Guided Math Quotes

Dreams are what guide us, art is what defines us, math is what makes it all possible, and love is what lights our way.

The best way to be appreciative for your life is to live it; don't die for any other reason but love. Dreams are what guide us, art is what defines us, math is makes it all possible, and love is what lights our way.

[The Old Astronomer to His Pupil] Reach me down my Tycho Brahe, I would know him when we meet, When I share my later science, sitting humbly at his feet; He may know the law of all things, yet be ignorant of how We are working to completion, working on from then to now. Pray remember that I leave you all my theory complete, Lacking only certain data for your adding, as is meet, And remember men will scorn it, 'tis original and true, And the obloquy of newness may fall bitterly on you. But, my pupil, as my pupil you have learned the worth of scorn, You have laughed with me at pity, we have joyed to be forlorn, What for us are all distractions of men's fellowship and smiles; What for us the Goddess Pleasure with her meretricious smiles. You may tell that German College that their honor comes too late, But they must not waste repentance on the grizzly savant's fate. Though my soul may set in darkness, it will rise in perfect light; I have loved the stars too fondly to be fearful of the night. What, my boy, you are not weeping? You should save your eyes for sight; You will need them, mine observer, yet for many another night. I leave none but you, my pupil, unto whom my plans are known. You 'have none but me,' you murmur, and I 'leave you quite alone'? Well then, kiss me, -- since my mother left her blessing on my brow, There has been a something wanting in my nature until now; I can dimly comprehend it, -- that I might have been more kind, Might have cherished you more wisely, as the one I leave behind. I 'have never failed in kindness'? No, we lived too high for strife,-- Calmest coldness was the error which has crept into our life; But your spirit is untainted, I can dedicate you still To the service of our science: you will further it? you will! There are certain calculations I should like to make with you, To be sure that your deductions will be logical and true; And remember, 'Patience, Patience,' is the watchword of a sage, Not to-day nor yet to-morrow can complete a perfect age. I have sown, like Tycho Brahe, that a greater man may reap; But if none should do my reaping, 'twill disturb me in my sleep So be careful and be faithful, though, like me, you leave no name; See, my boy, that nothing turn you to the mere pursuit of fame. I must say Good-bye, my pupil, for I cannot longer speak; Draw the curtain back for Venus, ere my vision grows too weak: It is strange the pearly planet should look red as fiery Mars,-- God will mercifully guide me on my way amongst the stars.

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.

Working an integral or performing a linear regression is something a computer can do quite effectively. Understanding whether the result makes sense—or deciding whether the method is the right one to use in the first place—requires a guiding human hand. When we teach mathematics we are supposed to be explaining how to be that guide. A math course that fails to do so is essentially training the student to be a very slow, buggy version of Microsoft Excel.

Looking at numbers as groups of rocks may seem unusual, but actually it's as old as math itself. The word "calculate" reflects that legacy -- it comes from the Latin word calculus, meaning a pebble used for counting. To enjoy working with numbers you don't have to be Einstein (German for "one stone"), but it might help to have rocks in your head.

If you understand the math behind compounding you realize the most important question is not “How can I earn the highest returns?” It’s “What are the best returns I can sustain for the longest period of time?” Little changes compounded for a long time create extraordinary changes. Same as ever.

Home is where heart is. Heart where cookie is. Math clear: Home is cookie.

The best way to be appreciative for your life is to live it; don’t die for any other reason but love. Dreams are what guide us, art is what defines us, math is what makes it all possible, and love is what lights our way.” ~ Mike Norton

People assume because I'm a coffee expert I drink lots of coffee. I can't. It takes me half an hour to brew my perfect cup. Do the math. I simply don't have time to drink more.

—things that seem hopelessly random and unpredictable when viewed in isolation often turn out to be lawful and predictable when viewed in aggregate.

open a basic math book, and make sure you are really good at multiplying, dividing, compounding, probability, and statistics.

Those of us who teach math should try to turn this bug into a feature. We should be up front about the fact that word problems force us to make simplifying assumptions. That’s a valuable skill—it’s called mathematical modeling.

(Actually, languages can be very tricky in this respect. The eminent linguistic philosopher J. L. Austin of Oxford once gave a lecture in which he asserted that there are many languages in which a double negative makes a positive but none in which a double positive makes a negative—to which the Columbia philosopher Sidney Morgenbesser, sitting in the audience, sarcastically replied, “Yeah, yeah.

It took twenty-five years from the prediction of the neutrino to its detection, almost fifty years to confirm the Higgs boson, a hundred years to directly detect gravitational waves. Now the time it takes to test a new fundamental law of nature can be longer than a scientist’s full career. This forces theorists to draw upon criteria other than empirical adequacy to decide which research avenues to pursue. Aesthetic appeal is one of them. In our search for new ideas, beauty plays many roles. It’s a guide, a reward, a motivation. It is also a systematic bias.

Mathematics is one of the major modern mysteries. Perhaps it is the leading one, occupying a place in our society similar to the religious mysteries of another age. If we want to know something about what our age is all about, we should have some understanding of what mathematics is, and of how the mathematician operates and thinks.

Even as we sit, we dance – In this cosmic saga of numbers and geometric patterns, continually coming alive.

