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Original question: What's the best thing you've heard your teacher say?The first day I walked into my 10th grade chemistry class, the lights were off. The teacher was waiting at the front of the room, holding a small candle in front of him. The candle's tiny flame was the only source of light in the room. The class murmured a bit, confused and excited. Once we’d all quieted down, the teacher said, “Tell me some of the observations you can make about this candle."We paused for a moment. That seemed a little too easy for chemistry class. "But keep in mind," the teacher added, "that observations are the things you observe empirically. Inferences are the things you conclude… and inferences can be wrong.”We stared at him, wary now.  At last, someone piped up, “Um...The flame is yellow?”“The flame gives off heat.""The wick is burning." "The wax is cylindrical!"The teacher raised his eyebrows at that one. "Are you sure?"We stared harder at the candle, not sure how to respond to that. It definitely looked cylindrical. The teacher shrugged. Then he put the candle into his mouth and started chewing.The class stared at him.At this point I was wondering if I ought to call the emergency room or something, but the teacher just kept chewing as though nothing was wrong. Eventually, he swallowed, shrugged again, and said, “Actually tastes okay.”As it turns out, the so-called “candle” wasn't made out of wax at all. It was actually just a slice of banana, with an almond slice for the wick.The student’s “wax” statement had actually been an inference.After explaining this to us, the teacher turned on the lights and said, "If there's one thing I want you to learn in this class, it's the first rule of scientific thinking: Never assume that your mind can’t lie to you."And that's the story of how my 10th-grade chemistry teacher swallowed a candle to teach us the first principle of scientific thinking. Here's to you, Mr. Miller.EDIT: A lot of readers seem to be confused about how the banana candle could be lit like a normal candle. The answer's pretty simple - you just light the sliver of almond on fire! If you want to know more, you can google "edible candle." Evan Zajc asked about this in the comments and then made one himself, and the result was pretty cool:Mr. Miller's candle looked a lot like Evan's, but without the awesome plate.

Did you know that a million dollars in U.S. currency weighs just ten kilograms? It's true. A freshly-minted $100 bill weighs in at slightly over a gram, and 100 of them is ten thousand dollars. 100 of those stacks, and there's your million.It’s not often that 10 kilograms - 22 lbs of anything can change your life. But on February 25th, 2014, that’s exactly what happened.  Day 1: $1,000,000 As the man in the gray suit walks away, I shout after him “Hey, come back here. Who are you? What’s this all about?” He does not look back and quickens his pace. Between the choice of chasing down a stranger, or securing what appeared to be stacks of currency, I chose the currency. We can resolve the issue of his identity later, but a loose sack of cash is, well, a loose sack of cash. I look through the contents again. Bundles of US$100 bills, stacked a hundred bills deep, wrapped in standard $10,000 bank bands. A quick count revealed that there were precisely a hundred of those stacks in the bag, and spot-check riffle-counts of the $10k bands suggest that there are no short-stacks within. These were full bands of $10,000 apiece of non sequential USD$100 bills, and I was holding what appears to be a million even in cash. And it feels like just as many question are swirling in my head, as I feel my pulse pounding in my skull. Who was that guy? Why me? What is this all about? But the most urgent thoughts swim past the dizzying deluge of unanswerable questions. Fakes. It’s one thing to inadvertently be the recipient of counterfeit currency; as you’re reading this very sentence, a clerk at a retail store somewhere in your city just accepted a counterfeit bill and made change from the real money in the till.  But to be in possession of a life-changing amount of counterfeit currency of the United States of America? Well, that’s sort of thing that can bring the full might and wrath of their law enforcement apparatus on your head. My emotions swing wildly between the elation of instantaneous wealth, and sheer terror that I was minutes away from being snatched from my home and corralled into a Federal holding cell, where I will grow old within its walls. Terror was the stronger of the two emotions, and I quickly went to work. First things first: the bag had to go. If there is a GPS tracking device embedded in its seams, it would take too long for me to root it out. Better to incinerate it, and make sure that whatever trail it was laying stops at a dead-end for its pursuers. I pour the stacks of bills into an empty duffle back from my garage, and lock the bag in my condo. There’s an abandoned marina just a mile from my home and I get in my car and drive straight to the docks, at the top of the posted speed limit. After pouring enough Kerosene on the bag to see the shimmering mist of petroleum evaporate above it, I lit a book of matches and threw it in the middle of the mass. A satisfying “Whoomph” lights up the fire, and I watch the edges of the bag curl and burn - sizzling in the midmorning sun. As the remnants of the bag’s embers swirl around the scorched mark on the docks, I drive back to my condo, pulse still pounding in my skull.I still haven’t figured out if the bills are real or not, but if this morning’s bag-drop was an attempt to pin a piece of deeply incriminating evidence bearing a tracking device … well that plan has been thwarted. Or delayed, at the very least. What do I do? What should I do? Call the authorities? Consider how it would sound: “Hi, Police? Somebody dropped a million dollars in cash at my home. I don’t know if it’s fake or not. Please help.” Would you believe such a ridiculous story? I wouldn't. Any reasonable law enforcement dispatcher would consider the caller legally insane, and I'd be arrested on the spot and sent to psychiatric care. If the money was real, it’d be seized and I'll never see it or spend it. If it was fake, they’d find a way to stick “possession of counterfeit currency” charge on me, and I'll be shoved into a Federal concrete box, draining the best years of my life away, only to be released when I can’t chew solid food any more. No. The only recourse is to handle this myself. I call an old college friend practicing criminal defense law in New York City: “Hey Roger, it’s Kai. How’ve you been?” “I'm cool. It's been a while. What’s up man.” “We should catch up soon in person. But I’m calling because I need something.” “Ok, shoot.” I swallow hard - it’s difficult to even say the words: “Who’s the best CrimDef lawyer you know in California, who defends against Federal charges?” A moment. His voice lowers noticeably. “Shit, man. You in some kind of trouble?” “I’m not sure yet.” I said, truthfully. “But I need someone experienced and smart ... someone who you’d hire, if you’re facing serious attention from the Feds." He lets out a long exhale. “Vincent King. Former rockstar DOJ prosecutor in D.C. Had a change of heart halfway through his rotation in Maryland, when he was securing Life sentences for “interstate drug transportation” charges on young Black kids who were busted muling for the cartels. Was offered a fast-track promotion straight to the U.S. Attorney’s office but went rogue. He set up independent shop in San Francisco, fighting Fed cases. Heavy hitter clients, but makes a point of refusing to represent anyone accused of murder or human trafficking. Intimate knowledge of Federal prosecutorial procedures and evidence-collection protocol. Smart. Methodical. Very expensive.” “Perfect.” “I did mention ‘very expensive?’” “You did.” “I’ll send his contact information now.” =================================“I’m sorry - Mr. King is in court all day and won’t be back in the office. His earliest appointment is tomorrow morning after a client meeting. Shall I book him for 11am for you?” “Yes, thank you Marta.” “We’ll see you tomorrow at 11 then.” I look at the digital clock in my kitchen -  it reads 10:44am. Just me and a stack of bills which may or may not be fake, no formal legal representation for over 24 hours. It’s going to be a long day. Taking even a few of these bills to a bank to corroborate their authenticity is out of the question. If a bank officer confirms they are fraudulent, I’ll be arrested on the spot, and since I haven’t hired counsel, I’d be at the mercy of the Public Defender’s Office - the most overworked and underpaid division of the American Criminal Justice system. No, thank you. The next number I dial is an old friend, Robert Kendrick, sole proprietor of ‘Secher Nbiw - The Golden Path,’ a gold bullion dealer with a whimsical Dune reference in the name of his shop. I’ve known Robert for over a decade; his business deals in large amounts of (mostly) legal cash. By necessity, he has a high-end currency counter/ counterfeit detection device in his office, which can swiftly count and verify large sums of money with precision. “Bobby, it’s me.” “Hey, what’s up.” “Can I come to your office - like right now?” “Sure, what do you need?” “I, uh, came into some money. Long story, and I really don’t want to get too much into the details … but I’m wondering if you’d be willing to run the bills through your counter for me? I’m not 100% sure they’re real, and I’d like a discreet way of verifying them. If they are, I’m going to pick up some bullion as well.” “Sure man. Happy to help. How much money are we talking about?”“$60,000” I flinch at that - I hate lying to friends, but at this point, I have no idea who to trust. Though if you want to be technical about it, I did come across $60,000. I am just simply not telling Kendrick about the other $940,000 that accompanied the $60k in the satchel that dropped into my life just three hours ago. “Come on by.” I pull apart a few $10,000 currency bands and start plucking random $100 bills from the middle of every 10k stack to assemble a randomized sample of the entire million. 