as a student he often felt stupid because he was among the slowest math-thinkers in class.

Chris Argyris, professor emeritus at Harvard Business School, wrote a lovely article in 1977,191 in which he looked at the performance of Harvard Business School graduates ten years after graduation. By and large, they got stuck in middle management, when they had all hoped to become CEOs and captains of industry. What happened? Argyris found that when they inevitably hit a roadblock, their ability to learn collapsed: What’s more, those members of the organization that many assume to be the best at learning are, in fact, not very good at it. I am talking about the well-educated, high-powered, high-commitment professionals who occupy key leadership positions in the modern corporation.… Put simply, because many professionals are almost always successful at what they do, they rarely experience failure. And because they have rarely failed, they have never learned how to learn from failure.… [T]hey become defensive, screen out criticism, and put the “blame” on anyone and everyone but themselves. In short, their ability to learn shuts down precisely at the moment they need it the most.192 [italics mine] A year or two after Wave, Jeff Huber was running our Ads engineering team. He had a policy that any notable bug or mistake would be discussed at his team meeting in a “What did we learn?” session. He wanted to make sure that bad news was shared as openly as good news, so that he and his leaders were never blind to what was really happening and to reinforce the importance of learning from mistakes. In one session, a mortified engineer confessed, “Jeff, I screwed up a line of code and it cost us a million dollars in revenue.” After leading the team through the postmortem and fixes, Jeff concluded, “Did we get more than a million dollars in learning out of this?” “Yes.” “Then get back to work.”193 And it works in other settings too. A Bay Area public school, the Bullis Charter School in Los Altos, takes this approach to middle school math. If a child misses a question on a math test, they can try the question again for half credit. As their principal, Wanny Hersey, told me, “These are smart kids, but in life they are going to hit walls once in a while. It’s vital they master geometry, algebra one, and algebra two, but it’s just as important that they respond to failure by trying again instead of giving up.” In the 2012–2013 academic year, Bullis was the third-highest-ranked middle school in California.194

In discussing California's "Whole Math" (as it is called) with one such leading critic (math professor Abby Thompson of the University of California at Davis), I once suggested that the program seemed designed to brake the progress of bright kids. Thompson replied that I underestimated its anti-elitist fervor: "I don't think this is quite correct. The idea [espoused by 'Whole' Math advocates] is more precisely that there are no smart kids, that all children are equally talented at math, and that any differences that appear are due to societal failure, and probably racism and sexism to boot."-'(

Here is your working practice list: 1. Accused of cheating on a test, Janis goes to visit her math professor with the goal of convincing him she did not cheat. 2. Searching for an embezzler, Calvin accosts the bank examiner with the goal of convincing the examiner to give him the name of the prime suspect. 3. Lost in the caverns, Billy explores a narrow shaft with the goal of finding his way out. (A hard one! No living opponent.) 4. Ted visits Jennifer with the goal of getting her to marry him. 5. Wanting to win permission to enter graduate school, Bari goes into the office of the graduate dean with the goal of convincing him to let her in. (If the dean is a male, there is a very obvious “Yes, but!” disaster possibility lurking at the end of this scene.)

packed in steamer trunks.” “Good. How many trunks?” She glanced at the nearby tables, which were empty. “A typical steamer trunk filled with hundred-dollar bills will hold about fifteen million dollars, and weigh about four hundred pounds.” “Okay . . . one in each hand, two people, that’s sixty million.” She ignored my math and said, “But there are also fifty-dollar bills, and twenties, so there are more than four trunks.” “How many?” “My grandfather said ten.” “Each weighing four hundred pounds?” “Yes. A twenty-dollar bill weighs the same as a hundred-dollar bill.” “Right. That’s four thousand pounds of steamer trunks.” “Give or take.” If I’d known this in Key West I would have gone to the gym. “How about the gold and jewels?” “The gold may be too heavy to take. But there are four valises of jewelry which we’ll take.” “Always room for jewelry. And how about the property deeds that you mentioned?” “That’s another steamer trunk.” I pointed out, “This could be a bit of a logistical problem. You know, getting the trunks out of the cave, onto a truck, then to the boat.” “Carlos has a plan.” “Well, thank God. Would you like another cup of coffee?” She stared at me. “We wouldn’t be doing this if we didn’t think we could do it.” “Right.” A pretty waitress cleared our plates and smiled at me. It was almost 8 A.M. and people from various tour groups were making their way toward the lobby. We stood and I left two CUCs on the table, and Sara said, “That’s three days’ pay.” “She worked hard.” “And she had a nice butt.” “Really?” The Yale group was already boarding and Sara and I got on the bus together, said good morning to José, Tad, Alison, Professor Nalebuff, and our travel mates as we made our way toward the rear and found a seat together. The efficient Tad did a head count and announced, “We’re all here.” Antonio hopped aboard and called out, “Buenos días!” Everyone returned the greeting so we could get moving. “We will have a beautiful day!” said Antonio. Sí, camarada. CHAPTER 20 The bus wound its way out of Havana and again I had the impression of a once vibrant city that was suffocating under the weight of a rotting corpse. Hemingway’s house, Finca Vigía, was a handsome Spanish Colonial located about fifteen kilometers from Havana,