Physicist Freeman Dyson once explained that what’s often attributed to the supernatural, or magic, or miracles, is actually just basic math. In any normal person’s life, miracles should occur at the rate of roughly one per month: The proof of the law is simple. During the time that we are awake and actively engaged in living our lives, roughly for eight hours each day, we see and hear things happening at a rate of one per second. So the total number of events that happen to us is about 30,000 per day, or about a million per month. If the chance of a “miracle” is one in a million, we should therefore experience one per month, on average.

let’s begin with the word “vector.” It comes from the Latin root vehere, “to carry,” which also gives us words like “vehicle” and “conveyor belt.” To an epidemiologist, a vector is the carrier of a pathogen, like the mosquito that conveys malaria to your bloodstream. To a mathematician, a vector (at least in its simplest form) is a step that carries you from one place to another.

It’s easy to forget that teaching is holy work. We forget that building up the intellect- teaching our children to really think- does not happen by the might of human reason, but rather by the grace of God. On an ordinary day, you and I likely have a set of tasks we've scheduled for our kids. But it's more than math. It's more than history. It's the building up of our children's minds and hearts, and we can only do that if we realize that this is how we thank Him for the graces He so lavishly pours out on us.

Listen carefully: you do not need to have a "productive" homeschool day to please the Savior. You do not need to have a clean house to please the Savior. You do not even need to have well-behaved kids to please Him. It doesn't matter if you hit every math problem, get through an entire spelling lesson, or whether your child loves learning the way you want him to. It doesn't matter! What matters is that we seek to imitate Christ. That we order our loves so that our hearts better reflect His.  Many days, checklists will go untouched, books will go unread, ducks will not line up in a row, no matter how much we strive. So cease striving. "It is our part to offer what we can, His to finish what we cannot." —— St. Jerome

In the early part of the ninth century, Muhammad ibn Musa al-Khwarizmi, a mathematician working in Baghdad, wrote a seminal textbook in which he highlighted the usefulness of restoring a quantity being subtracted (like 2, above) by adding it to the other side of an equation. He called this process al-jabr (Arabic for “restoring”), which later morphed into “algebra.” Then, long after his death, he hit the etymological jackpot again. His own name, al-Khwarizmi, lives on today in the word “algorithm.

Albert Einstein had Max Talmud, his first mentor. It was Max who introduced a ten-year-old Einstein to key texts in math, science, and philosophy. Max took one meal a week with the Einstein family for six years while guiding young Albert. No one is self-made.

A voice came from outside. "Give yourselves up. We have you surrounded." "You can't have us surrounded," Penelope yelled back. "The outhouse backs against a stone wall." "Fine. We have you semicircled, then." Penelope grimaced. "They're using math against us.

underneath perfectionism is always an emotion regulation struggle. Underneath “I am the worst artist in the world!” is a child who could envision the picture they wanted to paint and is disappointed in their final product; underneath “I stink at math” is a child who wants to feel capable and instead feels confused; underneath “I let down my team” is a child who can’t access all the moments they played well and is mired in their missed layup. In each case, that disappointment—or the mismatch between what a child wanted to happen and what actually happened—manifests as perfectionism. And, because perfectionism is a sign of an emotion regulation struggle, logic won’t help—we can’t convince a child that her art is great or that math concepts are hard for everyone or that one missed shot doesn’t define an athlete.

Today I think they don't teach the principle of least action in school because then everybody would go and study physics.

Nature—cue the theme from The Twilight Zone—somehow knows calculus.

In fact, it is virtually impossible for students to do their own research without someone guiding their work. Having an advisor is absolutely essential.

probably heard that math is the language of science, or the language of Nature is mathematics. Well, it’s true. The more we understand the universe, the more we discover its mathematical connections. Flowers have spirals that line up with a special sequence of numbers (called Fibonacci numbers) that you can understand and generate yourself. Seashells form in perfect mathematical curves (logarithmic spirals) that come from a chemical balance. Star clusters tug on

WE ALL DO IT, YOU know. Distract ourselves from noticing how time’s passing. We throw ourselves into our jobs. We focus on keeping the blight off our tomato plants. We fill up our gas tanks and top off our Metro cards and do the grocery shopping so that the weeks look the same on the surface. And then one day, you turn around, and your baby is a man. One day, you look in the mirror, and see gray hair. One day, you realize there is less of your life left than what you’ve already lived. And you think, How did this happen so fast? It was only yesterday when I was having my first legal drink, when I was diapering him, when I was young. When this realization hits, you start doing the math. How much time do I have left? How much can I fit into that small space? Some of us let this realization guide us, I guess. We book trips to Tibet, we learn how to sculpt, we skydive. We try to pretend it’s not almost over. But some of us just fill up our gas tanks and top off our Metro cards and do the grocery shopping, because if you only see the path that’s right ahead of you, you don’t obsess over when the cliff might drop off. Some of us never learn. And some of us learn earlier than others. —

Success comes from hard work and the accumulation of small numbers. Unlike yesterday, today’s prosperity can bloom from continuous intelligent production. For the first time in history, life as a full-time writer has become about simple math.

When a child shows a particular interest in one area—for example, movement, language, math, reading—it is known as a sensitive period. This describes a moment when the child is particularly attuned to learning a certain skill or concept and it happens with ease and without effort.

