Physics Electricity and Magnetism lecture notes

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Introductory Physics II Electricity, Magnetism and Optics by Robert G. Brown Duke University Physics Department Durham, NC 27708-0305 rgbphy.duke.eduPreliminaries See, Do, Teach Ifyouarereadingthis,Iassumethatyouareeithertakingacourseinphysicsorwishtolearnphysics on your own. If this is the case, I want to begin by teaching you the importance of your personal engagement in the learning process. If it comes right down to it, how well you learn physics, how good a grade you get, and how much fun you have all depend on how enthusiastically you tackle the learning process. If you remain disengaged, detatched from the learning process, you almost certainly will do poorly and be miserable while doing it. If you can find any degree of engagement – or open enthusiasm – with the learning process you will very likely do well, or at least as well as possible. Note that I use the term learning, not teaching – this is to emphasize from the beginning that learning is a choice and that you are in control. Learning is active; being taught is passive. It is up to you to seize control of your own educational process and fully participate, not sit back and wait for knowledge to be forcibly injected into your brain. You may find yourself stuck in a course that is taught in a traditional way, by an instructor that lectures, assigns some readings, and maybe on a good day puts on a little dog-and-pony show in the classroomwith some audiovisualaids or somedemonstrations. The standardexpectation in this class is to sit in your chair and watch, passive, taking notes. No real engagement is “required” by the instructor, and lacking activities or a structure that encourages it, you lapse into becoming a lecture transcription machine, recording all kinds of things that make no immediate sense to you and telling yourself that you’ll sort it all out later. You may find yourself floundering in such a class – for good reason. The instructor presents an ocean of material in each lecture, and you’re going to actually retain at most a few cupfuls of it functioning as a scribe and passively copying his pictures and symbols without first extracting their sense. And the lecture makes little sense, at least at first, and reading (if you do any reading at all) does little to help. Demonstrations can sometimes make one or two ideas come clear, but only at the expense of twenty other things that the instructor now has no time to cover and expects you to get from the readings alone. You continually postpone going over the lectures and readings to understand the material any more than is strictly required to do the homework, until one day a big test draws nigh and you realize that you really don’t understand anything and have forgotten most of what you did, briefly, understand. Doom and destruction loom. Sound familiar? On the other hand, you may be in a course where the instructor has structured the course with a balancedmix of open lecture (held as a freeformdiscussion wherequestions aren’tjust encouraged but required) and group interactive learning situations such as a carefully structured recitation and lab where discussion and doing blend together, where students teach each other and use what they have learned in many ways and contexts. If so, you’re lucky, but luck only goes so far. 56 Preliminaries Even in a course like this you may still be floundering because you may not understand why it is important for you to participate with your whole spirit in the quest to learn anything you ever choose to study. In a word, you simply may not give a rodent’s furry behind about learning the materialso that studying is alwaysa fight with yourselfto “make”yourselfdo it – so that no matter what happens, you lose. This too may sound very familiar to some. The importance of engagement and participation in “active learning” (as opposed to passively being taught) is not really a new idea. Medical schools were four year programs in the year 1900. They are four year programs today, where the amount of information that a physician must now master in those four years is probably ten times greater today than it was back then. Medical students are necessarily among the most efficient learners on earth, or they simply cannot survive. In medical schools, the optimal learning strategy is compressed to a three-step adage: See one, do one, teach one. See a procedure (done by a trained expert). Do the procedure yourself, with the direct supervision and guidance of a trained expert. Teach a student to do the procedure. See, do, teach. Now you are a trained expert (of sorts), or at least so we devoutly hope, because that’s all the training you are likely to get until you start doing the procedure over and over again withrealhumansandwithlimitedoversightfromanattendingphysicianwithtoomanyotherthings to do. So you practice and study on your own until you achieve real mastery, because a mistake can kill somebody. This recipe is quite general, and can be used to increase your own learning in almost any class. In fact, lifelong success in learning with or without the guidance of a good teacher is a matter of discovering the importance of active engagement and participation that this recipe (non-uniquely) encodes. Let us rank learning methodologies in terms of “probable degree of active engagement of the student”. By probable I mean the degree of active engagement that I as an instructor have observed in students over many years and which is significantly reinforced by research in teaching methodology, especially in physics and mathematics. Listening to a lecture as a transcription machine with your brain in “copy machine” mode is almost entirely passive and is for most students probably a nearly complete waste of time. That’s not to say that “lecture” in the form of an organized presentation and review of the material to be learned isn’t important or is completely useless It serves one very important purpose in the grand schemeoflearning,butbybeingpassiveduringlectureyoucauseittofailinits purpose. Its purpose is not to give you a complete, line by line transcription of the words of your instructor to ponder later and alone. It is to convey, for a brief shining moment, the sense of the concepts so that you understand them. It is difficult to sufficiently emphasize this point. If lecture doesn’t make sense to you when the instructorpresentsit, youwillhavetoworkmuchhardertoachievethesenseofthematerial“later”, if later ever comes at all. If you fail to identify the important concepts during the presentation and see the lecture as a string of disconnected facts, you will have to remember each fact as if it were an abstract string of symbols, placing impossible demands on your memory even if you are extraordinarily bright. If you fail to achieve some degree of understanding (or synthesis of the material, if you prefer) in lecture by asking questions and getting expert explanations on the spot, you will have to build it later out of your notes on a set of abstract symbols that made no sense to you at the time. You might as well be trying to translate Egyptian Hieroglyphs without a Rosetta Stone, and the best of luck to you with that. Readingisabitmoreactive–attheveryleastyourbrainismorelikelytobesomewhatengagedif youaren’t“just”transcribingthebookontoapieceofpaperorlettingthewordsandsymbolshappen in your mind – but is still pretty passive. Even watching nifty movies or cool-ee-oh demonstrationsPreliminaries 7 is basically sedentary – you’re still just sitting there while somebody or something else makes it all happen in your brain while you aren’t doing much of anything. At best it grabs your attention a bit better (on average) than lecture, but you are mentally passive. In all of these forms of learning, the single active thing you are likely to be doing is taking notes or moving an eye muscle from time to time. For better or worse, the human brain isn’t designed to learn well in passive mode. Parts of your brain are likely to take charge and pull your eyes irresistably to the window to look outside where active things are going on, things that might not be so damn boring With your active engagement, with your taking charge of and participating in the learning process, things change dramatically. Instead of passively listening in lecture, you can at least try to askquestionsandinitiatediscussionswheneveranideaispresentedthatmakesnointialsensetoyou. Discussion is an active process even if you aren’t the one talking at the time. You participate Even a tiny bit of participation in a classroom setting