This reminds me of an old story from the Harvard math department, concerning one of the grand old Russian professors, whom we shall call O. Professor O is midway through an intricate algebraic derivation when a student in the back row raises his hand. “Professor O, I didn’t follow that last step. Why do those two operators commute?” The professor raises his eyebrows and says, “Eet ees obvious.” But the student persists: “I’m sorry, Professor O, I really don’t see it.” So Professor O goes back to the board and adds a few lines of explanation. “What we must do? Well, the two operators are both diagonalized by . . . well, it is not exactly diagonalized but . . . just a moment . . .” Professor O pauses for a little while, peering at what’s on the board and scratching his chin. Then he retreats to his office. About ten minutes go by. The students are about to start leaving when Professor O returns, and again assumes his station in front of the chalkboard. “Yes,” he says, satisfied. “Eet ees obvious

During five ninety-minute home visits, some children heard about 4 math-oriented words while others heard more than 250. In one week, therefore, some children would hear 28 math words, others 1,799. Extrapolating to one year, this meant the 1,500 math words one child heard could be contrasted to almost 100,000 for others. An enormous difference. Professor Levine and her colleagues then used a test to determine whether those differences were predictive of a child’s math ability. Children just under four years of age were shown two cards, each with a different number of dots. The children were told a number then asked to point to the card with that number of dots on it. The researchers wanted to know, of course, if the children were able to match the word for the number to the actual number of dots. There was no doubt. Children who had been exposed to more math talk were predictably more likely to choose the cards with the corresponding number of dots. Their superior understanding of the essential cardinal principle of mathematics, compared to peers who had heard much less number talk, corroborated the power of parent talk.

The Polytechnic was a new sort of college dedicated to producing teachers and professors for various math or scientific disciplines, and it was one of the few universities in Europe to grant women degrees.

Innovations are happening in conventional schooling. Some people will read the chapters to come and respond that their own children’s schools are incorporating evidence-based changes, making them more like Montessori schools—eliminating grades, combining ages, using a lot of group work, and so on. One could take the view that over the years, conventional schooling has gradually been discovering and incorporating many of the principles that Dr. Montessori discovered in the first half of the 20th century. However, although schooling is changing, those changes are often relatively superficial. A professor of education might develop a new reading or math program that is then adopted with great fanfare by a few school systems, but the curricular change is minute relative to the entire curriculum, and the Lockean model of the child and the factory structure of the school environment still underlie most of the child’s school day and year. “Adding new ‘techniques’ to the classroom does not lead to the developmental of a coherent philosophy. For example, adding the technique of having children work in ‘co-operative learning’ teams is quite different than a system in which collaboration is inherent in the structure” (Rogoff, Turkanis, & Bartlett, 2001, p. 13). Although small changes are made reflecting newer research on how children learn, particularly in good neighborhood elementary schools, most of the time, in most U.S. schools, conventional structures predominate (Hiebert, 1999; McCaslin et al., 2006; NICHD, 2005; Stigler, Gallimore, & Hiebert, 2000), and observers rate most classes to be low in quality (Weiss, Pasley, Smith, Banilower, & Heck, 2003). Superficial insertions of research-supported methods do not penetrate the underlying models on which are schools are based. Deeper change, implementing more realistic models of the child and the school, is necessary to improve schooling. How can we know what those new models should be? As in medicine, where there have been increasing calls for using research results to inform patient treatments, education reform must more thoroughly and deeply implement what the evidence indicates will work best. This has been advocated repeatedly over the years, even by Thorndike. Certainly more and more researchers, educators, and policy makers are heeding the call to take an evidence-based stance on education. Yet the changes made thus far in response to these calls have not managed to address to the fundamental problems of the poor models. The time has come for rethinking education, making it evidence based from the ground up, beginning with the child and the conditions under which children thrive. Considered en masse, the evidence from psychological research suggests truly radical change is needed to provide children with a form of schooling that will optimize their social and cognitive development. A better form of schooling will change the Lockean model of the child and the factory structure on which our schools are built into something radically different and much better suited to how children actually learn.