So let’s scrap the GCSE altogether: what purpose does it serve? The nation requires a snapshot of performance in key skills at the point of the legal school-leaving age, so let us have a basic matriculation requirement in English, Maths, Science and maybe a modern language: no more than this.

Biologist Leslie Orgel used to say, “Evolution is cleverer than you are,” because whenever a critic says, “Evolution could never do that,” they usually just lacked imagination. It’s also easy to underestimate because of basic math. Evolution’s superpower is not just selecting favorable traits. That part is so tedious, and if it’s all you focus on you’ll be skeptical and confused. Most species’ change in any millennia is so trivial it’s unnoticeable. The real magic of evolution is that it’s been selecting traits for 3.8 billion years. The time, not the little changes, is what moves the needle.

At the end of our lives, He is going to look into hearts. What is it He will find there, I wonder? Will he find that we used the geography lesson, the dreaded math test, the teetering laundry pile and the boiling-over pot of soup to draw closer to Him? Did we use these gifts to teach our children to lift their eyes heavenward? Were the tedious details of a homeschooling day offered up as a way for us to love Him, or were they merely gotten through, checked off and accomplished? Did we even realize that every Monday, every Thursday, we were standing on holy ground? No task is too trivial, no assignment too small. Educating our children is an offering of love we make to the God Who was so gracious to bestow them upon us in the first place. Every moment of the daily grind in raising and teaching and loving on them is hallowed, because we do it for Him and because there would be no point of doing it without Him.

Focusing on individual nutrients, their identities, their contents in food, their tissue concentrations, and their biological mechanisms, is like using math and physics to catch balls. It’s not the way nature evolved, and it makes proper nutrition far more difficult than it needs to be. Our bodies use countless mechanisms, strategically placed throughout our digestion, absorption, and transport and metabolic pathways, to effortlessly ensure tissue concentrations consistent with good health—no database consultation required. But as long as we let reductionism guide our research and our understanding of nutrition, good health will remain unattainable.

Present new material in small steps with sufficient practice after each step. Ask a large number of questions and check the responses of all students. Provide models. Guide student practice. Check for student understanding. Obtain a high success rate. Provide scaffolds for difficult tasks. Require and monitor independent practice.

When a guitar string is plucked or when children jiggle a jump rope, the shape that appears is a sine wave. The ripples on a pond, the ridges of sand dunes, the stripes of a zebra—all are manifestations of nature’s most basic mechanism of pattern formation: the emergence of sinusoidal structure from a background of bland uniformity.

And because the PageRanks are defined as proportions, they have to add up to 1 when summed over the whole network. This conservation law suggests another, perhaps more palpable, way to visualize PageRank. Picture it as a fluid, a watery substance that flows through the network, draining away from bad pages and pooling at good ones. The algorithm seeks to determine how this fluid distributes itself across the network in the long run.

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.

We all do it, you know. Distract ourselves from noticing, how time's passing. We throw ourselves into our jobs...We fill up our gas tanks and top off our Metrocards and do the grocery shopping... And then one day, you look in the mirror and see gray hair. One day, you realize there is less of your life left than what you've already lived. And you think, how did this happen so fast ? ... ...When this realization hits, you start doing the math. How much time do I have left? How much can I fit into that small space? Some of us let this realization guide us... We book trips to Tibet...We try to pretend it's not almost over. But some of us just fill up our gas tanks and top off our Metrocards..., because if you only see the path, that's right ahead of you, you don't obsess over when the clip might drop off. Some of us never learn. And some of us learn earlier than others...

Euclid's Elements has been for nearly twenty-two centuries the encouragement and guide of that scientific thought which is one thing with the progress of man from a worse to a better state. The encouragement; for it contained a body of knowledge that was really known and could be relied on, and that moreover was growing in extent and application. For even at the time this book was written—shortly after the foundation of the Alexandrian Museum—Mathematics was no longer the merely ideal science of the Platonic school, but had started on her career of conquest over the whole world of Phenomena. The guide; for the aim of every scientific student of every subject was to bring his knowledge of that subject into a form as perfect as that which geometry had attained. Far up on the great mountain of Truth, which all the sciences hope to scale, the foremost of that sacred sisterhood was seen, beckoning for the rest to follow her.

This is a book for those of you who want to be parents as well as people, who don't want to feel guilty about taking some time for yourself or your relationship (if you're even in one). It's for those folks who might, every once in a while, want to get drunk, and have sloppy sex without worrying that they're going to roll over on their kid because you all sleep in a "family bed" since that's how they do it in Taiwan and they have the highest math scores of any country on earth.

Numbers written on restaurant checks within the confines of restaurants do not follow the same mathematical laws as numbers written on any other pieces of paper in any other parts of the Universe. This single statement took the scientific world by storm. It completely revolutionized it. So many mathematical conferences got held in such good restaurants that many of the finest minds of a generation died of obesity and heart failure and the science of math was put back by years. Slowly,

You are teaching your living, breathing, made-in-the-Image-of-God students. The resources are there to help you do that. It's that simple, we just forget when we get all wrapped up in "getting through" all the math lessons before the end of May, or finishing every science experiment in the book before we call it good and move on. It doesn't really matter how far in the book we get. What matters is what happens in the mind and heart of our student, and for that matter- in ourselves. You know this. I know this. But we've got to start living it. We are all spinning our wheels because we're frantically trying to "get through" published curriculum as if turning the last page in the book by the beginning of summer vacation will somehow mean that our children learned something. Truth is, they do learn something from that. But it's not at all the message we want them to internalize. We are teaching people, not books. We need to understand the limitations of curriculum. We need to stop trying to make it something that it's not, expecting it to yield what it was never intended to deliver.