100 bills, wrap it up. 100 bills, wrap it up. 100 bills, wrap it up. Three bands, thirty thousand dollars, randomized and fully assembled to be tested for authenticity. “Half” of my alleged $60k windfall. The rest of the loose bills are refolded back so there remains 97 stacks of $10k racks, re-wrapped and properly sorted. In 30 minutes, I will figure out if I’m rich, or holding on to enough illicit contraband to send me to Federal Prison for the rest of my life. =============================The Golden Path, like most bullion dealers, work out of small, highly secured office covered by multiple layers of security. At any given moment, Robert may have several hundred thousand dollars in cash or gold, silver and platinum bullion on the premise, it pays to be careful. One of the few civilians in California with a Concealed Carry Weapons permit, Kendrick and I met on pistol gun range ten years ago; we bonded over shooting .45 ACP slugs down-range. He and I spent countless hours debating the relative merits of his preference for single-action 1911s, vs my bias toward double-action SIG-Sauer P220s. In the bullion business, you learn to know the boundaries of money-laundering laws, and know how to walk right up to the edge without triggering reporting thresholds. Drop US$10,000 in cash or more at a car dealership, bank or bullion dealer in a single day’s transaction, and the U.S. authorities gets very interested in the source of your funds. By law, these business that receive such sums of cash must fill out invasive forms to tie the transaction to you and your Social Security Number. Keep cash transactions below US$10,000, and you can avoid much of that intense scrutiny. “Welcome back man. I haven’t seen you in a while.” A discreet man, Kendrick does not inquire further about the source of the cash. In the business of buying and selling gold bullion, you learn to comply with the letter of the law, while avoiding conversational topics that can jeopardize one’s own plausible deniability. While his clientele is mostly legitimate, I’m certain the most lucrative of his customers are criminals - and he smart enough to know not to ask the sort of questions that open up a line of liability for him.  So long as the proper theatrics of anti-money-laundering protocols are observed, everyone is technically in the clear. I hand him the three $10k stacks and he pulls the bands off them and puts the entire block in his high-speed currency counter. After a second, the machine spools up and the digital counter swiftly runs from zero to three hundred. Thirty thousand dollars. “It’s real.” It’s real. His words hang in the air for a moment, and it takes a moment for them to sink in. One million dollars. Genuine currency of the United States of America, the most recognized and accepted form of money in the world - denominated in crisp, non-sequential bills. I hold my face as neutral as possible, but my excitement made me slightly dizzy, and I am glad I was sitting down. “What’s the spot price of Gold today?” Kendrick’s eyes drift to his laptop computer, where the current day’s commodities prices were fed to him via a live stream. “$1334 Ask, $1335 Bid.” I nodded my understanding.Precious metals bullion trade in troy ounces, and prices are quoted on a per troy oz basis; depending on the specific type of bullion (bars, coins, make), there are different markups from the quoted price. Depending on the specific form, Gold is typically marked up by USD$20 to $60 over the day’s quoted Bid price, and sells for $5~10 over the Ask. “What do you have in inventory right now for gold?” “The usual. South African Kugerrands. American Eagles. Canadian Maples. Oh, I do have a lovely Credit Suisse 5oz bar that somebody just sold to me, and I’m happy to let it go for $25/oz over spot.” I quickly did the mental math calculation. With the hard-cap spending limit of $10,000 before I trigger any mandatory anti-money-laundering paperwork, $1335/oz works out to about seven troy ounces of bullion I can buy, without forcing Robert to fill out invasive forms about me and my identity. “I’ll take the 5oz Credit Suisse bar, and two American Gold Eagles.” Kendrick pulls out a calculator and taps in the numbers, “So five troy ounces at 25 over spot plus Eagles at $50 over spot works out to nine thousand six hundred and -“ “Take ten grand and keep the change.” I interrupt. “I will be back for more.” He raises his eyebrow, but says nothing. “Thank you. I’ll be right back.” He counts back $20,000 and hands it to me, taking the $10,000 in the back room of his office and returning with the 5oz Swiss bar and two heavy 1oz American Eagles, along with a receipt for US$9675. I pause for a moment and hand him back one of the $10,000 stacks. “I know the limit is $10k in transactions per day. Consider this pre-payment for a purchase tomorrow. Your call, on a mix of anything up that totals up to $9500. Keep the rest for you and Katie.” A barely-perceptible smile flickers across his face, then his face was clear again. “Sure thing.” There’s nothing like the feeling of holding physical gold - the density, color and heft of the metal is like no other substance on earth, and it is no wonder that since its discovery, every culture on Earth treated gold with awe and respect. With 18 hours left before I can understand my legal options, there’s only two things I know for certain: 1. The money is real. 2. At least one person knows exactly where I live, and where the money was dropped off. I need to get mobile. I need to get mobile and off the grid ASAP.... to be continuedIf you'd like to be the first to get updates to this story, please add me kai chang 張敦楷 (kaichang) on Twitter. Part 2 (of 10) is being written right now, will be announced on Twitter. Please follow for updates on the saga of the Quora Millionaire! :D

Indeed!!! This story will make you laugh!!!Yeah, one time, I decided to try out for the TV game show “Who Wants to Be a Millionaire.” I filled out the online application form, and took the online quiz, and — Lo and Behold! — I passed! Yes!!!! I passed!!! I was accepted for an audition in New York City!!!Now feeling quite smug about my own superior intelligence, I patted myself on the back and gave myself a hearty “YES!!!” (with thumbs up) every time I sat on my bed, in the middle of the afternoon, watching “Who Wants To Be a Millionaire” with my cat.Ha!!! Ha!!! Yes!!!!!!!!!!!! I guessed the right answer once again! HA! HA! HA!!!!!!!!!!!!!!!!I am brilliant!!! I am GENIUS!!!! AHA!!!!Then came the big day — try outs — in New York City.I had flown there the day before, taken a taxi to my cheap little pathetic (expensive) hotel in the middle of downtown Manhattan, where I was kept awake most of the night from the bright lights and sounds of “the City that never sleeps.” Also, the smell of roach spray in the hallways and in my pathetic little room, with the shared bathroom down at the end of the hallway, did not produce feelings of sleepiness, either.In the morning, I got up early and went to find a beauty salon. I wanted to look my best. I wanted to look like the winning contest during my audition.New York City beauty salons are not cheap, I discovered! The hair cut and style and manicure cost me well over $250, but, hey, when you’re about to become a millionaire, I realized that little expensive was just a drop in the bucket.The next taxi dropped me off at the audition location. I had my audition acceptance letter in hand, as I joined the crowd standing in the long line.The long line soon queued up. There were literally hundreds waiting in the long line.Soon, the doors opened, and we were ushered in, like cattle, and directed to be seated at various tables where we were given name tags. The guy sitting across from me had already been a contestant on Jeopardy. “This will be a piece of cake,” he stated, smugly.There were many pencils on the table that had “Who Wants To Be A Millionaire” engraved on them.Soon, we were given a test sheet. Our sheets were all unique, so no one could cheat off of anyone’s paper. We were instructed to use those “millionaire” pencils to fill in the correct little multiple test circles.When the bell rang, we were given 10 minutes to answer thirty questions. Some of the questions were complicated math word problems, which I could figure out, if given enough time. Other questions were about pop culture, country music, or history, science, etc.I was on about the seventh question when the test administrator announced, “You have five more minutes.”Five minutes???I instantly felt panic-stricken, absolutely panicked! I looked at the Jeopardy guy across from me and noticed that he was already on Question 29!Before time was called, he had already finished, and had turned over his paper, and was now just sitting there, leaning forward with his hands clasped together, with that smug little smile on his face.“One more minute!” the test administrator announced.At this point, with more than half of the questions still unanswered, I just simply decided to hastily mark them all “C.” Right? Didn’t I, the genius, read somewhere that “C” is the best guess?????C, C, C, C, C, C, C, C, C, C . . . .DING! Time’s up!We were instructed to leave all of our test papers face down and to just sit and wait.While we waited to hear the results of who “made” it, and hear which auditioners would be invited for the Next Step (to participate in a mock game), the man across from me smugly bragged about how “easy” the test had been for him.Sure enough, out of a room of about three hundred applicants, about five names were called. Yep! Only five people made it to the next step out of about three hundred applicants.The guy across from me was one of the five.Back in Gainesville, Florida, my friends were eager to know when I would be appearing on Millionaire.“Well, I will not be appearing on Millionaire,” I informed them, “but here’s a pencil from the try outs.” Everyone LOVED their “Who Wants to Be a Millionnaire” pencil.I should have kept one for myself. Those free pencils — that we were invited to take off the tables — had cost me more than $1,000 — between the plane ticket, the new outfits, roach spray smelling dingy hotel room, beauty salon, etc.The following day, I sat on my bed, watching an episode of Millionaire with my cat.Wouldn’t you know it! I got an answer right that the contestant missed!!!Dang!!!