Professor Stephen Hawking predicted that the Earth could become inhabitable by as soon as 2600. Let's do the math: the average life expectancy exceeds 72 years. We are, at least mathematically, just a few generations away from mass extinction. So finding practicable alternatives to an environmentally harmful industry like ours is not only desirable. It's mandatory, at this point

Sometimes I feel compelled to do something, but I can only guess later why it needed to done, and I question whether I am drawing connections where none really exist. Other times I see an event – in a dream or in a flash of “knowing” – and I feel compelled to work toward changing the outcome (if it’s a negative event) or ensuring it (when the event is positive). At the times I am able to work toward changing or ensuring the predicted event, sometimes this seems to make a difference, and sometimes it doesn’t seem to matter. Finally, and most often, throughout my life I have known mundane information before I should have known it. For example, one of my favourite games in school was to guess what numbers my math teacher would use to demonstrate a concept, or to guess the words on a vocabulary test before the test was given. I noticed I was not correct all the time, but I was correct enough to keep playing the game. Perhaps partially because of the usefulness of this mundane skill, I was an outstanding student, getting straight As and graduating from college with highest honours in neuroscience and a minor in computer science. I was a modest drinker even in college, but I found I could ace tests when I was hungover after a night of indulgence. Sometimes I think I even did better the less I paid attention to the test and the more I felt sick or spacey. It was like my unconscious mind could take over and put the correct information onto the page without interruption from my overly analytical conscious mind. At graduate school in neuroscience, I focused on trying to understand human experience by studying how the brain processes pain and stress. I wanted to know the answer to the question: what’s going on inside people’s heads when we suffer? Later, as I finished my PhD in psychoacoustics, which is all about the psychology of sound, I became fascinated with timing. How do we figure out the order of sounds, even when some sounds take longer to process than others? How can drummers learn to decode time differences of 1/1,000 of a second, when most people just can’t hear those kinds of subtle time differences? At this point, I was using my premonitions as just one of the tools in my day-to-day toolkit, but I wasn’t thinking about them scientifically. At least not consciously. Sure, every so often I’d dream of the slides that would be used by one of my professors the next day in class. Or I’d realize that the data I was recording in my experiments followed the curve of an equation I’d dreamed about a year before. But I thought that was just my quirky way of doing things – it was just my good student’s intuition and it didn’t have anything to do with my research interests or my life’s work. What was my life’s work again?

I learnt that in teaching young children the concept of number, you should start with the concrete, then move to the pictorial, before finally representing numbers in the abstract. I learnt that children should be encouraged to articulate their processes, and feed back to each other on whether they are right or wrong, and why. And I learnt that this is so children understand number concepts, not just procedures, because (though not only because) the PSLE tests understanding, not just memorisation. As I was chatting to the professor in the car as she gave me a lift to the station, she also expounded on the importance of teacher-student relationships – 'you can't touch their brain until you have touched their heart'.

I think of all the nights I lay in bed, wondering what it might be like if things were different, if I hadn’t taken the branch in the road that made me a father and mediocre physics professor instead of a luminary in my field. I suppose it all comes down to wanting what I didn’t have. What I perceived might have been mine through a different set of choices. But the truth is, I did make those different choices. Because I am not just me. My understanding of identity has been shattered—I am one facet of an infinitely faceted being called Jason Dessen who has made every possible choice and lived every life imaginable. I can’t help thinking that we’re more than the sum total of our choices, that all the paths we might have taken factor somehow into the math of our identity. But none of the other Jasons matter. I don’t want their lives. I want mine. Because as fucked as everything is, there is no place I’d rather be than with this Daniela, this Charlie. If one tiny thing were different, they wouldn’t be the people I love.