Investor Howard Marks once talked about an investor whose annual results were never ranked in the top quartile, but over a fourteen-year period he was in the top 4 percent of all investors. If he keeps those mediocre returns up for another ten years he may be in the top 1 percent of his peers—one of the greatest of his generation despite being unremarkable in any given year…If you understand the math behind compounding you realize the most important question is not ‘How can I earn the highest returns?’ It’s ‘What are the best returns I can sustain for the longest period of time?

Tell me this- if you could have a guarantee that your child would be a National Merit Scholar and get into a prestigious college, have good work habits and a successful career, but that your relationship with him would be destroyed in the process, would you do it? Why not? Because you are made to love, that's why. We care about our relationships more than about our accomplishments. That's the way God made us. Then why don't we live that way? Why, come a damp and gloomy day in March, do we yell over a  math lesson or lose our temper over a writing assignment? Why do we see the lessons left to finish and get lost in an anxiety-ridden haze? We forget that we are dealing with a soul, a precious child bearing the Image of God, and all we can see is that there are only a few months left to the school year and we are still only halfway through the math book. When you are performing mommy triage- that is, when you have a crisis moment and have to figure out which fire to put out first- always choose your child. It's just a math lesson. It's only a writing assignment. It's a Latin declension. Nothing more. But your child? He is God's. And the Almighty put him in your charge for relationship. Don't damage that relationship over something so trivial as an algebra problem. And when you do (because you will, and so will I), repent. We like to feed our egos. When our children perform well, we can puff up with satisfaction and pat ourselves on the back for a job well done. But as important as it is to give our children a solid education (and it is important, don't misunderstand me), it is far more important that we love them well.  Our children need to know that the most important thing about them is not whether they finished their science curriculum or score well on the SAT. Their worth is not bound up in a booklist or a test score. Take a moment. Take ten. Look deep into your child's eyes. Listen, even when you're bored. Break out a board game or an old picture book you haven't read in ages. Resting in Him means relaxing into the knowledge that He has put these children in our care to nurture. And nurturing looks different than charging through the checklist all angst-like. Your children are not ordinary kids or ordinary people, because there are no ordinary kids or ordinary people. They are little reflections of the

WE ALL DO IT, YOU know. Distract ourselves from noticing how time’s passing. We throw ourselves into our jobs. We focus on keeping the blight off our tomato plants. We fill up our gas tanks and top off our Metro cards and do the grocery shopping so that the weeks look the same on the surface. And then one day, you turn around, and your baby is a man. One day, you look in the mirror, and see gray hair. One day, you realize there is less of your life left than what you’ve already lived. And you think, How did this happen so fast? It was only yesterday when I was having my first legal drink, when I was diapering him, when I was young. When this realization hits, you start doing the math. How much time do I have left? How much can I fit into that small space? Some of us let this realization guide us, I guess. We book trips to Tibet, we learn how to sculpt, we skydive. We try to pretend it’s not almost over. But some of us just fill up our gas tanks and top off our Metro cards and do the grocery shopping, because if you only see the path that’s right ahead of you, you don’t obsess over when the cliff might drop off. Some of us never learn. And some of us learn earlier than others.

The Roman general wanted to spare Archimedes, because he was so valuable—sort of like the Einstein of the ancient world—but some stupid Roman soldier killed him.” “There you go again,” Hazel muttered. “Stupid and Roman don’t always go together, Leo.” Frank grunted agreement. “How do you know all this, anyway?” he demanded. “Is there a Spanish tour guide around here?” “No, man,” Leo said. “You can’t be a demigod who’s into building stuff and not know about Archimedes. The guy was seriously elite. He calculated the value of pi. He did all this math stuff we still use for engineering. He invented a hydraulic screw that could move water through pipes.” Hazel scowled. “A hydraulic screw. Excuse me for not knowing about that awesome achievement.” “He also built a death ray made of mirrors that could burn enemy ships,” Leo said. “Is that awesome enough for you?” “I saw something about that on TV,” Frank admitted. “They proved it didn’t work.” “Ah, that’s just because modern mortals don’t know how to use Celestial bronze,” Leo said. “That’s the key. Archimedes also invented a massive claw that could swing on a crane and pluck enemy ships out of the water.” “Okay, that’s cool,” Frank admitted. “I love grabber-arm games.” “Well, there you go,” Leo said. “Anyway, all his inventions weren’t enough. The Romans destroyed his city. Archimedes was killed.