where students are constantly asking questions, where the instructor is constantly answering them and asking the students questions in turn makes a huge difference. Humans being social creatures, it also makes the class a lot more fun 1 In summary, sitting on your ass and writing meaningless (to you, so far) things down as some- body says them in the hopes of being able to “study” them and discover their meaning on your own later is boring and for most students, later never comes because you are busy with many classes, because you haven’t discovered anything beautiful or exciting (which is the reward for figuring it all out – if you ever get there) and then there is partying and hanging out with friends and having fun. Even if you do find the time and really want to succeed, in a complicated subject like physics you are less likely to be able to discover the meaning on your own (unless you are so bright that learning methodology is irrelevant and you learn in a single pass no matter what). Most introduc- tory students are swamped by the details, and have small chance of discovering the patterns within those details that constitute “making sense” and make the detailed information much, much easier to learn by enabling a compression of the detail into a much smaller set of connected ideas. Articulation of ideas, whether it is to yourself or to others in a discussion setting, requires you to create tentative patterns that might describe and organizeall the details you are being presented with. Using those patterns and applying them to the details as they are presented, you naturally encounter places where your tentative patterns are wrong, or don’t quite work, where something “doesn’t make sense”. In an “active” lecture students participate in the process, and can ask questions and kick ideas around until they do make sense. Participation is also fun and helps you pay far more attention to what’s going on than when you are in passive mode. It may be that this increased attention, this consideration of many alternatives and rejecting some while retaining others with social reinforcement, is what makes all the difference. To learn optimally, even “seeing” must be an active process, one where you are not a vessel waiting to be filled through your eyes but rather part of a team studying a puzzle and looking for the patterns together that will help you eventually solve it. Learningisincreasedstillfurtherbydoing, theveryessenceofactivityandengagement. “Doing” varies from course to course, depending on just what there is for you to do, but it always is the application of what you are learning to some sort of activity, exercise, problem. It is not just a recapitulation of symbols: “looking over your notes” or “(re)readingthe text”. The symbols for any given course of study (in a physics class, they very likely will be algebraic symbols for real although I’m speaking more generally here) do not, initially, mean a lot to you. If I write F = q(v×B) on the board,it means a greatdealto me, but if youaretaking this coursefor thefirsttime it probably means zilch to you, and yet I pop it up there, draw some pictures, make some noises that hopefully make sense to you at the time, and blow on by. Later you read it in your notes to try to recreate that sense, but you’ve forgotten most of it. Am I describing the income I expect to make selling B 1 I mean, of course, your donkey. What did you think I meant?8 Preliminaries tons of barley with a market value ofv and a profit margin of q? To learn this expression (for yes, this is a force law of nature and one that we very much must learn this semester) we have to learn what the symbols stand for – q is the charge of a point-like object in motion at velocity v in a magnetic field B, and F is the resulting force acting on the particle. We have to learn that the× symbol is the cross product of evil (to most students at any rate,atleastatfirst). Inordertogetagut feelingforwhatthisequationrepresents,forthedirections associated with the cross product, for the trajectories it implies for charged particles moving in a magnetic field in a variety of contexts one has to use this expression to solve problems, see this expression in action in laboratory experiments that let you prove to yourself that it isn’t bullshit and that the world really does have cross product force laws in it. You have to do your homework that involves this law, and be fully engaged. The learning process isn’t exactly linear, so if you participate fully in the discussion and the doing while going to even the most traditional of lectures, you have an excellent chance of getting to the point where you can score anywhere from a 75% to an 85% in the course. In most schools, say a C+ to B+ performance. Not bad, but not really excellent. A few students will still get A’s – they either work extra hard, or really like the subject, or they have some sort of secret, some way of getting over that barrier at the 90’s that is only crossed by those that really do understand the material quite well. Here is the secret for getting yourself over that 90% hump, even in a physics class (arguably one of the most difficult courses youcantakein college), even if you’renot a super-genius (or havenever managed in the past to learn like one, a glance and you’re done): Work in groups In fact, a really good course (in my opinion) is one where the entire learning process is organized around student teams, basically carefully contructed, semi-permanent groups where each member is at least partly responsible for the effective learning of all the team members, not just themselves That’s it. Nothing really complex or horrible, just get together with your friends who are also taking the course and do your homework together. In a well designed physics course (and many courses in mathematics, economics, and other subjects these days) you’ll have some aspects of the class, such as a recitation or lab, where you are required to work in groups/teams, and the teams and team activities may be highly structured or freeform. “Studio” or “Team Based Learning” for teaching physics have even interleaved the lecture itself withteam-basedactivelearning,soeverythingisdoneinteams. Thismakesititnearly impossible tobedisengagedandsitpassivelyinclasswaitingforlearningto“happen”. Italsoyieldsmeasureable improvements (all things being equal) on at least some objective instruments for measurement of learning, although (long story) measuring learning is a lot harder than you might think... If you take chargeof your own learning, though, you will quickly see that in any course, however it is formally organized and taught, you can study in a group This is true even in a course where “thehomework”istobedonealonebyfiatofthe(unfortunatelyignorantandmisguided)instructor. Juststudy“around”theactualassignment–assignyourselvesproblems“like”theactualassignment – mosttextbooks haveplentyofextraproblems andthenthereis the Internetandother textbooks– anddotheminagroup,then(afterwards) breakupanddoyouractualassignmentalone. Notethat if you use a completely different textbook to pick your group problems from and do them together before looking at your assignment in your textbook, you can’t even be blamed if some of the ones you pick turn out to be ones your instructor happened to assign. Oh, and not-so-subtly – give the instructor a (link to a) PDF copy of this book (it’s as free for instructors as it is for students, after all, just a click away on the Internet). Who knows? Maybe they will give some of these ideas a try Let’s understand in more detail why working on hard problems in teams often has a dramatic effect on learning. What happens when a team works together? Well, a lot of discussion happens,Preliminaries 9 because humans working on a common problem like to talk. There is plenty of doing going on, presuming that the group has a common task list to work through, like a small mountain of really difficult problems that nobody can possibly solve working on their own and are barely within their abilities working as a group backed up by the course instructor Finally, in team-based learning everybody has the opportunity to teach The importance of teaching – not only seeing the lecture presentation with your whole brain actively engaged and participating in an ongoing discussion so that it makes sense at the time, not only doing lots of homework problems and exercises that apply the material in some way, but articulating what you have discovered in this process and answering questions that force you to consider and reject alternative solutions or pathways (or not) cannot be overemphasized. Teaching