The other article was by Lois Weiner, a professor who prepared urban teachers at New Jersey City University. Weiner was a parent activist at P.S. 3 in District 2, which she described as a highly progressive alternative school with an unusual degree of parent involvement. She claims that district administrators were stifling teachers and parents at P.S.3 by mandating "constructivist" materials and specific instructional strategies ... She [Weiner] continued, "The degree of micromanagement is astounding." Those who challenged the district office's mandates, she said, risked getting an unsatisfactory rating or being fired. Weiner contended that "opposition from parents is building against the new math curriculum," which was supposed to be field-tested with control groups, but instead was mandated for every classroom." Teachers were expressly prohibited from using other math textbooks or materials, and some were clandestinely "photocopying pages of now-banned workbooks.

While some patients find it easier to suppress their swearing in public (where anxiety levels are high, but the social penalties for swearing are higher), others find it easier to suppress their swearing among friends and family, in settings where they are more relaxed. Stress might increase the severity of the urges that some TS patients feel, which would explain the greater severity of tics in high-stress situations. For others, the fear of social consequences acts as a very strong motivator to suppress swearing and other tics, no matter what the cost. To make sense of this, Professor Conelea set up an experiment to differentiate between situational stress and social pressure. She set TS patients time-limited mental arithmetic problems in order to induce a baseline level of stress. She told these patients that they could swear or tic as much as they wanted. When allowed to tic freely, these patients’ tic rates were slightly lower while doing the maths challenge than when they were just resting, possibly because of the amount of concentration that the mental arithmetic required. She then asked them to try again, but this time to try to suppress their tics. As a result, the number of tics per minute almost doubled under stress. The very stress of trying not to act on ticcish urges made it much more likely that the volunteers would suffer from tics.

During the 1919 solar eclipse, people go out to measure the positions of the stars and they find exactly what Einstein predicted. Einstein gets a telegram saying this, and somebody asked him, Professor Einstein, what would you have said if the observations didn’t agree with what your prediction of general relativity said should be happening? And Einstein said, “I’d be sorry for the dear lord; the theory is correct.” What he meant by that is the math is just so elegant, so beautiful, so powerful, that almost seemingly it can’t possibly be wrong.

mathematics professor in Russia slugging it out with another mathematics professor in India, kilobyte for kilobyte, over some stupefyingly arcane detail in prime number theory, while an eighteen-year-old, tube-fed math prodigy in Cambridge jumps in every few days with an even more stupefying explanation of why they are both wrong.

The trouble was that the rankings were self-reinforcing. If a college fared badly in U.S. News, its reputation would suffer, and conditions would deteriorate. Top students would avoid it, as would top professors. Alumni would howl and cut back on contributions. The ranking would tumble further. The ranking, in short, was destiny.

My name is Charlie Chucky, I am in the sixth grade, I love playing Minecraft, and I am learning to become a Super Spy. My Dad is the world’s best Super Spy, and he is starting to teach me all his tricks. Lately, I’ve been battling invisible giants, crazy zombie teachers, and super ninjas! Life has been pretty crazy, and I’ve enjoyed every second of it. My best friend Harley is different to me. He doesn’t want to become a Super Spy. He doesn’t want to battle bad guys and save the world each week. Nope. He wants to sit indoors and stare at numbers all day. Harley’s dream is to become the world’s greatest math professor. He loves school, he loves studying, and he absolutely loves math tests.  He goes mad for them. It is the one thing he is really good at. He just loves numbers.  Numbers are like candy for him – he can’t get enough of it. He even asked Mrs. Jackson for extra math homework last night. Mrs. Jackson then decided to give the whole class extra math homework. Let’s just say Harley wasn’t that popular after school.  This is Harley. Mrs. Jackson always says that someday math will save our lives, but I can’t see how it will. Maybe one day, four giant numbers will attack our school, and I will defeat them using an algebra equation… or maybe the numbers in my textbook will go bad, and start attacking all the words on the pages, and I will stop them using a calculator!