Albert Einstein, considered the most influential person of the 20th century, was four years old before he could speak and seven before he could read. His parents thought he was retarded. He spoke haltingly until age nine. He was advised by a teacher to drop out of grade school: “You’ll never amount to anything, Einstein.” Isaac Newton, the scientist who invented modern-day physics, did poorly in math. Patricia Polacco, a prolific children’s author and illustrator, didn’t learn to read until she was 14. Henry Ford, who developed the famous Model-T car and started Ford Motor Company, barely made it through high school. Lucille Ball, famous comedian and star of I Love Lucy, was once dismissed from drama school for being too quiet and shy. Pablo Picasso, one of the great artists of all time, was pulled out of school at age 10 because he was doing so poorly. A tutor hired by Pablo’s father gave up on Pablo. Ludwig van Beethoven was one of the world’s great composers. His music teacher once said of him, “As a composer, he is hopeless.” Wernher von Braun, the world-renowned mathematician, flunked ninth-grade algebra. Agatha Christie, the world’s best-known mystery writer and all-time bestselling author other than William Shakespeare of any genre, struggled to learn to read because of dyslexia. Winston Churchill, famous English prime minister, failed the sixth grade.

(Hablando del Universo) Población: Ninguna. Es sabido que existe un número infinito de mundos, sencillamente porque hay una cantidad infinita de espacio para que todos se asienten en él. Sin embargo, no todos están habitados. Por tanto, debe haber número finito de mundos habitados. Un número finito dividido por infinito se aproxima lo suficiente a la nada para que no haya diferencia, de manera que puede afirmarse que la población media de todos los planetas del Universo es cero. De ello se desprende que la población media de todo el Universo también es cero, y que todas las personas con que uno pueda encontrarse de vez en cuando no son más que el producto de una imaginación trastornada.

Economics ought to be a magpie discipline, taking in philosophy, history and politics. But heterodox approaches have long since been banished from most faculties, claims Tony Lawson. In the 1970s, when he started teaching at Cambridge, the economics faculty still boasted legends such as Nicky Kaldor and Joan Robinson. "There were big debates, and students would study politics, the history of economic thought." And now? "Nothing. No debates, no politics or history of economic thought and the courses are nearly all maths." How do elites remain in charge? If the tale of the economists is any guide, by clearing out the opposition and then blocking their ears to reality. The result is the one we're all paying for.

Watching violent, arousing shows may actually contribute to suppressing your immune system. As you identify with the anger you see on the screen or read about, stress chemicals called catecholamine and cortisone are released that can adversely affect your immune system. The effect of exposure to both anger and love on the immune system was shown in research by Harvard scientist David Mclelland, and later reproduced by the Heart Math Institute in California (Bhat 1995). Watching an anger-provoking movie suppressed the immune system (as measured by chemicals in the saliva) for five to six hours in study subjects. However, watching a movie about the compassionate work of Mother Teresa caused elevation of the immune level in the participants.

Here’s a deep paradox about learning: the more we embrace not-knowing and mistakes and struggles, the more we set the stage for growth, success, and achievement. This is true for adults and kids alike, and it’s a critical reminder about the importance of normalizing difficulties, embracing mistakes as an opportunity to learn, and building frustration tolerance. After all, the more a child can tolerate frustration, the longer they can stick with a hard puzzle, work on a tough math problem, or stay engaged while writing an essay. And, of course, these skills translate outside of academics as well, because tolerating frustration is key to managing disappointments, communicating effectively with people with different opinions, and sticking with personal goals.

Circuitry for self-confidence depends on a child’s ability to locate identity over observable behavior; this comes from growing up in a family that focuses more on what’s “inside” a child (enduring qualities, feelings, ideas) than what is “outside” (accomplishments, outcomes, labels). In regard to your child’s sports team, for example, inside stuff might be her effort in practice, her attitude when winning and losing, and her willingness to try new things; outside stuff might be her number of goals or home runs, or labels like “most valuable player.” When it comes to academics, inside stuff might be willingness to try a bonus math problem, spending time on studying, and showing enthusiasm about a subject; outside stuff might be a grade, a test score, or a label like “smartest kid in class.

Homework _Yes _No 1. Did you make a serious effort to understand the text? (Just hunting for relevant worked-out examples doesn’t count.) _Yes _No 2. Did you work with classmates on homework problems, or at least check your solutions with others? _Yes _No 3. Did you attempt to outline every homework problem solution before working with classmates? Test Preparation The more “Yes” responses you recorded, the better your preparation for the test. If you recorded two or more “No” responses, think seriously about making some changes in how you prepare for the next test. _Yes _No 4. Did you participate actively in homework group discussions (contributing ideas, asking questions)? _Yes _No 5. Did you consult with the instructor or teaching assistants when you were having trouble with something? _Yes _No 6. Did you understand ALL of your homework problem solutions when they were handed in? _Yes _No 7. Did you ask in class for explanations of homework problem solutions that weren’t clear to you? _Yes _No 8. If you had a study guide, did you carefully go through it before the test and convince yourself that you could do everything on it? _Yes _No 9. Did you attempt to outline lots of problem solutions quickly, without spending time on the algebra and calculations? _Yes _No 10. Did you go over the study guide and problems with classmates and quiz one another? _Yes _No 11. If there was a review session before the test, did you attend it and ask questions about anything you weren’t sure about? _Yes _No 12. Did you get a reasonable night’s sleep before the test? (If your answer is no, your answers to 1–11 may not matter.) _Yes _No TOTAL

When the subjects arrived at the psychology lab, they were sent into individual dressing rooms with full-length mirrors. Half of the dressing rooms contained bathing suits (one-piece for the women, trunks for the men) and half contained sweaters, all of which were available in a wide range of sizes. Once the subjects put on the assigned clothing, they were told to hang out in the dressing room for fifteen minutes before they filled out a questionnaire about whether or not they would want to purchase the item. While they waited, they were asked, in order to help the researchers use the time efficiently, to complete a math test “for an experimenter in the Department of Education.” As you’ve already guessed, the psychologists weren’t helping their colleagues in the Department of Education. They were measuring whether taking a math test while wearing a bathing suit would affect the women’s scores.