each other in a peer setting (ideally with mentorship and oversight to keep you from teaching each other mistakes) is essential Thisproblemyou“get”,andteachothers (andactuallylearnitbetter fromteachingitthanthey do from your presentation – never begrudge the effort required to teach your fellow team members even if some of them are very slow to understand). The next problem you don’t get but some other group member does – they get to teach you. In the end you all learn far more about every problem as a consequence of the struggle, the explorationof false paths, the discoveryand articulationof the correct path, the process of discussion, resolution and agreement in teaching whereby everybody in the team hopefully reaches full understanding. Note that success in this last key metric depends on you and you alone. No teaching/learning approachwillhelpyoulearnifyouquithalfwaythere. Someapproachesmakeiteasier,someharder, butintheendyou beartheultimateresponsibilityforyourownactive,engagedlearning. Whenyou have completed see, do, teach, you have achieved a critical milestone on the path to comprehension. I would assert that it is all but impossible for someone to become a (halfway decent) teacher of anything without learning along the way that the absolute best way to learn any set of material deeply is to teach it – it is the very foundation of Academe and has been for two or three thousand years. It is, as we have noted, built right into the intensive learning process of medical school and graduate school in general. For some reason, however, we don’t incorporate a teaching component in most undergraduate classes, which is a shame, and it is basically nonexistent in nearly all K-12 schools, which is an open tragedy. As an engaged student you don’t have to live with that Put it there yourself, by incorporating group study and mutual teaching into your learning process with or without the help or permission of your teachers A really smart and effective team soon learns to iterate the teaching – I teach you, and to make sure you got it you immediately use the material I taught you and try to articulate it back to me. Eventually everybody on the team understands, everybody on the team benefits, everybody on the team gets the best possible grade on the material. This process will actually make you (quite literally) more intelligent. You may or may not manage to lock down an A, but you will get the best grade you are capable of getting, for your given investment of effort. This is close to the ultimate in engagement – highly active learning, with all cylinders of your brain firing awayon the process. You can see why learning is enhanced. It is simply a bonus, a sign of a just and caring God, that it is also a lot more fun to work in a team, especially in a relaxed context with food and drink present. Yes, I’m encouraging you to have “physics study parties” (or historystudyparties,orpsychologystudyparties). Holdcontests. Givesillyprizes. See. Do. Teach. Other Conditions for Learning Learning isn’t only dependent on the engagement pattern implicit in the See, Do, Teach rule. Let’s absorb a few more True Facts about learning, in particular let’s come up with a handful of things10 Preliminaries that can act as “switches” and turn your ability to learn on and off quite independent of how your instructor structures your courses. Most of these things aren’t binary switches – they are more like dimmer switches that can be slid up between dim (but not off) and bright (but not fully on). Some of these switches, or environmental parameters, act together more powerfully than they act alone. We’llstartwiththemostimportantpair,a pairthatresearchhasshownworktogetherto potentiate or block learning. Insteadof just telling you what they are, arguing that they areimportant for a paragraphor six, and moving on, I’m going to give you an early opportunity to practice active learning in the context ofreadinga chapter onactivelearning. Thatis, I wantyouto participatein a tinymini-experiment. It works a little bit better if it is done verbally in a one-on-onemeeting, but it should still workwell enough even if it is done in this text that you are reading. I going to give you a string of ten or so digits and ask you to glance at it one time for a count of three and then look away. No fair peeking once your three seconds are up Then I want you to do something else for at least a minute – anything else that uses your whole attention and interrupts your ability to rehearse the numbers in your mind in the way that you’ve doubtless learned permits you to learn other strings of digits, such as holding your mind blank, thinking of the phone numbers of friends or your social security number. Even rereading this paragraph will do. Attheendoftheminute,trytorecallthenumberIgaveyouandwritedownwhatyouremember. Then turn back to right here and compare what you wrote down with the actual number. Ready? (No peeking yet...) Set? Go Ok, here it is, in a footnote at the bottom of the page to keep your eye from naturally reading 2 ahead to catch a glimpse of it while reading the instructions above . How did you do? If you are like most people, this string of numbers is a bit too long to get into your immediate memory or visual memory in only three seconds. There was very little time for rehearsal, and then you went and did something else for a bit right awaythat was supposed to keep you from rehearsing whatever of the string you did manage to verbalize in three seconds. Most people will get anywhere from the first three to as many as seven or eight of the digits right, but probably not in the correct order, unless... ...they are particularly smart or lucky and in that brief three second glance have time to notice that the number consists of all the digits used exactly once Folks that happened to “see” this at a glance probably did better than average, getting all of the correct digits but maybe in not quite the correct order. People who are downright brilliant (and equally lucky) realized in only three seconds (without cheatinganextrasecondorthree, youknowwhoyouare)thatitconsistedofthe stringofodddigits in ascending order followed by the even digits in descending order. Those people probably got it all perfectly right even without time to rehearse and “memorize” the string Look again at the string, see the pattern now? The moral of this little mini-demonstration is that it is easy to overwhelm the mind’s capacity for processing and remembering “meaningless” or “random” information. A string of ten measly (apparently) random digits is too much to remember for one lousy minute, especially if you aren’t given time to do rehearsal and all of the other things we have to make ourselves do to “memorize” meaningless information. Of coursethings changed radically the instant I pointed out the pattern At this point you could very likely go away and come back to this point in the text tomorrow or even a year from now and haveanexcellent chanceofrememberingthisparticulardigitstring,becauseitmakes senseofasort, 2 1357986420 (one, two, three, quit and do something else for one minute...)Preliminaries 11 and there are plenty of cues in the text to trigger recall of the particular pattern that “compresses and encodes” the actual string. You don’t have to remember ten random things at all – only two and a half – odd ascending digits followed by the opposite (of both). Patterns rock This example has obvious connections to lectureand class time, and is one reasonretentionfrom lecture is so lousy. For most students, lecture in any nontrivial college-level course is a long-running litany of stuff they don’t know yet. Since it is all new to them, it might as well be random digits as far as their cognitive abilities are concerned, at least at first. Sure, there is pattern there, but you have to discover the pattern, which requires time and a certain amount of meditation on all of the information. Basically, you have to have a chance for the pattern to jump out of the stream of information and punch the switch of the damn light bulb we all carry around inside our heads, the one that is endlessly portrayed in cartoons. That light bulb experience is real – it actually exists, in more than just a metaphorical sense – and if you study long enough and hard enough to obtain a sudden, epiphinaic realization in any topic you are studying, however trivial or complex (like the pattern exposed above) it is quite likely to be accompanied by a purely mental flash of “light”. You’ll know it when it happens to you, in other words, and it feels great. Unfortunately, the instructor doesn’tusuallygivestudents