I can only speak for myself.I visited India four times for work and a tiny bit of travel. Here's what I think of India:Best food in the world hands down.Mess. India is messy. The roads seem impossible to navigate. The non-stop honking looks pointless - honking is supposed to draw attention to danger, but if everybody is honking all the time, how can anybody notice it? And so many people walking the streets everywhere. The busiest streets of Tel-Aviv and Jerusalem can't compare to an average street in Mumbai, Pune or Bangalore (the cities I visited). Yet somehow everything works out.I keep hearing a lot about hostility between Hindus and Muslims in India, but I see Hindu buildings and Muslim buildings everywhere side by side without any apparent problems between them. More broadly, India has many religions and ethnic groups, and even though not everything is smooth, Indians don't kill each other much because of that. I envy this.It's strange and sad to me that the native languages of India are not used more in communications, education, advertisement, government etc. More English makes it comfortable for tourists, of course, but you should feel more pride about your own languages. You should stop thinking that English is so important. From what I heard, all higher education in India is in English, and it's almost impossible to find a decent job without knowing this foreign language. My Indian friends are very surprised every time I tell them that we speak Hebrew at work, even in software companies, and that almost all of our schools and universities are in Hebrew medium. Although we could do much better ourselves, India could still learn some things from us about respecting the native language. Letting people use their languages more will unleash India's huge human and economic potential.Bureaucracy. Israeli bureaucracy can easily make a reasonable person desperate, but Indian bureaucracy often goes beyond ridiculous. The consulate promises a business visa for five years and forces me to fill a lot of weird forms and eventually gives me a six-month visa. The consulate clerk asks me whether I have a permission from the government of India to have the business meetings; surprised, I ask whether I need a permission to have a meeting and he says that I don't. And so on.The head bobble is baffling. I have no idea whether it's a yes or a no. Can be both.Weird internal racism is taken for granted. I went over personal marriage announcements in newspapers and I was shocked: They were meticulously sorted by language, religion and caste, and if that's not enough, bright skin of both boys and girls was presented as a positive thing. At most, I would imagine that it's fine to mention skin color in a neutral way, as an innate characteristic of a person, but definitely not as a positive or a negative thing. Such a thing wouldn't fly in western countries these days.The taxi drivers don't know where to go. I would think that the whole point of being a taxi driver is knowing all the streets in the city, but taxi drivers in India don't.Best food in the world hands down!An odd and somewhat unpleasant point: A surprising number of Indians see Hitler positively. To almost all people in Europe, Russia and America, and especially in Israel, Hitler is the worst monster ever, World War II is the most destructive event ever, which hurt all the countries that participated in it, including Germany, and "Mein Kampf" is a disgusting and hateful book. In India, apparently, some people think that Hitler was a powerful and good leader, and "Mein Kampf" is a bestseller, and they ignore his responsibility for the death of many millions of people. It may be due to lack of education, or maybe for some other reason.India is so cheap. A taxi for an hour in Mumbai costs like a taxi for five minutes in Jerusalem. A whole lunch in Pune costs less than a sandwich in a coffee shop in Tel-Aviv.Cricket, cricket, cricket everywhere. A lot of people in Israel love football and basketball, but my impression is that it's nowhere near the passion that India has for cricket. Everybody loves it. Young, old, men, women, Muslims, Hindus, everybody.I frequently think that even though Israel is a heavily militarized country, India is even more so. But maybe that's because when I'm in India, I spend most of my time in Pune, with all its military camps.I have this strange feeling that people in India don't really understand each other even when they speak in their own language, be it Kannada, Hindi, Marathi, Tamil or anything else.Though some of these things may read as kind of negative and strange, it doesn't change the most important part: Indians are friendly, well-meaning and clever people.Oh, and:Best food in the world hands down!!!Note: I don’t want to disable comments, but before you write a comment, please check the existing ones by clicking “All” in the comments section. This answer has a lot of comments, and it’s quite possible that somebody has already written what you want to write. In that case, you may want to just upvote that one instead of writing a new one. But if you have something new to say, go for it.

Spend less time using your blocked Internet to ask questions on Quora, andTravel back in time to when there was no curfew and you were playing Super Mario Kart, and instead, fill out your exam form.

You arrive at lecture and sit perched on the edge of your seat, notebook open to a clean page and freshly-sharpened pencil in hand. You follow every word the professor says. Well, maybe you zone out a few times in the middle, but who doesn't? Besides, you're copying everything down and can review it later.That weekend, you diligently read the textbook. Maybe you skip a few parts since it's a busy week, but you definitely study the chapter summary and read all the examples. You do the homework problems, even starting three days early. When you're stuck, you go to office hours and ask the TA for help until they show you how to do it.Before the exam, you study your notes and the published homework solutions. You try the practice exam, and it seems the pieces are finally falling into place. You can solve most of the problems and remember most of the formulas and derivations! At last you take the final, referencing the single allowed sheet of notes you prepared at length the night before. You get almost every question right, or at least partial credit, and take home a well-deserved A.Three months later, you can hardly remember what the class was all about. What's going on? Why did you forget so much? Are you the only one? Should you have memorized more and worked even harder?The answer is no. A student who memorizes the entire physics curriculum is no more a physicist than one who memorizes the dictionary is a writer. Studying physics is about building skills, specifically the skills of modeling novel situations and solving difficult problems. The results in your textbook are just the raw material. You're a builder. Don't spend all your time collecting more materials. Collect a few, then build things. Here's how.The Cathedral and the StonesWhile delivering his famous set of freshman lectures on physics, Richard Feynman held a few special review sessions. In the first of these, he discussed the problem of trying to memorize all the physics you've learned:It will not do to memorize the formulas, and to say to yourself, "I know all the formulas; all I gotta do is figure out how to put 'em in the problem!"Now, you may succeed with this for a while, and the more you work on memorizing the formulas, the longer you'll go on with this method - but it doesn't work in the end.You might say, "I'm not gonna believe him, because I've always been successful: that's the way I've always done it; I'm always gonna do it that way."You are not always going to do it that way: you're going to flunk - not this year, not next year, but eventually, when you get your job, or something - you're going to lose along the line somewhere, because physics is an enormously extended thing: there are millions of formulas! It's impossible to remember all the formulas - it's impossible!And the great thing that you're ignoring, the powerful machine that you're not using, is this: suppose Figure 1 - 19 is a map of all the physics formulas, all the relations in physics. (It should have more than two dimensions, but let's suppose it's like that.)Now, suppose that something happened to your mind, that somehow all the material in some region was erased, and there was a little spot of missing goo in there. The relations of nature are so nice that it is possible, by logic, to "triangulate" from what is known to what's in the hole. (See Fig. 1-20.)And you can re-create the things that you've forgotten perpetually - if you don't forget too much, and if you know enough. In other words, there comes a time - which you haven't quite got to, yet - where you'll know so many things that as you forget them, you can reconstruct them from the pieces that you can still remember. It is therefore of first-rate importance that you know how to "triangulate" - that is, to know how to figure something out from what you already know. It is absolutely necessary. You might say, "Ah, I don't care; I'm a good memorizer! In fact, I took a course in memory!"That still doesn't work! Because the real utility of physicists - both to discover new laws of nature, and to develop new things in industry, and so on - is not to talk about what's already known, but to do something new - and so they triangulate out from the known things: they make a "triangulation" that no one has ever made before. (See Fig. 1-21.)In order to learn how to do that, you've got to forget the memorizing of formulas, and to try to learn to understand the interrelationships of nature. That's very much more difficult at the beginning, but it's the only successful way.Feynman's advice is a common theme in learning. Beginners want to memorize the details, while experts want to communicate a gestalt.Foreign language students talk about how many words they've memorized, but teachers see this as the most trivial component of fluency. Novice musicians try to get the notes and rhythms right, while experts want to find their own interpretation of the piece's aesthetic. Math students want to memorize theorems while mathematicians seek a way of thinking instead. History students see lists of dates and facts while professors see personality, context, and narrative. In each case, the beginner is too overwhelmed by details to see the whole. They look at a cathedral and see a pile of 100,000 stones.One particularly clear description of the difference between the experts' and beginners' minds comes from George Miller's 1956 study "The magical number seven, plus