In a calm moment, introduce the idea of you and your child both having a “Perfect Kid” or “Perfect Girl” or “Perfect Boy” inside. It might sound like this: “Do you know that I have a Perfect Girl in me? Yes! She often tells me things have to be perfect or else they’re not worth doing! I think you have one too! I think she popped up when you were doing your math homework. Anyway, there’s no problem with having a Perfect Voice. A lot of people have them! But sometimes Perfect Girl, for me, she just gets so loud and she makes it hard for me to focus. I’ve found that talking to her nicely can help . . .” Now pause. See how your child responds. Often a child will take to this immediately and say, “What do you mean?” Continue: “Well, Perfect Girl isn’t a problem unless she’s so loud that I can’t hear the other voices in me. So when she’s getting loud, I just say to her, ‘Oh, hi, Perfect Girl. You again! I know, you always say, “Perfect, perfect, must be perfect, if it’s not perfect I have to stop.” I hear you! And also, I’m going to ask you to step back. I am going to take a deep breath and find my “I can do hard things” voice because I know that’s in there too.’ Then I can hear a quieter voice telling me it’s okay that things are hard and I can do hard things.

I BET I CAN guess your favorite math subject in high school. It was geometry.

resist

Proofs can cause dizziness or excessive drowsiness. Side effects of prolonged exposure may include night sweats, panic attacks, and, in rare cases, euphoria. Ask your doctor if proofs are right for you.

So as convenient as technologies can be, they also have the potential of making us stupid.

The acknowledgement of mathematics as a creative, exploratory and a faillible human endeavor is not fatalistic.

Upholding the value of intellectual independence doesn't mean that we need to refrain from group learning and other participatory activities.

Taking responsibility in choosing our own math materials is better than getting struck with resources which we don't find particularly useful.

Without the guiding influence of the heart, we easily fall prey to reactive emotions such as insecurity, anger, fear, and blame, as well as other energy-draining reactions and behaviors.

Certainly prevents some believers from being compassionate, sympathetic, or even tolerant of others who are not as certain in their faith. Their arrogance turns them into the "frozen chosen," consciously or unconsciously excluding others from their cozy, believing world. This is the crabbed, joyless, and ungenerous religiosity that Jesus spoke against: spiritual blindness. There is a more subtle danger for this group: a complacency that makes one's relationship with God stagnate. Some people cling to ways of understanding their faith learned in childhood that might not work for an adult. For example, you might cling to a childhood notion of a God who will never let anything bad happen. When tragedy strikes, since your youthful image of God is not reflected in reality, you may abandon the God of your youth. Or you may abandon God completely. An adult life requires an adult faith. Think of it this way: you wouldn't consider yourself equipped to face life with a third-grader's understanding of math. Yet people often expect the religious instruction they had in grammar school to sustain them in the adult world. In his book A Friendship Like No Other, the Jesuit spiritual writer William A. Barry invites adults to relate to God in an adult way. Just as an adult child needs to relate to his or her parent in a new way, he suggest, so adult believers need to relate t God in new ways as they mature. Otherwise, one remains stuck in a childlike view of God that prevents fully embracing a mature faith.

As young girls, my friends and I often played “school” or “house,” cooperative role-playing games in which we acted out scenarios like math class or making dinner. As long as I got to be the teacher or the mother, I loved these games. They played into my need for control and my love of organizing. If I didn’t get to be the teacher or the mother, the game usually ended in a nasty fight between me and the girl who got that role. Other kids’ rules made no sense to me. They felt all wrong. I had to be in charge or I wasn’t playing. Dr. Tony Attwood describes this as “god mode”—the way that autistic kids need to control every aspect of a social situation to make it safe for them to interact. For whatever reason, a few friends tolerated my god mode and hung around, though not all the time. I remember sometimes daily shouting matches that left me without a playmate for the rest of the day. Unlike boys’ games, where there tend to be winners and losers, girls’ games are often based on how well a girl cooperates with the group to create an enjoyable role-playing scenario. Boys

Math is not a constructor, or even a function. It’s an object. As you know, Math is a built-in object that you can use to do things like get the value of pi (with Math.PI) or generate a random number (with Math.random). Think of Math as just like an object literal that has a bunch of useful properties and methods in it, built-in for you to use whenever you write JavaScript code. It just happens to have a capital first letter to let you know that it’s built-in to JavaScript.