a chance to experiencethisin lecture. No sooneris oneseeminglyrandomfactoidlaidoutonthetable thanalongcomesa new, apparently disconnected one that pushes it out of place long before we can either memorize it the hard way or make sense out of it so we can remember it with a lot less work. This isn’t really anybody’s fault, of course; the light bulb is quite unlikely to go off in lecture just from lecture no matter what you or the lecturer do – it is something that happens to the prepared mind at the end of a process, not something that just fires away every time you hear a new idea. ThehumbleandunsurprisingconclusionIwantyouto drawfromthissillylittlemini-experiment is that things are easier to learn when they make sense A lot easier. In fact, things that don’t make sense to you are never “learned” – they are at best memorized. Information can almost always be compressed when you discover the patterns that run through it, especially when the patterns all fit together into the marvelously complex and beautiful and mysterious process we call “deep understanding” of some subject. Thereis one moreexampleI liketo use to illustrate howimportantthis informationcompression is to memory and intelligence. I play chess, badly. That is, I know the legal moves of the game, and have no idea at all how to use them effectively to improve my position and eventually win. Ten moves into a typical chess game I can’t recall how I got myself into the mess I’m typically in, and at the end of the game I probably can’t remember any of what went on except that I got trounced, again. A chess master, on the other hand, can play umpty games at once, blindfolded, against pitiful fools like myself and when they’ve finished winning them all they can go back and recontruct each one move by move, criticizing each move as they go. Often they can remember the games in their entirety days or even years later. This isn’t just because they are smarter – they might be completely unable to derivethe Lorentz group from first principles, and I can, and this doesn’t automatically make me smarter than them either. Itisbecausechessmakessensetothem–they’veachievedadeepunderstandingofthegame, as it were – and they’ve built a complex meta-structure memory in their brains into which they can poke chess moves so that they can be retrieved extremely efficiently. This gives them the attendant capability of searching vast portions of the game tree at a glance, where I have to tediously work through each branch, one step at a time, usually omitting some really important possibility because I don’t realize that that particular knight on the far side of the board can affect things on this side where we are both moving pieces. This sort of “deep” (synthetic) understanding of physics is verymuch the goalof this course(the one in the textbook you are reading, since I use this intro in many textbooks), and to achieveit you12 Preliminaries must not memorize things as if they are random factoids, you must work to abstract the beautiful intertwining of patterns that compress all of those apparently random factoids into things that you can easily remember offhand, that you can easily reconstruct from the pattern even if you forget the details, and that you can search through at a glance. But the process I describe can be applied to learning pretty much anything, as patterns and structure exist in abundance in all subjects of interest. There are even sensible rules that govern or describe the anti-pattern of pure randomness There’s one more important thing you can learn from thinking over the digit experiment. Some of you reading this verylikely didn’t do what I asked, youdidn’t play along with the game. Perhaps it was too much of a bother – you didn’t want to waste a whole minute learning something by actually doing it, just wanted to read the damn chapter and get it over with so you could do, well, whatever the hell else it is youwereplanning to do todaythat’s moreimportant to youthan physics or learning in other courses. If you’re one of these people, you probably don’t remember any of the digit string at this point from actually seeing it – you never even tried to memorize it. A very few of you may actually be so terribly jaded that you don’t even remember the little mnemonic formula I gave above for the digit string (although frankly, people that are that disengaged are probably not about to do things like actually read a textbook in the first place, so possibly not). After all, either way the string is pretty damn meaningless, pattern or not. Pattern and meaning aren’t exactly the same thing. There are all sorts of patterns one can find in random number strings, they just aren’t “real” (where we could wax poetic at this point about information entropy and randomness and monkeys typing Shakespeare or seeing fluffy white sheep in the clouds if this were a different course). So why bother wasting brain energy on even the easy way to remember this string when doing so is utterly unimportant to you in the grand scheme of all things? From this we can learn the second humble and unsurprising conclusion I want you to draw from this one elementary thought experiment. Things are easier to learn when you care about learning them In fact, they are damn near impossible to learn if you really don’t care about learning them. Let’s put the two observations together and plot them as a graph, just for fun (and because graphs help one learn for reasons we will explore just a bit in a minute). If you care about learning whatyouarestudying,andtheinformationyouaretryingtolearnmakessense(ifonlyforamoment, perhaps during lecture), the chances of your learning it are quite good. This alone isn’t enough to guarantee that you’ll learn it, but it they are basically both necessary conditions, and one of them is directly connected to degree of engagement. On the other hand, if you care but the information you want to learn makes no sense, or if it makes sense but you hate the subject, the instructor, your school, your life and just don’t care, your chances of learning it aren’t so good, probably a bit better in the first case than in the second as if you care you have a chance of finding someone or some way that will help you make sense of whatever it is you wish to learn, where the person who doesn’t cares, well, they don’t care. Why should they remember it? If you don’t give a rat’s ass about the material and it makes no sense to you, go home. Leave school. Do something else. You basically have almost no chance of learning the material unless you are gifted with a transcendent intelligence (wasted on a dilettante who lives in a state of perpetual ennui) and are miraculously gifted with the ability learn things effortlessly even when they make no sense to you and you don’t really care about them. All the learning tricks and study patterns in the world won’t help a student who doesn’t try, doesn’t care, and for whom the material never makes sense. If we worked at it, we could probably find other “logistic” controlling parameters to associate withlearning–thingsthatincreaseyourprobabilityoflearningmonotonicallyastheyvary. SomeofPreliminaries 13 Figure 1: Relation between sense, care and learning themarealreadyapparentfromthediscussionabove. Let’slistafewmoreofthemwithexplanations just so that you can see how easy it is to sit down to study and try to learn and have “something wrong” that decreases your ability to learn in that particular place and time. Learning is actual work and involves a fair bit of biological stress, just like working out. Your brain needs food – it burns a whopping 20-30% of your daily calorie intake all by itself just living dayto day,evenmorewhen youarereallyusing itor aresomewhatsedentaryin yourphysical habits so your consumption in the form of physical motion is smaller than normal or healthy. Note thatyourbrainrunsonpure,energy-richglucose,sowhenyourbloodsugardropsyourbrainactivity drops right along with it. This can happen (paradoxically) because you just ate a carbohydrate rich 3 meal. Abalanceddietcontainingfoodswithalowerglycemicindex tendstobehardertodigestand provides a longer period of sustained energy for your brain. A daily multivitamin (and sometimes various antioxidant or metabolic supplements such as alpha lipoic acid) can also help maintain your body’s energy release mechanisms at the cellular level. Blood sugar is typically lowest first thing in the morning, so this is a lousy time to actively study. Ontheother hand, a goodheartybreakfast,eatenatleastanhour beforeplunging into your studies, is a great idea and is a far better habit to develop for a lifetime than eating no breakfast 4 and instead eating a huge meal right before bed Learning requires adequate sleep. Sure this is tough to manage at college – there are no parents 3 Wikipedia: http://www.wikipedia.org/wiki/glycemic index. 