or minus two." Miller presented chess boards to both master-level chess players and to novices. He found that the masters could memorize an entire board in just five seconds, whereas the novices were hopeless, getting just a few pieces. However, this was only true when the participants were memorizing positions from real chess games. When Miller instead scattered the pieces at random, he found the masters' advantage disappeared. They, like the novices, could only remember a small portion of what they'd seen.The reason is that master-level chess players have "chunked" chess information. They no longer have to remember where each pawn is; they can instead remember where the weak point in the structure lies. Once they know that, the rest is inevitable and easily reconstructed.I played some chess in high school, never making it to a high level. At a tournament, I met a master who told me about how every square on the chess board was meaningful to him. Whereas, when writing down my move, I would have to count the rows and columns to figure out where I had put my knight ("A-B-C, 1-2-3-4, knight to C4") he would know instantaneously because the target square felt like C4, with all the attendant chess knowledge about control of the center or protection of the king that a knight on C4 entails.To see this same principle working in yourself right now, memorize the following. You have two seconds:首先放花生酱，然后果冻Easy, right? Well, it would be if you were literate in Chinese. Then you’d know it’s the important maxim, “first the peanut butter, then the jelly”.You can remember the equivalent English phrase no problem, but probably don't remember the Chinese characters at all (unless you know Chinese, of course). This is because you automatically process English to an extreme level. Your brain transforms the various loops and lines and spaces displayed on your screen into letters, then words, then a familiar sandwich-related maxim, all without any effort. It's only this highest-level abstraction that you remember. Using it, you could reproduce the detail of the phrase "first the peanut butter, then the jelly" fairly accurately, but you would likely forget something like whether I capitalized the first letter or whether the font had serifs.Remembering an equally-long list of randomly-chosen English words would be harder, a random list of letters harder still, and the seemingly-random characters of Chinese almost impossible without great effort. At each step, we lose more and more ability to abstract the raw data with our installed cognitive firmware, and this makes it harder and harder to extract meaning.That is why you have such a hard time memorizing equations and derivations from your physics classes. They aren't yet meaningful to you. They don't fit into a grand framework you've constructed. So after you turn in the final, they all start slipping away.Don't worry. Those details will become more memorable with time. In tutoring beginning students, I used to be surprised at how bad their memories were. We would work a problem in basic physics over the course of 20 minutes. The next time we met, I'd ask them about it as review. Personally, I could remember what the problem was, what the answer was, how to solve it, and even details such as the minor mistakes the student made along the way and the similar problems to which we'd compared it last week. Often, I found that the student remembered none of this - not even what the problem was asking! What had happened was, while I had been thinking about how this problem fit into their understanding of physics and wondering what their mistakes told me about which concepts they were still shaky on, they had been stressed out by what the sine of thirty degrees is and the difference between "centrifugal" and "centripetal".Imagine an athlete trying to play soccer, but just yesterday they learned about things like "running" and "kicking". They'd be so distracted by making sure they moved their legs in the right order that they'd have no concept of making a feint, much less things like how the movement pattern of their midfielder was opening a hole in the opponent's defense. The result is that the player does poorly and the coach gets frustrated.Much of a technical education works this way. You are trying to understand continuum mechanics when Newton's Laws are still not cemented in your mind, or quantum mechanics when you still haven't grasped linear algebra. Inevitably, you'll need to learn subjects more than once - the first time to grapple with the details, the second to see through to what's going on beyond.Once you start to see the big picture, you'll find the details become meaningful and you'll manipulate and remember them more easily. Randall Knight's Five Easy Lessons describes research on expert vs. novice problem solvers. Both groups were given the same physics problems and asked to narrate their thoughts aloud in stream-of-consciousness while they solved them (or failed to do so). Knight cites the following summary from Reif and Heller (1982)Observations by Larkin and Reif and ourselves indicate that experts rapidly redescribe the problems presented to them, often use qualitative arguments to plan solutions before elaborating them in greater mathematical detail, and make many decisions by first exploring their consequences. Furthermore, the underlying knowledge of such experts appears to be tightly structured in hierarchical fashion.By contrast, novice students commonly encounter difficulties because they fail to describe problems adequately. They usually do little prior planning or qualitative description. Instead of proceeding by successive refinements, they try to assemble solutions by stringing together miscellaneous mathematical formulas from their repertoire. Furthermore, their underlying knowledge consists largely of a loosely connected collection of such formulas.Experts see the cathedral first, then the stones. Novices grab desperately at every stone in sight and hope one of them is worth at least partial credit.In another experiment, subjects were given a bunch of physics problems and asked to invent categories for the problems, then put the problems in whatever category they belonged. Knight writes:Experts sort the problems into relatively few categories, such as "Problems that can be solved by using Newton's second law" or "Problems that can be solved using conservation of energy." Novices, on the other hand, make a much larger number of categories, such as "inclined plane problems" and "pulley problems" and "collision problems." That is, novices see primarily surface features of a problem, not the underlying physical principles.The "Aha!" FeelingIt is clear that your job as a student is to slowly build up the mental structures that experts have. As you do, details will get easier. Eventually, many details will become effortless. But how do you get there?In the Mathoverflow question I linked about memorizing theorems, Timothy Gowers wroteAs far as possible, you should turn yourself into the kind of person who does not have to remember the theorem in question. To get to that stage, the best way I know is simply to attempt to prove the theorem yourself. If you've tried sufficiently hard at that and got stuck, then have a quick look at the proof -- just enough to find out what the point is that you are missing. That should give you an Aha! feeling that will make the step far easier to remember in the future than if you had just passively read it.Feynman approached the same questionThe problem of how to deduce new things from old, and how to solve problems, is really very difficult to teach, and I don't really know how to do it. I don't know how to tell you something that will transform you from a person who can't analyze new situations or solve problems, to a person who can. In the case of the mathematics, I can transform you from somebody who can't differentiate to somebody who can, by giving you all the rules. But in the case of the physics, I can't transform you from somebody who can't to somebody who can, so I don't know what to do.Because I intuitively understand what's going on physically, I find it difficult to communicate: I can only do it by showing you examples. Therefore, the rest of this lecture, as well as the next one, will consist of doing a whole lot of little examples - of applications, of phenomena in the physical world or in the industrial world, of applications of physics in different places - to show you how what you already know will permit you to understand or to analyze what's going on. Only from examples will you be able to catch on.This sounds horribly inefficient to me. Feynman and Gowers both signNowed the highest level of achievement in their domains, and both are renowned as superb communicators. Despite this, neither has any better advice than "do it a lot and eventually expertise will just sort of happen." Mathematicians and physicists talk about the qualities of "mathematical maturity" and "physical insight". They're essential to moving past the most basic level, but it seems that no one knows quite where they come from.Circular ReasoningThere are certainly attempts to be more systematic than Feynman or Gowers, but before we get to that, let's take a case study. I recall that as a college freshman, I knew that the formula for the acceleration of a ball orbiting in a circle was [math]a = v^2/r[/math]. I wanted to know why, so I drew a picture:I imagined a small ball starting on the right side of the circle, heading upwards where the blue velocity vector [math]v_1[/math] is drawn. The ball moves around the circle, goes counter-clockwise over the top and then heads downwards on the left hand side, where the red velocity [math]v_2[/math] is. The ball's velocity changed, which means it accelerated. The acceleration is[math]a = \frac{\Delta v}{\Delta t}[/math][math]\Delta v[/math] is clearly [math]2v[/math], and [math]\Delta t[/math] is the time it takes to go half way around the circle, which is [math]\frac{\text{distance}}{\text{speed}} = \frac{\pi r}{v}[/math]. Hence, the acceleration is[math]a = \frac{2v}{\pi r/v} = \frac{2 v^2}{\pi r} \approx 0.64 \frac{v^2}{r}[/math]This isn't quite right. The answer is supposed to be [math]v^2/r[/math]. Somehow there is an extra factor of [math]2/\pi[/math] floating around.If you already understand calculus, this is a silly and obvious mistake. But for me it took quite some time - weeks, I think - until I understood that I had found the average acceleration, but the formula I was trying to derive was the instantaneous acceleration.The way I broke out of this mental rut was to think about the case where the ball has gone one quarter of the way around, like this:Then the same approach