For example, one time we were putting some numbers into a formula, and got to 48 squared. I reach for the Marchant calculator, and he says, “That’s 2,300.” I begin to push the buttons, and he says, “If you want it exactly, it’s 2,304.” The machine says 2,304. “Gee! That’s pretty remarkable!” I say. “Don’t you know how to square numbers near 50?” he says. “You square 50—that’s 2,500—and subtract 100 times the difference of your number from 50 (in this case it’s 2), so you have 2,300. If you want the correction, square the difference and add it on. That makes 2,304.

Maybe ET will use math to get our attention—a string of prime numbers, perhaps, or the apparently omnipresent, omni-important Pythagorean theorem. High school trig teachers should rebrand it the alien communication formula. That would’ve kept you from daydreaming in class, right?

Warren Buffett puts it all rather well: “What the human being is best at doing is interpreting all new information so that their prior conclusions remain intact.

Because most people are intimidated by math and can’t independently critique it, they overvalue opinions backed with math/pseudoscience.

Focus on the next step, not the next thousand steps. Use your willpower to move your attention away from the overwhelming aspects of a project and narrow it down to the next actionable task you can get started on. Lower your perfectionistic standards. Decrease your initial resistance to getting started by lowering your standards. For example, aim to meditate for one minute, not 20 minutes. Follow the two-minute rule. If a task takes less than two minutes to complete, do it immediately. Set an implementation intention. Use the formula, “If situation X arises, then I will perform response Y.” A common example: “If I get home after work, then I’ll immediately start studying for my upcoming math exam.” Focus on the process, not the outcome. Set a timer for 20 to 30 minutes and focus on the process of working on a dreaded task for that predetermined amount of time.

Reading science, math, and philosophy one hour per day will likely put you at the upper echelon of human success within seven years.

The concepts are simple, but that doesn’t mean the process is easy. It’s not. That’s because money is not just about math; it’s about behavior. Personal finance is only 20 percent head knowledge. The other 80 percent—the bulk of the issue—is behavior. And it’s our behaviors with money that can get us into the biggest trouble or lead us into the biggest successes. Behavior is the key to the whole deal, and we’ll unpack how that plays out through all these different areas as we work through this book.

Does physics underlie even the spontaneous, beautiful displays of life on earth? The answer depends, in a sense, on whether you believe math is discovered or invented; whether it's a pervasive force, guiding every action in this universe, or whether logic is imposed by the human brain.

The academic task is about service—each of us contributing to the centuries-old great conversation in our field of study, using our voices to expand the symphony of ideas, for the flourishing of all people (my people, your people), everywhere. “What can I do where I am?” is the clarifying goal for a robust, meaningful education. Too often, schooling feels more like this: “I have to pass the math test so I can take another math test.

What are the most efficient ways to build new mental models? Read a lot—just read. [2] Reading science, math, and philosophy one hour per day will likely put you at the upper echelon of human success within seven years.

can

In the current technological era we are more vulnerable to manipulation via confirmation bias than ever. Take the YouTube algorithm. It recommends videos to you on the basis of those you’ve already watched by analyzing the viewing habits of people who clicked on the same video. It predicts that you are more likely to enjoy their favorite content than a completely random selection of videos, which turns out to be a fair assumption. The trouble is, there are concerns that it can have the effect of plunging the viewer into a confirmation bias odyssey. If you watched a video about aliens visiting rural America and intrusively probing local farmhands, then you are fairly likely to be interested in other bonkers conspiracy theories, such as that the Earth is flat, and that vaccines cause autism. Before long, you may find yourself presented with videos of people telling you that school shootings in the US were faked, and that the 9/11 terrorist attacks on the World Trade Center towers were perpetrated by the US government. People who believe such things are more likely to distrust the government anyway, and to be on the political right. In the run-up to the US presidential election in 2016, virulently anti-Hillary Clinton videos were viewed six times more than anti-Donald Trump videos.

Many flying insects see in the ultraviolet. And flowers, which need insects to pollinate them, know this well; their pretty petals often have ultraviolet runways pointing down toward their juicy nectar, and the all-important sexual organs. Honeybees see in the UV very well, though not in the red part of the spectrum like we do. Their eyes, which have a higher flicker threshold and can also perceive iridescence, are superbly evolved to spot flowers while hurtling along on the wing. Just as long as they’re not red.

The reason why we can’t see our eyes moving with our own eyes is because our brains edit out the bits between the saccades—a process called saccadic suppression. Without it, we’d look at an object and it would be a blurry mess. What we perceive as vision is the director’s cut of a film, with your brain as the director, seamlessly stitching together the raw footage to make a coherent reality. Perception is the brain’s best guess at what the world actually looks like. Immense though the computing power of that fleshy mass sitting in the darkness of our skulls is, if we were to take in all the information in front of our eyes, our brains would surely explode.** Instead, our eyes sample bits and pieces of the world, and we fill in the blanks in our heads. This fact is fundamental to the way that cinema works. A film is typically 24 static images run together every second, which our brain sees as continuous fluid movement—that’s why it’s called a movie. The illusion of movement actually happens at more like 16 frames per second. At that speed, a film projection is indistinguishable from the real world, at least to us. It was the introduction of sound that set the standard of 24 frames per second with The Jazz Singer in 1927, the first film to have synchronized dialogue. The company