4 ...which is, alas, my own pattern unless I’m careful, made into a habit back in college. It seemed to work a lot better at age 20 than it does at age 60...14 Preliminaries to tell you to go to bed, lots of things to do, and of course you’re in class during the day and then you study, so late night is when you have fun. Unfortunately, learning is clearly correlated with engagement, activity, and mental alertness, and all of these tend to shut down when you’re tired. Furthermore, the formation of long term memory of any kind from a day’s experiences has been shown in both animal and human studies to depend on the brain undergoing at least a few natural sleep cycles of deep sleep alternating with REM (Rapid Eye Movement) sleep, dreaming sleep. Rats taught a maze and then deprived of REM sleep cannot run the maze well the next day; rats that are taught the same maze but that get a good night’s of rat sleep with plenty of rat dreaming can run the maze well the next day. People conked on the head who remain unconscious for hours and are thereby deprived of normal sleep often have permanent amnesia of the previous day – it never gets turned into long term memory. Wikipedia: http://www.wikipedia.org/wiki/Sleep Apnea is also a great undiagnosed epidemic (e.g. 24% of all males by late middle age, most of them untreated) and can seriouslyaffect learning. Indeed, if you have any variation of Attention Deficit Disorder (ADD) and snore, or have any symptoms of interrupted sleep due to breathing interruption or e.g. restless legs you 5 should probably read about the co-morbidity of sleep disorders and ADD and talk to your doctor to make sure that you really have ADD and are not suffering from a sleep disorder, as the two can actually result in nearly identical daytime symptoms, including difficulty learning This is hardly surprising. Pure common sense and experience tell you that your brain won’t work too well if it is hungry and tired or oxygen deprived. Common sense (and yes, experience) will rapidly convince you that learning generally works better if you’re not stoned or drunk when you study. Learning works much better when you have time to learn and haven’t put everything 6 off to the last minute. In fact, all of Maslow’s hierarchy of needs are important parameters that contribute to the probability of success in learning. There is one more set of very important variables that strongly affect our ability to learn, and they are in some ways the least well understood. These are variables that describe you as an individual, that describe your particular brain and how it works. Pretty much everybody will learn better if they are self-actualized and fully and actively engaged, if the material they are trying to learn is available in a form that makes sense and clearly communicates the implicit patterns that enable efficient information compressionand storage,and above all if they care about what they are studying and learning, if it has value to them. But everybody is not the same, and the optimal learning strategy for one person is not going to be what works well, or even at all, for another. This is one of the things that confounds “simple” empirical researchthat attempts to find benefit in one teaching/learningmethodologyover another. Somestudentsdoimprove,evendramaticallyimprove–whenthisorthatteaching/learningmethod- ology is introduced. In others there is no change. Still others actually do worse. In the end, the beneficial effect to a selected subgroup of the students may be lost in the statistical noise of the study and the fact that no attempt is made to identify commonalities among students that succeed or fail. Thepoint is that finding anoptimalteaching and learningstrategyis technically anoptimization problem on a high dimensional space. We’ve discussed some of the important dimensions above, isolating a few that appear to havea monotonic effect on the desired outcomein atleast somerange (relying on common sense to cut off that range or suggest trade-offs – one cannot learn better by simply discussing one idea for weeks at the expense of participating in lecture or discussing many other ideas of equal and coordinated importance; sleeping for twenty hours a day leaves little time 5 A Clinical Overview of Sleep and Attention-Deficit/Hyperactivity Disorder in Children and Adolescents 6 Wikipedia: http://www.wikipedia.org/wiki/Maslow’s hierarchy of needs. In a nutshell, in order to become self- actualized and realize your full potential in activities such as learning you need to have your physiological needs met, you need to be safe, you need to be loved and secure in the world, you need to have good self-esteem and the esteem of others. Only then is it particularly likely that you can become self-actualized and become a great learner and problem solver.Preliminaries 15 forexperiencetofixintolongtermmemorywithallofthatsleep). We’veomittedonethatiscrucial, however. That is your brain Your Brain and Learning Yourbrainismorethanjustauniqueinstrument. Insomesenseitisyou. Youcouldimaginehaving your brain removed from your body and being hooked up to machinary that provided it with sight, 7 sound, and touch in such a way that “you” remain . It is difficult to imagine that you still exist in any meaningful sense if your brain is taken out of your body and destroyed while your body is artificially kept alive. Your brain, however, is an instrument. It has internal structure. It uses energy. It does “work”. It is, in fact, a biologicalmachine of sublime complexity and subtlety, oneof the true wondersof the world Note that this statement can be made quite independent of whether “you” are your brain per se or a spiritual being who happens to be using it (a debate that need not concern us at this time, however much fun it might be to get into it) – either way the brain itself is quite marvelous. Forallofthat,fewindeedarethepeoplewhobothertolearntoactuallyusetheirbraineffectively as an instrument. It just works, after all, whether or not we do this. Which is fine. If you want to get the most mileage out of it, however, it helps to read the manual. So here’s at least one user manual for your brain. It is by no means complete or authoritative, but it should be enough to get you started, to help you discover that you are actually a lot smarter than you think, or that you’ve been in the past, once you realize that you can change the way you think and learn and experience life and gradually improve it. In the spirit of the learning methodology that we eventually hope to adopt, let’s simply itemize 8 in no particular order the various features of the brain that bear on the process of learning. Bear in mind that such a minimal presentation is more of a metaphor than anything else because simple (andextremelycommon)generalizationssuchas“creativityisaright-brainfunction”arenotstrictly true as the brain is far more complex than that. 9 • The brain is bicameral: it has two cerebral hemispheres , right and left, with brain functions asymmetrically split up between them. • The brain’s hemispheres are connected by a networked membrane called the corpus callosum that is how the two halves talk to each other. • Thehumanbrainconsistsoflayers witha structurethatrecapitulatesevolutionaryphylogeny; that is, the core structures are found in very primitive animals and common to nearly all vertebrate animals, with new layers (apparently) added by evolution on top of this core as the various phyla differentiated, fish, amphibian, reptile, mammal, primate, human. The outermost layer where most actual thinking occurs (in animals that think) is known as the cerebral cortex. 10 • The cerebral cortex – especially the outermost layer of it called the neocortex – is where “higher thought” activities associated with learning and problem solving take place, although the brain is a very complex instrument with functions spread out over many regions. 11 • Animportantbrainmodelisaneuralnetwork . Computersimulatedneuralnetworksprovide us with insight into how the brain can remember past events and process new information. 7 Imagine very easily if you’ve ever seen The Matrix movie trilogy... 8 Wikipedia: http://www.wikipedia.org/wiki/brain. 9 Wikipedia: http://www.wikipedia.org/wiki/cerebral hemisphere. 10 Wikipedia: http://www.wikipedia.org/wiki/Cerebral cortex. 