gives[math] a = \frac{\Delta v}{\Delta t} = \frac{2\sqrt{2}v^2}{\pi r} \approx 0.90 \frac{v^2}{r}[/math],which is closer to the right value. If you try it when the ball goes 1/8 the way around, you get[math]a = \frac{4 \sqrt{2 - \sqrt{2}}v^2}{\pi r} \approx 0.97 \frac{v^2}{r}[/math]and you're getting the idea that what you have to do is take the limit as the ball goes an infinitesimal fraction of the way around. (By the way, if I had been clever, maybe I'd have discovered Viète's formula this way, or something like it. I only recognized this now because I remembered encountering Viete's formula. So memory certainly has its place in allowing you to make connections. It's just not as central as beginners typically believe.)How do you do that "infinitesimal fraction of the way around" thing? Well, if the ball travels an angle [math]\theta[/math] around the circle, we can draw the before and after velocities asand[math] \Delta v = 2 \sin (\theta/2) v[/math]which in the limit [math]\theta \to 0[/math] becomes[math] \Delta v = \theta v[/math]and[math] a = \frac{\Delta v}{\Delta t} = \frac{\theta v}{\theta r/v} = \frac{v^2}{r}[/math]But all of this took a long time to come together in my mind, assembling gradually, but in discrete chunks with each small epiphany. As I walk through it now, I can see there are many concepts involved, and in fact if you're a beginning student it's likely that the argument isn't clear because I skipped some steps.The main idea in that argument is calculus - we're looking at an infinitesimal displacement of the ball. To understand the entire argument, though, we also need to do a fair amount of geometry, develop the idea of sliding velocity vectors around in space so they originate at the same point, introduce the concept of an arbitrary angle of rotation [math]\theta[/math], find the time it takes to rotate by that angle for a given [math]r[/math] and [math]v[/math], use the small-angle approximation of the sine function, and maybe a couple other things I'm not seeing.That's a lot of mental exercise. It's no wonder that working all this out for yourself is both harder and more effective than reading it in the book. Just reading it, you'll skip over or fail to appreciate how much goes into the derivation. The next time you try to understand something, you want those previously-mastered ideas about geometry and calculus already there in your mind, ready to be called up. They won't be if you let a book do all the work.Today, I can solve this problem in other ways. For example, I could write down the rectangular coordinates and differentiate, describe the motion in the complex plane as [math]r e^{i\omega t}[/math] and differentiate that, or transform to a rotating reference frame and note the centrifugal force on the stationary ball and conclude it must be accelerating in an inertial frame. A cute one is to write down the position and velocity vectors by intuition, and notice that going from position to velocity you rotate 90 degrees and multiply the length by [math]v/r[/math]. To go from velocity to acceleration is mathematically identical, so rotating another 90 degrees and multiply by [math]v/r[/math] again we obtain the answer.I can argue from dimensional analysis that the only way to get something with units of acceleration is [math]v^2/r[/math], or heuristically point out that if you increase the velocity, the velocity vectors get bigger, but we also go from one to the next in less time, so the acceleration ought to scale with [math]v^2[/math], etc.I also see aspects of the problem that I didn't back then, such as that this isn't really a physics problem. There are no physical laws involved. It would become a physics problem if we included that the ball is circling due to gravitational forces and used Newton's gravitational law, for example, but as it stands this problem is just a little math.So yes, I can easily memorize this result and provide a derivation for it. I can do that for most of the undergrad physics curriculum, including the pendulum and Doppler formulas you mentioned, and I think I could ace, or at least beat the class average, on the final in any undergraduate physics course at my university without extra preparation. But I can do that because I built up a general understanding of physics, not because I remember huge lists of equations and techniques.How to Chunk ItI can do these things now because of years' of accumulated experience. Somehow, my mind built chunks for thinking about elementary physics the same way chess players do for chess. I've taught classes, worked advanced problems, listened to people, discussed with people, tutored, written about physics on the internet, etc. It's a hodgepodge of activities and approaches, and there's no way for me to tease from my own experience what was most important to the learning process. Fortunately, people from various fields have made contributions to understanding how we create the cognitive machinery of expertise. Here is a quick hit list.George Pólya's How to Solve It examines the problem-solving process as a series of stages, and suggests the student ask themselves specific questions like, "Is it clear that there enough information to solve the problem?"Scott H Young, Cal Newport, and many others give specific advice on study skills: how to take notes, how to diagram out the connections between ideas, how to test your knowledge, how to fit what you're learning into the larger scheme of things, etc.When you do need to memorize things, spaced repetition software like Anki takes an algorithmic, research-backed approach to helping you remember facts with the minimum of time and effort.K. Anders Ericsson has tried to find the key factors that make some forms of practice better than others - things like getting feedback as you go and having clear goals. He refined these into the concept of Deliberate Practice. He also believes there is no shortcut. Even if you practice effectively, it usually takes around 10,000 hours of hard work to signNow the highest levels in complex fields like physics or music.Chunking and assigning meaning are your mind's ways of dealing with the information overload of the minutiae that inevitably pop up in any field. Another approach, though, is to try to expand your mind's ability to handle those minutiae. If you can push your "magical number" from seven to ten, you'll be able to remember and understand more of your physics work because it takes a bit longer to fill your cognitive buffer. Dual N-Back exercises are the most popular method of working on this. Nootropic drugs may also provide benefits to some people. Low-hanging fruit first, though. If you aren't sleeping 8-9 hours a day, getting a few hours of exercise a week, and eating healthy food for most meals, you're probably giving up some of your mind's potential power already. (There is individual variation, though.)Howard Gardner is one champion of the idea of multiple intelligences, or different learning types. When working on electric fields, for example, Gardner might advise you to study Maxwell's equations, draw pictures of vector fields and intuit their curls, get up and use your body, pointing your arms around to indicate electric field vectors, write or speak about what you're studying, learn with a friend or tutor, or maybe even create musical mnemonics to help you study, depending on where your personal strengths lie. Certainly, all students should build facility with drawing sketches, plotting functions, manipulating equations, visualizing dynamics, and writing and speaking about the material.Psychologist Carol Dweck's research studies the effect of your attitude towards learning on how much you learn, finding, for example, that children praised for their hard work are likely to press on further and learn more when given tough problems, whereas children praised for their intelligence are more likely to give up.Productivity guru David Allen helps people organize their lives and defeat procrastination with specific techniques, such as dividing complicated tasks into small, specific "next actions" and deciding when to do them, then organizing them in a planner system.Mihály Csíkszentmihályi believes that people operate best in a state of "flow", where they are so focused on the task they find it enjoyable and engrossing to the point they're innately motivated to continue. He emphasizes, for example, that the task needs to be the right level of difficulty - not too hard and not too easy - to find the flow state. (Some people think this state doesn't jibe with deliberate practice; others contend it's possible to achieve both simultaneously.)Taken together, this yields enough practical advice to chew on for months or years. To summarize, when you are learning something new:Try to figure it out for yourselfIf you get stuck, take a peek at your textbook to get the main ideaTeach the idea to someone elseOnce you've learned something, repeat the entire reasoning behind it for yourself, working through each detailAsk yourself Pólya's questions when you're stuckUse Young and Newport's techniques to map out the ideas of your class and relate them to your prior knowledgeMake Anki decks and review them a few minutes a day to retain what you've learnedMake sure your study sessions include all the principles of deliberate practice, especially feedback, challenge, and attentionBuild an image of yourself as someone motivated by learning and proud of having worked hard and effectively rather than as someone proud of being smart or renowned.Find a organizational system that lets you handle all the details of life smoothly and efficiently.Search for the flow state, notice when you enter it, and put yourself in position to find flow more and more often.Work on different subjects, reviewing both advanced and basic material. They will eventually all form together in your mind, and you're likely to have to take at least two passes at any subject before you understand it well.Take care of your physical health.This list does not include reading every page of the textbook or solving every problem at the end of the chapter. Those things aren't necessarily bad, but they can easily become rote. Building the material up for yourself while dipping into reference materials for hints is likely to be more effective and more engaging, once you learn to do it. It is a slow, difficult process. It can be frustrating, sitting there wracking your brain and feeling incredibly stupid for not understanding something you know you're supposed to have down. And strangely, once you have it figured out, it will probably seem completely obvious! That's your reward. Once the thing is obvious, you've chunked it, and you can move on. (Though you still need to