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It has the link above for the form and a link for the instructions. Good luck. 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! 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.
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. 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! 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.
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 , stones. One particularly clear description of the difference between the experts' and beginners' minds comes from George Miller's study "The magical number seven, plus or minus two. 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, , 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. Well, it would be if you were literate in Chinese.
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. 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 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. The "Aha!
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! 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. 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.
The ball's velocity changed, which means it accelerated. 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. 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? 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. 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. 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.
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. 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, 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 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.
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.
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. 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? 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. 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. Pinker claims that pouring merely lets a liquid move under gravity's influence, whereas loading is motion determined by the human agent. But we've stumbled upon these great categories of cognition. 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.
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