11 network. Wikipedia: http://www.wikipedia.org/wiki/Neural16 Preliminaries • Thefundamentaloperationalunitsofthebrain’sinformationprocessingfunctionalityarecalled 12 neurons . Neurons receive electrochemical signals from other neurons that are transmitted 13 14 through long fibers called axons Neurotransmitters are the actual chemicals responsible for the triggeredfunctioning of neurons and hence the neural networkin the cortex that spans the halves of the brain. • Parts of the cortex are devoted to the senses. These parts often contain a map of sorts of the worldasseenbytheassociatedsensemechanism. Forexample,thereexistsatopographicmap in the brain that roughly corresponds to points in the retina, which in turn are stimulated by an image of the outside world that is projected onto the retina by your eye’s lens in a way we will learn about later in this course There is thus a representation of your visual field laid out inside your brain • Similar maps exist for the other senses, although sensations from the right side of your body are generally processed in a laterally inverted way by the opposite hemisphere of the brain. Whatyourrighteyesees,what yourrighthand touches,is ultimately transmitted to a sensory area in your left brain hemisphere and vice versa, and volitional muscle control flows from these brain halves the other way. • Neurotransmitters require biological resources to produce and consume bioenergy (provided as glucose) in their operation. You can exhaust the resources, and saturate the receptors for the various neurotransmitters on the neurons by overstimulation. • You can also block neurotransmittersby chemical means, put neurotransmitter analoguesinto yoursystem, and alterthe chemicaltriggerpotentials ofyour neuronsby taking variousdrugs, poisons, or hormones. The biochemistry of your brain is extremely important to its function, and (unfortunately) is not infrequently a bit “out of whack” for many individuals, resulting in e.g. attention deficit or mood disorders that can greatly affect one’s ability to easily learn while leaving one otherwise highly functional. 15 • Intelligence , learning ability, and problem solving capabilities are not fixed; they can vary (often improving) over your whole lifetime Your brain is highly plastic and can sometimes even reprogram itself to full functionality when it is e.g. damaged by a stroke or accident. On the other hand neither is it infinitely plastic – any given brain has a range of accessible capabilities and can be improved only to a certain point. However, for people of supposedly “normal” intelligence and above, it is by no means clear what that point is Note well that intelligence is an extremely controversial subject and you should not take things like your own measured “IQ” too seriously. • Intelligence is not even fixed within a population over time. A phenomenon known as “the 16 Flynn effect” (after its discoverer)suggests that IQ tests have increased almost six points a decade, on average, over a timescale of tens of years, with most of the increases coming from thelowerhalfofthedistributionofintelligence. Thisisanactiveareaofresearch(asonemight well imagine) and some of that research has demonstrated fairly conclusively that individual intelligences can be improved by five to ten points (a significant amount) by environmentally correlated factors such as nutrition, education, complexity of environment. • The best time for the brain to learn is right before sleep. The process of sleep appears to “fix” long term memories in the brain and things one studies right before going to bed are retained much better than things studied first thing in the morning. Note that this conflicts 12 Wikipedia: http://www.wikipedia.org/wiki/Neurons. 13 Wikipedia: http://www.wikipedia.org/wiki/axon. . 14 Wikipedia: http://www.wikipedia.org/wiki/neurotransmitters. 15 Wikipedia: http://www.wikipedia.org/wiki/intelligence. 16 effect. Wikipedia: http://www.wikipedia.org/wiki/flynnPreliminaries 17 directly with the party/entertainment schedule of many students, who tend to study early in the evening and then amuse themselves until bedtime. It works much better the other way around. 17 • Sensory memory corresponds to the roughly 0.5 second (for most people) that a sensory impressionremainsin the brain’s“activesensoryregister”,the sensorycortex. It cantypically hold less than 12 “objects” that can be retrieved. It quickly decays and cannot be improved by rehearsal, although there is some evidence that its object capacity can be improved over a longer term by practice. • Short term memory is where some of the information that comes into sensory memory is transferred. Just which information is transferred depends on where one’s “attention” is, and the mechanics of the attention process are not well understood and are an area of active research. Attentionactslikeafilteringprocess,asthereisawealthofparallelinformationinour sensory memory at any given instant in time but the thread of our awareness and experience of time is serial. We tend to “pay attention” to one thing at a time. Short term memory lasts from a few seconds to as long as a minute without rehearsal, and for nearly all people it holds 18 4−5 objects . However, its capacity can be increased by a process called “chunking” that is basically the information compression mechanism demonstrated in the earlier example with numbers – grouping of the data to be recalled into “objects” that permit a larger set to still fit in short term memory. • Studies of chunking show that the ideal size for data chunking is three. That is, if you try to remember the string of letters: FBINSACIAIBMATTMSN with the usualthree secondlook you’llalmost certainlyfind it impossible. If, however,I insert the following spaces: FBI NSA CIA IBM ATT MSN It is suddenly much easier to get at least the first four. If I parenthesize: (FBI NSA CIA) (IBM ATT MSN) so that you can recognize the first three are all governmentagencies in the general categoryof “intelligence and law enforcement” and the last three are all market symbols for information technologymega-corporations,you can once againrecall the informationa daylater with only themostcursoryofrehearsals. You’vetakeneighteen”random”objectsthatweremeaningless and could hence be recalled only through the most arduous of rehearsal processes, converted them to six “chunks” of three that can be easily tagged by the brain’s existing long term memory (note that you are not learning the string FBI, you are building an association to the already existing memory of what the string FBI means, which is much easier for the brain to do), and chunking the chunks into two objects. Eighteenobjectswithoutmeaning–difficultindeed Thosesameeighteenobjectswithmeaning – umm, looks pretty easy, doesn’t it... Short term memory is still that – short term. It typically decays on a time scale that ranges from minutes for nearly everything to order of a day for a few things unless the information can be transferred to long term memory. Long term memory is the big payoff – learning is associated with formation of long term memory. • Now we get to the really good stuff. Long term is memory that you form that lasts a long time in human terms. A “long time” can be days, weeks, months, years, or a lifetime. Long 17 Wikipedia: http://www.wikipedia.org/wiki/memory. Several items in a row are connected to this page. 18 From this you can see why I used ten digits, gave you only a few seconds to look, and blocked rehearsal in our earlier exercise.18 Preliminaries termmemory is encoded completely differently from shortterm or sensory/immediatememory 19 – it appears to be encoded semantically , that is to say, associatively in terms of its meaning. Thereisconsiderableevidenceforthis,anditisonereasonwefocussomuchontheimportance of meaning in the previous sections. Tomiraculouslytransformthingswetrytorememberfrom“difficult”tolearnrandomfactoids that have to be brute-force stuffed into disconnected semantic storage units created as it were one at a time for the task at hand into “easy” to learn factoids, all we have to do is discover meaning associations with things we already know, or create a strong memory of the global meaning or conceptualization of a subject that serves as an associativehome for all those little factoids. A characteristic of this as a successful process is that when one works systematically to learn by means of the latter process, learning gets easier as time goes on. Every factoid you add to the semantic structure of the global conceptualization strengthens it, and makes it even easier to add new factoids. In fact, the mind’s extraordinary rational capacity permits it to interpolate and extrapolate, to fill