review with spaced repetition.) This is the opposite of the usual pattern of sitting in lectures and feeling you understand everything quite clearly, only to find it all evaporated the next day, or acing a final only to find your knowledge is all gone the next month.That, I believe, summarizes the practical knowledge and advice about the learning process. Memorizing equations and derivations is difficult and ineffective because they are just the details. You can only handle a few details before your mind gets swamped. To cope, train yourself to the point where you process equations and physical reasoning automatically. This will free your conscious effort up to take in the big picture and see what the subject is all about.It Just Gets In The Way, You SeeSomehow, I've developed a "this is calculus" instinct, so that if I see the problem about acceleration in circular motion, or any other problem about rates of change, I know that it's talking about a limit of some kind. Where does this instinct exist in my brain? What form does it take? How does it get called up at the right time?George Lakoff believes that almost everything we understand is via metaphor. Any sort of abstract concept is understood by linking it to concrete concepts we've previously understood. For example, in Where Mathematics Comes From, Lakoff and coauthor Rafael Nuñez argue that we think of the mathematical concept of a "set" as a sort of box or container with things stacked in it. We reason about sets using our intuition about boxes, then later go back and support our conclusions with the technical details. Learning to reason about sets, then, is learning to think about the box metaphor and translate it back and forth into the formal language of axioms and theorems. This seems to fit with the introspective reports of many mathematicians, who say they build intuitive or visual models of their mathematics when finding results, then add in the deltas and epsilons at the end.This may be why we so often see beginning students asking things like, "but what is the electron, really?" If they were told it is just a tiny little ball, that would work, because it's a very easy metaphor. But instead, they're told it's not a ball, not a particle, not a wave, not spinning even though it has spin, etc. In fact, they're told to dismiss all prior concepts entirely! This is something Lakoff believes is simply impossible. No wonder students are bobbing in an ocean of confused thought bubbles, with nothing but mixed metaphors to grasp at until the last straw evaporates, across the board.Linguists like Steven Pinker believe that the language we use tells us how our mind works. Physicists certainly do have a specialized lexicon, and the ability to use it correctly correlates pretty well to general physics intuition, in my experience. In his review of Pinker's The Stuff of Thought, Douglas Hofstadter summarizes:Pinker shows, for example, how subtle features of English verbs reveal hidden operations of the human mind. Consider such contrasting sentences as "The farmer loaded hay into the wagon" and "The farmer loaded the wagon with hay." In this pair, the verb "load" has two different kinds of objects: the stuff that gets moved and the place it goes. Also, in the first sentence, the destination is the object of one preposition; in the second, the stuff is the object of another. Pinker sees these "alternations" as constituting a "microclass" of verbs acting this way, such as "spray" ("spray water on the roses" versus "spray the roses with water"). Where does this observation lead him? To the idea that we sometimes frame events in terms of motion in physical space (moving hay; moving water) and sometimes in terms of motion in state-space (wagon becoming full; roses becoming wet).Moreover, there are verbs that refuse such alternations: for instance, "pour." We can say "I poured water into the glass" but not "I poured the glass with water." What accounts for this curious difference between "load" and "pour"? Pinker claims that pouring merely lets a liquid move under gravity's influence, whereas loading is motion determined by the human agent. "Pour" and "load" thus belong to different microclasses, and these microclasses reveal how we construe events. "[W]e have discovered a new layer of concepts that the mind uses to organize mundane experience: concepts about substance, space, time, and force," Pinker writes. " . . . [S]ome philosophers consider [these concepts] to be the very scaffolding that organizes mental life. . . . But we've stumbled upon these great categories of cognition . . . by trying to make sense of a small phenomenon in language acquisition."If correct, then in order to think about physics the way an expert does, we should learn to speak the way experts do. If we try to solve physics problems using the words "load" and "pour", we may be carrying around a bunch of distracting anthropocentric baggage. If we don't recognize that, we'll get stuck, saying the problem "doesn't make sense", when really it's our linguistically-instilled expectations that are wrong. To combat this, it may be just as helpful to gain facility with the language of physics as with its equations.Five Easy Lessons provides a clear example of such difficulties: the case study of "force". As I type this, my laptop is sitting on a desk which exerts an upward force on it. Few beginning students believe this is really a force, even after they've been browbeaten into drawing arrows for the "normal force" on exam diagrams.The problem is in the way we use "force":"The robber forced the door open." "Your apology sounded forced.""...the force of the explosion..." "...the force of righteousness...""I'm being forced to take physics even though I'll never use it."Literally or figuratively, we think of "force" as implying not only motion, but intent or purpose, and also control. Force is for people pushing on things, or maybe for cars and projectiles. These things are using energy and will run down if left alone. But the desk under my laptop? It's just sitting there, totally passive. How could it be "exerting a force" when it doesn't even get tired? Needing some sort of rationalization for why the laptop doesn't fall, beginners say that it's not that the desk exerts a force on the laptop, the desk just provides something for the laptop to sit on. Or if something falls on the desk, the desk didn't exert a force to stop it. It just got in the way is all. Why doesn't the professor understand this obvious difference? A desk exerting a force? Come on...Five Easy Lessons describes how students only overcome this difficulty after seeing a classroom demonstration where, using a laser pointer and a mirror laid on the desk top, the professor demonstrates how when a heavy cinder block is laid on the desk, the surface responds by bending out of its natural shape, exerting force on the cinder block like a compressed spring would.You may need to find many such visualizations before you can reconcile your colloquial use of words with their use in physics. But this might also be dangerous, because although finding a way to make physics obey your idea about what a word means works decently in this case, in other instances it's your expectations for the word that ought to change. (Relativity, with words like "contraction", "slowing down", etc. is a good example.)Mythologist Joseph Campbell believes that we understand the world primarily through story. Perhaps we understand derivations, experimental evidence, and the logic behind physical conclusions as a sort of story, and it's in building this story that our cognitive chunks are formed.Mind The Neural Gap JunctionsYou are the pattern of neural activity in your brain. When a part of you changes, building a new memory, installing a new habit, or constructing a tool to approach a class of problems, that change must be reflected somewhere in your brain.Lesswrong user kalla724 describes this process in "Attention control is critical for changing/increasing/altering motivation"First thing to keep in mind is the plasticity of cortical maps. In essence, particular functional areas of our brain can expand or shrink based on how often (and how intensely) they are used. A small amount of this growth is physical, as new axons grow, expanding the white matter; most of it happens by repurposing any less-used circuitry in the vicinity of the active area. For example, our sense of sight is processed by our visual cortex, which turns signals from our eyes into lines, shapes, colors and movement. In blind people, however, this part of the brain becomes invaded by other senses, and begins to process sensations like touch and hearing, such that they become signNowly more sensitive than in sighted people. Similarly, in deaf people, auditory cortex (part of the brain that processes sounds) becomes adapted to process visual information and gather language clues by sight.But, they caution, these neural changes occur primarily to those parts of our minds to which we pay conscious attention:A man is sitting in his living room, in front of a chessboard. Classical music plays in the background. The man is focused, thinking about the next move, about his chess strategy, and about the future possibilities of the game. His neural networks are optimizing, making him a better chess player.A man is sitting in his living room, in front of a chessboard. Classical music plays in the background. The man is focused, thinking about the music he hears, listening to the chords and anticipating the sounds still to come. His neural networks are optimizing, making him better at understanding music and hearing subtleties within a melody.A man is sitting in his living room, in front of a chessboard. Classical music plays in the background. The man is focused, gritting his teeth as another flash of pain comes from his bad back. His neural networks are optimizing, making the pain more intense, easier to feel, harder to ignore.You need to pay attention not just to doing physics, but to the right parts of doing physics - the parts most related to intuition.James Nearing gave his advice on how to do this in Mathematical Tools for PhysicistsHow do you learn intuition?When you've finished a problem and your answer agrees with the back of the book or with your friends or even a teacher, you're not done. The way do get an intuitive understanding of the mathematics and of the physics is to analyze your solution thoroughly. Does it make sense? There are almost always several parameters that enter the problem, so what happens to your solution when you push these parameters to their limits? In a mechanics problem, what if one mass is much larger than another? Does your solution do the right thing? In electromagnetism, if you make a couple of parameters equal to each other does it reduce everything to a simple, special case? When you're doing a surface integral should the answer be positive or negative and does your answer agree?When you address these questions to every problem you ever solve, you do several things. First, you'll find your own mistakes before someone else does. Second, you acquire an intuition about how the equations ought to behave and how the world that they describe ought to behave. Third, It makes all your later efforts easier because you will then have some clue about why the equations work the way they do. It reifies the algebra.Does it take extra time? Of course. It will however be some of the most valuable extra time you can spend.Is it only the students in my classes, or is it a widespread phenomenon that no one is willing to sketch a graph? (\Pulling teeth" is the cliche that comes to mind.) Maybe you've never been taught that there are a few basic methods that work, so look at section 1.8. And keep referring to it. This is one of those basic tools that is far more important than you've ever been told. It is astounding how many problems become simpler after you've sketched a graph. Also, until you've sketched some graphsof functions you really don't know how they behave.