The real guide to reality It’s not that pigeons are stupid. Well, they are, but that’s not the point. This gulf between how we see the world and how a pigeon sees the same world reveals something fundamental about our relationship with reality, and how we understand our place in the cosmos. Our eyes powerfully illustrate the fact that our experience is a heavily edited version of reality. Evolution has found a way for us to harvest, process and interpret elementary packets of light in the dark cavities of our skulls. Our minds navigate the many constraints of anatomy to make it work—frame rates, blind spots, faulty cones, colorless peripheral vision. We don’t even notice the limits of our eyes as we construct our subjective world view in our heads. Like all creatures on Earth, our bodies are carefully tuned to ensure our continued survival. But it would be a pointless waste of ego to think that they make us capable of experiencing reality as it really is. We are each locked into our own umwelt, profoundly limited by our senses, constrained by our biology, shackled by the inescapable bounds of our evolutionary history. We’re hopelessly tethered to what we can uncover while stuck on (or perhaps near) this planet, a speck of dust in the vastness of the cosmos. We see only the merest sliver of reality. We’re peering at the universe through a keyhole.

linear regression is something a computer can do quite effectively. Understanding whether the result makes sense—or deciding whether the method is the right one to use in the first place—requires a guiding human hand. When we teach mathematics we are supposed to be explaining how to be that guide. A math course that fails to do so is essentially training the student to be a very slow, buggy version of Microsoft Excel.

In God we trust, and AI models must disclose their training data.

AI is set to universalize the access to intellectual tools that otherwise require years of study, training and practice.

Your life will in certain ways be a long sequence of different kinds of homework. Horribly, maths or French is the easiest version: a beginner’s guide, almost a pleasure.

Script for “Did I Ever Tell You About the Time . . . ?” Identify the essence of your child’s struggle. (Is it hard for her to feel happy for other people’s accomplishments? Hard to stay engaged when math feels hard and frustrating?) Take on the problem as your own: remember a moment, in the recent past or when you were a child, when you struggled with something similar. Talk to your child not in the heat of the moment but when things are calm, starting with, “Did I ever tell you about the time . . . ?,” and share a story about yourself having a similar struggle. Engage your child in this story, ideally one where you didn’t come up with a quick fix but struggled and just kind of got through it. Do not end your story by directly relating it to your child. There’s no need to spell out, “Isn’t that just like when you . . . ?” Allow the story and moment to stand on their own, trusting that it will reach the part of your child that needed connection.

people fear sobriety because without booze, you’re forced to see Clockwork Orange style that you’re bad at relationships, and bad at vulnerability, and bad at honesty, and your rants about life’s injustices are generally not cute. and if you see all that you have to give up the dream that some brave soul is gonna show up and extend themselves to you even though your arms are crossed and your back is turned. you have to stop pretending and actually start earning your keep, it’s basic human math, you gotta bring something to the table, you can’t just show up empty-handed and expect to be fed.

Investor Howard Marks once talked about an investor whose annual results were never ranked in the top quartile, but over a fourteen-year period he was in the top 4 percent of all investors. If he keeps those mediocre returns up for another ten years he may be in the top 1 percent of his peers—one of the greatest of his generation despite being unremarkable in any given year…If you understand the math behind compounding you realize the most important question is not ‘How can I earn the highest returns?’ It’s ‘What are the best returns I can sustain for the longest period of time?’ Little changes compounded for a long time create extraordinary changes.

don’t do every last problem you see in this guide or in the associated online materials. Instead, do what you need. Try a couple problems of each type to see how it goes. If

Calculus is the mathematics of change. It describes everything from the spread of epidemics to the zigs and zags of a well-thrown curveball. The subject is gargantuan—and so are its textbooks.

At the most basic level, the main differences between classical and quantum physics boil down to four things: wave-particle duality, the uncertainty principle, quantum entanglement, and quantization.

Surprisingly, and almost unbelievably, when a photon is fired through the double slit apparatus, it will end up landing in one of the constructive interference areas of the screen on the other side. If you fire enough photons, you eventually will reform the waveform from the double slit experiment, too – the result is a series of vertical bars, just like before. The same result can be achieved not only with photons, but electrons, whole atoms, and even entire molecules.

My brain doesn’t process information as normal people do, and my short attention span. I can’t stand listening to many things at once, like numbers or letters, which means that it is really hard for me to do the math and read at the same time”(

You can’t always choose your future. Not in a world of risk and uncertainty. No matter what the self-help gurus tell you. You can only attempt to guide it in the right direction, like a willful horse, but accept there will be times when it will gallop off in a direction not of your choosing. No one can tell you what lies ahead with one hundred percent accuracy. If your doctor tells you ninety-nine out of one hundred people die of your disease, you most likely will die, but you might also be the one who beats the odds, and if you do, you will believe yourself special and blessed, and your loved ones will believe the fervency of their prayers for you paid dividends, but it’s just math. It’s all just math.

The most effective way to see if you understand a concept is to first review it and they try to explain it in your own words as if you were teaching an imaginary class or a five year old.

The key to getting better and understanding mathematics is deliberate practice.

When in doubt, do the math: The Wizard of Oz = Al Lundy = God. Why not? Everything makes sense to a dead guy in a lion suit.