in parts of the structure on its own without effort and in many cases without even being exposed to the information that needs to be “learned” • Onearea wherethis extrapolationis particularlyevident and powerfulis in mathematics. Any timewecanlearn,ordiscoverfromexperienceaformulaforsomephenomenon,amathematical pattern, we don’t have to actually see something to be able to “remember” it. Once again, it is easy to find examples. If I give you data from sales figures over a year such as January = 1000, October = 10,000, December = 12,000, March=3000, May = 5000, February = 2000, September = 9000, June = 6000, November = 11,000, July = 7000, August = 8000, April = 4000, at first glance they look quite difficult to remember. If you organize them temporally by month and look at them for a moment, you recognizethat sales increased linearly by month, starting at 1000 in January, and suddenly you can reduce the whole series toasimplementalformula(straightline)andacouplepieces ofinitialdata(slopeandstarting point). One amazing thing about this is that if I asked you to “remember” something that you have not seen, such as sales in February in the next year, you could make a very plausible guess that they will be 14,000 Note that this isn’t a memory, it is a guess. Guessing is what the mind is designed to do, as it is part of the process by which it “predicts the future” even in the most mundane of ways. WhenIputtendollarsinmypocketandreachinmypocketforitlater,I’mbasicallyguessing, on the basis of my memory and experience, that I’ll find ten dollars there. Maybe my guess is 20 wrong – my pocket could have been picked , maybe it fell out through a hole. My concept of object permanence plus my memory of an initial state permit me to make a predictive guess about the Universe This is, in fact, physics This is what physics is all about – coming up with a set of rules (like conservationof matter) that encode observations of object permanence, more rules (equations of motion) that dictate how objects move around, and allow me to conclude that “I put a ten dollar bill, at rest, into my pocket, and objects at rest remain at rest. The matter the bill is made of cannot be created or destroyed and is bound together in a way that is unlikely to come apart over a period of days. Therefore the ten dollar bill is still there” Nearly anything that you do or that happens in your everyday life can be formulated as a predictive physics problem. 21 • The hippocampus appears to be partly responsible for both forming spatial maps or visual- izations of your environment and also for forming the cognitive map that organizes what you know and transforms short term memory into long term memory, and it appears to do its job 19 Wikipedia: http://www.wikipedia.org/wiki/semantics. 20 With three sons constantly looking for funds to attend movies and the like, itisn’tas unlikely as you might think 21 Wikipedia: http://www.wikipedia.org/wiki/hippocampus.Preliminaries 19 (as notedabove)in your sleep. Sleep deprivationprevents the formation of long term memory. Being rendered unconscious for a long period often produces short term amnesia as the brain loses short term memory before it gets put into long term memory. The hippocampus shows evidence of plasticity – taxi drivers who have to learn to navigate large cities actually have larger than normal hippocampi, with a size proportional to the length of time they’ve been driving. This suggests (once again) that it is possible to deliberately increase the capacity of your own hippocampus through the exercise of its functions, and consequently increase your ability to store and retrieve information, which is an important component (although not the only component) of intelligence • Memory is improved by increasing the supply of oxygen to the brain, which is best accom- plished by exercise. Unsurprisingly. Indeed, as noted above, having good general health, good nutrition, good oxygenation and perfusion – having all the biomechanism in tip-top running order–is perfectlyreasonablylinkedto being ableto performatyourbestinanything,mental activity included. 22 • Finally, the amygdala is abrainorganinour limbic system(partofour “old”,reptilebrain). The amygdala is an important part of our emotional system. It is associated with primitive survivalresponses,withsexualresponse,andappearstoplayakeyroleinmodulating(filtering) the process of turning short term memory into long term memory. Basically, any sort term memory associated with a powerful emotion is much more likely to make it into long term memory. There are clear evolutionary advantages to this. If you narrowly escape being killed by a saber-toothed tiger at a particular pool in the forest, and then forget that this happened by the next day and return again to drink there, chances are decent that the saber-tooth is still there and you’ll get eaten. On the other hand, if you come upon a particular fruit tree in that same forest and get a free meal of high quality food and forget about the tree a day later, you might starve. We see that both negative and positive emotional experiences are strongly correlated with learning Powerful experiences, especially, are correlated with learning. This translates into learning strategies in two ways, one for the instructor and one for the student. For the in- structor, there are two general strategies open to helping students learn. One is to create an atmosphereof fear, hatred, disgust, anger – powerfulnegativeemotions. The other is to create an atmosphere of love, security, humor, joy – powerful positive emotions. In between there is a great wasteland of bo-ring, bo-ring, bo-ring where students plod along, struggling to form memories because there is nothing “exciting” about the course in either a positive or negative way and so their amygdala degrades the memory formation process in favor of other more “interesting” experiences. Now, in my opinion, negative experiences in the classroom do indeed promote the formation of long term memories, but they aren’t the memories the instructor intended. The student is likely to remember, and loath, the instructor for the rest of their life but is not more likely to rememberthematerialexceptsporadicallyinassociationwithparticularlytraumaticepisodes. They may well be less likely, as we naturally avoid negative experiences and will study less and work less hard on things we can’t stand doing. For the instructor, then, positive is the way to go. Creating a warm, nurturing classroom environment, ensuring that the students know that you care about their learning and about them as individuals helps to promote learning. Making your lectures and teaching processes fun – and funny – helps as well. Manysuccessful lecturers makea powerful positive impression on the students, creating an atmosphere of amazement or surprise. A classroom experience should really be a joy in order to optimize learning in so many ways. 22 Wikipedia: http://www.wikipedia.org/wiki/amygdala.20 Preliminaries For the student, be aware that your attitude matters As noted in previous sections, caring is an essential component of successful learning because you have to attach value to the process in order to get your amygdala to do its job. However, you can do much more. You can see how many aspects of learning can be enhanced through the simple expedient of making it a positive experience Working in groups is fun, and you learn more when you’re having fun (or quavering in abject fear, or in an interesting mix of the two). Attending an interesting lecture is fun, and you’ll retain more than average. Participation is fun, especially if you are “rewarded” in some way that makes a moment or two special to you, and you’ll remember more of what goes on. From all of these little factoids (presented in a way that I’m hoping helps you to build at least the beginnings of a working conceptual model of your own brain) I’m hoping that you are coming to realize that all of this is at least partially under your control Even if your instructor is scary or boring, the material at first glance seems dry and meaningless, and so on – all the negative-neutral things that make learning difficult, you can decide to make it fun and exciting, you can ferret out the meaning, you can adopt study strategies that focus on the formation of cognitive maps and organizing structures first and then on applications, rehearsal, factoids, and so on, you can learn to study right before bed, get enough sleep, become aware of your brain’s learning biorhythms. Finally, you can learn to increase your functional learning capabilities by a significant amount. Solving