(To see the advice on graphs, along with a detailed step-by-step example, see his book, free online)Brown Big SpidersOne of the difficulties with chunks is that they're mostly subconscious. We may ultimately know of their existence, as did the chess master who told me he knew how each square of the chess board felt, but their precise nature and the process of their creation are almost immune to introspection. The study methods I've talked about above are empirically useful in creating chunks, so we have guidelines for how to make new chunks in general, but we usually don't know which ones we are creating.Lesswrong user Yvain comments on the essay Being a teacherI used to teach English as a second language. It was a mind trip.I remember one of my students saying something like "I saw a brown big spider". I responded "No, it should be 'big brown spider'". He asked why. Not only did I not know the rule involved, I had never even imagined that anyone would ever say it the other way until that moment.Such experiences were pretty much daily occurrences.In other words, the chunkiest cognitive process we have - language - develops largely without our awareness. (In retelling this story, I've met a surprising number of people who actually did know about adjective order in English, but most of them either learned English as a second language or had studied it in psychology or linguistics course.)This makes it incredibly difficult for physics teachers or textbook writers to communicate with beginners. It's inevitable that beginners will say that a certain lecturer or book just doesn't explain it clearly enough, or needs to give more examples. Meanwhile, the lecturer has no idea why what they said wasn't already perfectly clear and thinks the example was completely explicit. Neither party can articulate the problem, the student because they can't see the incorrect assumption they're making, the professor because they don't realize they've already made such an assumption.For example, once I was proctoring a test in a physics class for biology majors. A question on the test described a certain situation with light going through a prism and asked, "What is the sign of the phase shift?" A student came up to ask for clarification, and it wasn't until they'd asked their question three times that I finally got it. They thought they were supposed to find the "sign" as in a signpost, or marker. There would be some sort of observable behavior that would indicate that a phase shift had occurred, and that was the "sign of the phase shift." Until then, I was only able to think of "sign" as meaning positive or negative - did the wave get advanced or retarded?If you want to learn a language with all those rules you don't even know about, you need to immerse yourself. Endless drills and exercises from a book won't be enough, as millions of Americans a decade out of high school straining to remember, "Dondé esta el baño?" can attest. You need to read, speak, see, and hear that language all around you before it takes.To learn physics, then, read, speak, and hear it all around you. Attend colloquia. Read papers. Solve problems. Read books. Talk to professors and TA's, and expose yourself to all the patterns of thought that are the native language of the field.As you learn, you will build the right chunks to think about physics without realizing what they are. But there's a flip side to this problem, which is that when you're not doing physics, you can build the wrong chunks. They can get in the way, and again you don't realize it.In Drawing on the Right Side of the Brain, Betty Edwards discusses an exercise she gave her art students:One day, on impulse, I asked the students to copy a Picasso drawing upside down. That small experiment, more than anything else I had tried, showed that something very different is going on during the act of drawing. To my surprise, and to the students' surprise, the finished drawings were so extremely well done that I asked the class, "How come you can draw upside down when you can't draw right-side up?" The students responded, "Upside down, we didn't know what we were drawing."When we see a recognizable image, unconscious chunking immediately gets to work, interpreting, imparting meaning, and inevitably distorting. Learning to draw, according to Edwards, involves circumventing harmful chunks as much as building helpful ones.So it is with physics. The ideas about force, animation, and intent discussed in the laptop-and-desk example seem to illustrate just this problem. Five Easy Lessons lists many of the known misconceptions that students have somehow taught themselves in each topic of introductory physics - for example that electric current gets used up as it goes around a circuit. But I think it's likely that there are many more such obstructive thought patterns that we don't yet know exist. These might be more general notions about such things as cause and effect, what nature "wants" to accomplish, etc.I Feel DumbEducators are perpetually frustrated by what seems like an outrageous pattern. They explain something clearly. The students all claim to understand perfectly, and can even solve quantitative problems. Still, when you ask the students to answer basic conceptual questions, they get it all wrong. How is this possible?In this YouTube video, Veritasium explores what happens when you explain something clearly:Amazingly, the clearer the explanation, the less students learn. Humans have a huge array of cognitive biases. In general, these various biases work so that we'll keep believing whatever it was we believed to begin with, unless there's a really good reason not to. Someone giving a clear, authoritative physics lecture does not register in your mind as a good reason to check your beliefs, so you listen happily and rave about what a great lecture it was, all while maintaining your wrong ideas.However, with the right stimulus you can get your brain to throw out the old, wrong ideas. Entering such a state is a prerequisite to true learning, and fortunately we can detect it in ourselves. We call it confusion.Confusion is a message from your emotional mind (the part that tells your analytical mind what decisions to start justifying). It's saying, "Hey, something about our beliefs is very wrong, and this is actually important. Pay attention and figure it out."A great lecturer, instead of being clear, will confuse students by asking them to predict ahead of time what a demonstration will show, then do it, and the opposite actually occurs. Or they will ask students to solve questions that sound straightforward, but in fact the students can't figure out. Only after confusion sets in will the teacher reveal the trick.You want to defeat your biases, toss out your wrong beliefs, and learn physics to the Feynman level - the level where you create the knowledge as you go along. Even many specialists never fully get there, instead rising to increasingly-sophisticated levels of rehashing the same memorized arguments in a way that can carry them quite far and trick most people. The only way to avoid this is to spend many, many hours thoroughly confused.Have you ever lost an argument, only to think of the perfect retort two days later when stopped at a traffic light? This shows how your mind will continue working on hard problems in the background. It eventually comes up with a great answer, but only if you first prime it with what to chew on. This works for physics problems just as well as for clever comebacks, once you find good problems to grapple with. I conjecture that engaging this subconscious system requires a strong emotional connection to the problem, such as the frustration or embarrassment of being dumbstruck in an argument or the confusion of being stumped by a hard problem.Confusion is essential, but often also unpleasant. When you repeatedly feel frustrated or upset by your confusion, your mind unconsciously learns to shy away from hard thinking. You develop an ugh field.This could happen for different reasons. A common one arises in people who judge themselves by their intellect. Confusion for such people is a harsh reminder of just how limited they are; it's a challenge to their very identity. Whether for this reason or some other, it's common for students and academics to fall into patterns of procrastination and impostor syndrome when navigating the maze of confusion that come with their chosen path.I don't have the answer for this. I have heard many people tell their stories, but I have yet to figure out my own. Sometimes confusion feels awful, and my story in physics is a jerky, convoluted one because of how I've dealt with that. But once in a while a problem is so good that none of that matters. When I find one of these problems, it hijacks my mind like Cordyceps in a bullet ant, jerking me back to a fresh piece of scratch paper again and again, sometimes for days. If you signNow this state over and over, you'll know Feynman meant by, "What I cannot create I do not understand"Get confused. Solve problems. Repeat. The universe is waiting for you.ReferencesIn order of appearance in this answerFeynman's Tips on Physics: Richard P. Feynman, Michael A. Gottlieb, Ralph Leighton: 9780465027972: Amazon.com: Bookssoft question - Memorizing theorems - MathOverflowThe Magical Number Seven, Plus or Minus Two (wikipedia)The Magical Number Seven (original paper)Google Translate (Chinese