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. 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. 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. 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.
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. 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. 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.
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? Maybe you've never been taught that there are a few basic methods that work, so look at section 1. 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? 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.
If you want to learn a language with all those rules you don't even know about, you need to immerse yourself. 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?
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. It's going to be just some friendly chat if you are lucky. If your potential team is working on something like machine learning, expect that they are going to ask you questions about machine learning, courses related to machine learning you have and relevant experience projects, internship.
Of course you have to study that before the interview. Take as long time as you need if you feel rusty. It takes some time to get ready for the host matching it's less than the technical interview but it's worth it of course. Article 4, Section 3 of the Constitution states:New States may be admitted by the Congress into this Union; but no new State shall be formed or erected within the Jurisdiction of any other State; nor any State be formed by the Junction of two or more States, or Parts of States, without the Consent of the Legislatures of the States concerned as well as of the Congress.
This is really not feasible without direct assent from the state being separated from. Both Maine and Kentucky split based on the assent of Massachusetts and Virginia under the Constitution.
West Virginia was a wartime ploy, an act that held up due to the acceptance as fact that enough representatives of the state of Virginia showed up at the Wheeling Conventions to justify a state legislature.
Lincoln and Congress accepted it, and in , West Virginia was accepted as a state separate from Virginia. In peacetime, this arrangement would never have worked. To get the obstinate California legislature to go along with the notion of several of their prosperous counties just leave is just impossible. Just register on the admission portal and during registration you will get an option for the entrance based course.
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