phrase)Knight, Randall. Five Easy Lessons pp 37Reif and Heller, 1982 Viète's formulaHow To Solve It: A New Aspect of Mathematical Method (Amazon)How To Solve It (summary)How to Solve It (Wikipedia)Learn Faster with the Feynman Technique (Scott Young. His page is start to get spammy.)Study Hacks " About (Cal Newport)Anki - powerful, intelligent flashcardsSpaced repetition (review by Gwern)K. Anders Ericsson (Wikipedia)The Role of Deliberate Practice in the Acquisition of Expert PerformanceDual N-Back FAQ (gwern)Food Rules An Eater`s Manual (Amazon, how to eat)Core Performance Essentials (Amazon, exercise) Exercise is an interesting case because not everyone responds very well. For the majority of people it's worth the time.Howard Gardner (wikipedia)The Unschooled Mind: How Children Think And How Schools Should Teach: Howard E. Gardner (Amazon)The Perils and Promises of Praise (article by Dweck)Mindset, Dweck's book.Flow (psychology) (Wikipedia)Flow: The Psychology of Optimal Experience: Mihaly Csikszentmihalyi: 9780061339202: Amazon.com: Books David Allen, Getting Things Done® and GTD® Online to-do list and task management (One possible GTD software)How to Setup Remember The Milk for GTD George Lakoff (professional site)George Lakoff (Wikipedia)Where Mathematics Come From: How The Embodied Mind Brings Mathematics Into Being: George Lakoff, Rafael Nuñez: 9780465037711: Amazon.com: BooksLoaded sentences (Hofstadter reviews Pinker)The Stuff of Thought: Language as a Window into Human Nature: Steven Pinker: 9780143114246: Amazon.com: Books The Power of Myth: Joseph Campbell, Bill Moyers: 9780385418867: Amazon.com: BooksAttention control is critical for changing/increasing/altering motivationMathematical Tools for Physics (Nearing)Being a teacher - Less WrongDrawing on the Right Side of the Brain: The Definitive, 4th Edition: Betty Edwards: 9781585429202: Amazon.com: BooksVeritasium (channel)List of cognitive biases (wikipedia)Dunning–Kruger effect (wikipedia) Ugh fields - Less WrongUseful Quora AnswersAnonymous's answer to What is it like to understand advanced mathematics? Does it feel analogous to having mastery of another language like in programming or linguistics?Satvik Beri's answer to How do math geniuses understand extremely hard math concepts so quickly?Qiaochu Yuan's answer to Why is it almost impossible to learn a mathematical concept on Wikipedia? They are very difficult to follow, especially if one doesn't have a solid background in the subject.Christopher VanLang's answer to What should I do if my PhD advisor and lab colleagues think I'm stupid?What did Richard Feynman mean when he said, "What I cannot create, I do not understand"?Debo Olaosebikan's answer to What are some words, phrases, or expressions that physicists frequently use in ordinary conversation?Paul King's answer to How does the arbitrary become meaningful? How does the human mind convert things like art into emotion and experience?What are some English language rules that native speakers don't know, but still follow?User's answer to What's an efficient way to overcome procrastination?Further ReadingI feel a little sleazy writing this answer because when I mention, for example, Carol Dweck doing research on the psychology of mindsets or K. Anders Ericsson studying deliberate practice, in fact there are thousands of people working in those fields. The ones I've mentioned are simply the most public figures or those I've come across by chance. I haven't even read the original research in most of these cases, relying on summaries instead.The answer is also preliminary and incomplete. There's lots of research left to be done, and I'm not an expert in what's out there. Still, here is a guide to some further resources that have informed this answer.For an overview of the psychology of learning, I like Monisha Pasupathi's audio course How We Learn from The Teaching Company. It covers many clever experiments designed to help you build a model of what happens in your mind as you learn.Bret Victor explores software solutions to visualizing the connection between physical world, mathematical representation, and mental models. Check outThe Ladder of AbstractionExplorable ExplanationsI think it's helpful to build an innate impression of your mind as not perceiving the world directly, but as concocting its own, tailored interpretation from sense data. All your consciousness ever gets to experience is the highly-censored version. The books of Oliver Sacks are great for making this clear by illustrating what happens with people for whom some of the processing machinery breaks down.The LessWrong Sequences were, for me, a powerful introduction to the quirks of human thought, preliminary steps towards how to work best with the firmware we've got, and what it means to seek truth.Selected BibliographyThese are some physics books to which have helped me so far. I'm not choosing them for clear exposition or specialty knowledge in a certain subject, but for how I think they helped me understand the way to think about physics generally.Blandford and Thorne, Applications of Classical PhysicsEpstein, Thinking PhysicsFeynman, Lectures on Physics------------ The Character of Physical Law------------ QED: The Strange Theory of Light and Matter------------ Tips on PhysicsGeroch, General Relativity from A to BLevi, The Mathematical MechanicLewin, Walter "Classical Mechanics", "Electricity and Magnetism" (video lectures with demonstrations on MIT OpenCourseWare)Mahajan, Street-Fighting MathematicsMorin, Introduction to Classical MechanicsNearing, Mathematical Tools for PhysicsPurcell, Electricity and Magnetism----------, Back of the Envelope ProblemsSchey, Div, Grad, Curl, and All ThatThomas and Raine, Physics to a DegreeThompson, Thinking Like a PhysicistWeisskopf, "The Search for Simplicity" (articles in Am. J. Physics)ImagesFeynman's Tips on Physics, Feynman, Gottlieb, LeightonArchitectural detail- cut stone wallFile:NotreDameI.jpg

Indeed!!! This story will make you laugh!!!Yeah, one time, I decided to try out for the TV game show “Who Wants to Be a Millionaire.” I filled out the online application form, and took the online quiz, and — Lo and Behold! — I passed! Yes!!!! I passed!!! I was accepted for an audition in New York City!!!Now feeling quite smug about my own superior intelligence, I patted myself on the back and gave myself a hearty “YES!!!” (with thumbs up) every time I sat on my bed, in the middle of the afternoon, watching “Who Wants To Be a Millionaire” with my cat.Ha!!! Ha!!! Yes!!!!!!!!!!!! I guessed the right answer once again! HA! HA! HA!!!!!!!!!!!!!!!!I am brilliant!!! I am GENIUS!!!! AHA!!!!Then came the big day — try outs — in New York City.I had flown there the day before, taken a taxi to my cheap little pathetic (expensive) hotel in the middle of downtown Manhattan, where I was kept awake most of the night from the bright lights and sounds of “the City that never sleeps.” Also, the smell of roach spray in the hallways and in my pathetic little room, with the shared bathroom down at the end of the hallway, did not produce feelings of sleepiness, either.In the morning, I got up early and went to find a beauty salon. I wanted to look my best. I wanted to look like the winning contestant during my audition.New York City beauty salons are not cheap, I discovered! The hair cut and style and manicure cost me well over $250, but, hey, when you’re about to become a millionaire, I realized that little expensive was just a drop in the bucket.The next taxi dropped me off at the audition location. I had my audition acceptance letter in hand, as I joined the crowd standing in the long line.The long line soon queued up. There were literally hundreds waiting in the long line.Soon, the doors opened, and we were ushered in, like cattle, and directed to be seated at various tables where we were given name tags. The guy sitting across from me had already been a contestant on Jeopardy. “This will be a piece of cake,” he stated, smugly.There were many pencils on the table that had “Who Wants To Be A Millionaire” engraved on them.Soon, we were given a test sheet. Our sheets were all unique, so no one could cheat off of anyone’s paper. We were instructed to use those “millionaire” pencils to fill in the correct little multiple test circles.When the bell rang, we were given 10 minutes to answer thirty questions. Some of the questions were complicated math word problems, which I could figure out, if given enough time. Other questions were about pop culture, country music, or history, science, etc.I was on about the seventh question when the test administrator announced, “You have five more minutes.”Five minutes???I instantly felt panic-stricken, absolutely panicked! I looked at the Jeopardy guy across from me and noticed that he was already on Question 29!Before time was called, he had already finished, and had turned over his paper, and was now just sitting there, leaning forward with his hands clasped together, with that smug little smile on his face.“One more minute!” the test administrator announced.At this point, with more than half of the questions still unanswered, I just simply decided to hastily mark them all “C.” Right? Didn’t I, the genius, read somewhere that “C” is the best guess?????C, C, C, C, C, C, C, C, C, C . . . .DING! Time’s up!We were instructed to leave all of our test papers face down and to just sit and wait.While we waited to hear the results of who “made” it, and hear which auditioners would be invited for the Next Step (to participate in a mock game), the man across from me smugly bragged about how “easy” the test had been for him.Sure enough, out of a room of about three hundred applicants, about five names were called. Yep! Only five people made it to the next step out of about three hundred applicants.The guy across from me was one of the five.Back in Gainesville, Florida, my friends were eager to know when I would be appearing on Millionaire.“Well, I will not be appearing on Millionaire,” I informed them, “but here’s a pencil from the try outs.” Everyone LOVED their “Who Wants to Be a Millionnaire” pencil.I should have kept one for myself. Those free pencils — that we were invited to take off the tables — had cost me more than $1,000 — between the plane ticket, the new outfits, roach spray smelling dingy hotel room, beauty salon, etc.The following day, I sat on my bed, watching an episode of Millionaire with my cat.Wouldn’t you know it! I got an answer right that the contestant missed!!!Dang!!!

It's best to ask homeschoolers in your state. Every state has different laws. What works in one may not work in another.This looks like the information you need: Notice of Intent (NOI)

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