puzzles, playingmentalgames, doingcrosswordpuzzles or sudoku, workinghomeworkprob- lems,writingpapers,arguinganddiscussing,justplainthinkingaboutdifficultsubjectsandproblems even when you don’t have to all increase your active intelligence in initially small but cumulative ways. Youtoocanincreasethesizeofyourhippocampus,learntoengageyouramygdalabychoosing in a self-actualized way what you value and learning to discipline your emotions accordingly, and createmoreconceptualmaps within yourbrainthat canbe sharedas components acrossthevarious things you wish to learn. The more you know about anything, the easier it is to learn everything and vice versa This is the pure biology underlying the value of the liberal arts education. Use your whole brain, exercise it often, don’t think that you “just” need math and not spatial relations, visualization, verbal skills, a knowledge of history, a memory of performing experiments with your hands or mind or both – you need it all Remember, just as is the case with physical exercise (which you should get plenty of), mental exercise gradually makes you mentally stronger, so that you can eventually do easily things that at first appear insurmountably difficult. You can learn to learn three to ten times as fast as you did in high school, to have more fun while doing it, and to gain tremendous reasoning capabilities along the way just by trying to learn to learn more efficiently instead of continuing to use learning strategies that worked (possibly indifferently) back in elementary and high school. The next section, at long last, will make a very specific set of suggestions for one very good way to study physics (or nearly anything else) in a way that maximally takes advantage of your own volitional biology to make learning as efficient and pleasant as it is possible to be. How to Do Your Homework Effectively By now in your academic career (and given the information above) it should be very apparent just where homework exists in the grand scheme of (learning) things. Ideally, you attend a class where a warm and attentive professor clearly explains some abstruse concept and a whole raft of facts in some moderately interactive way that encourages engagement and “being earnest”. Alas, there are too many facts to fit in short term/immediate memory and too little time to move most of them through into long term/working memory before finishing with one and moving on to the next one. The material may appear to be boring and random so that it is difficult to pay full attention to the patterns being communicated and remain emotionally enthusiastic all the while to help the processPreliminaries 21 along. As a consequence, by the end of lecture you’ve already forgotten many if not most of the facts, but if you were paying attention, asked questions as needed, and really cared about learning the material you would remember a handful of the most important ones, the ones that made your brief understanding of the material hang (for a brief shining moment) together. This conceptual overview, however initially tenuous, is the skeleton you will eventually clothe withfactsandexperiencesto transformitintoanentiresystemofassociativememoryandreasoning where you can work intellectually at a high level with little effort and usually with a great deal of pleasure associated with the very act of thinking. But you aren’t there yet. You now know that you are not terribly likely to retain a lot of what you are shown in lecture without engagement. In order to actually learn it, you must stop being a passive recipient of facts. You must actively develop your understanding, by means of discussing the material and kicking it around with others, by using the material in some way, by teaching the material to peers as you come to understand it. To help facilitate this process, associated with lecture your professor almost certainly gave you anassignment. Amazinglyenough,its purposeis notto tormentyouorto bethe basisofyourgrade (although it may well do both). It is to give you some concrete stuff to do while thinking about the material to be learned, while discussing the material to be learned, while using the material to be learned to accomplish specific goals, while teaching some of what you figure out to others who are sharing this whole experience while being taught by them in turn. The assignment is much more important than lecture, as it is entirely participatory, where real learning is far more likely to occur. Youcould, onceyoulearnthe trickofit, blowoff lectureanddo fine ina coursein allother respects. If you fail to do the assignments with your entire spirit engaged, you are doomed. In other words, to learn you must do your homework, ideally at least partly in a group setting. The only question is: how should you do it to both finish learning all that stuff you sort-of-got in lecture and to re-attain the moment(s) of clarity that you then experienced, until eventually it becomes a permanent characteristic of your awareness and you know and fully understand it all on your own? There are two general steps that need to be iterated to finish learning anything at all. They are a lot of work. In fact, they are far more work than (passively) attending lecture, and are more importantthanattendinglecture. Youcanlearnthematerialwiththesestepswithouteverattending lecture, as long as you have access to what you need to learn in some media or human form. You in all probability will never learn it, lecture or not, without making a few passes through these steps. They are: a) Review the whole (typically lecture, textbooks and/or notes, the Internet, videos...) b) Work on the parts (do homework, and otherwise try to use what you are learning for something) (iterate until you thoroughly understand whatever it is you are trying to learn). Let’s examine these steps. Thefirstis prettyobvious. Yougenerallydon’t“getit” (where“it” is almostanythingnontrivial you are trying to learn) from one lecture, from reading one textbook one time. There is too much material, and it doesn’t initially make sense to you. If you are lucky and well prepared and blessed with a good instructor, perhaps you graspsome of it for a moment (and if your instructor is poor or you are particularly poorly prepared you may not manage even that) but what you do momentarily understand is fading, flitting further and further away with every moment that passes. You need to reviewtheentiretopic,asawhole,aswellasallitsparts. Asetofgoodsummarynotesmightcontain all the relative factoids, but there are relations between those factoids – a temporal sequencing, mathematical derivations connecting them to other things you know, a topical association with22 Preliminaries other things that you know. They tell a story, or part of a story, and you need to know that story in broad terms, not try to memorize it word for word. Reviewingthematerialshouldbedoneinlayers,skimmingthetextbookandyournotes,creating anewsetofnotesoutofthetextincombinationwithyourlecturenotes,maybereadinginmoredetail to understand some particular point that puzzles you, reworking a few of the examples presented. Lots of increasingly deep passes through it (starting with the merest skim-reading or reading a summary of the whole thing) are much better than trying to work through the whole text one line at a time and not moving on until you understand it. Many things you might want to understand will only come clear from things you are exposed to later, as it is not the case that all knowledge is ordinal, hierarchical, and derivatory. You especially do not have to work on memorizing the content. In fact, it is not desireable to try to memorize content at this point – you want the big picture first so that facts have a place to live in your brain. If you build them a house, they’ll move right in without a fuss, where if you try to grasp them one at a time with no place to put them, they’ll (metaphorically) slip away again as fast as you try to take up the next one. Let’s understand this a bit. Aswe’veseen,yourbrainisfabulouslyefficientatstoringinformationinacompressed associative form. Italsotendstorememberthingsthatareimportant–whateverthatmeans–andforgetthings thataren’timportanttomakeroomformoreimportantstuff, asyourbrainstructuresworktogether in understandable ways on the process. Building the cognitive map, the “house”, is what it’s all about. But as it turns out, building this house takes time. This is the goal of your iterated review process. At first youare memorizing things the hard way, trying to connect what you learn to very simple hierarchicalconcepts such as this step comes before that step. As you do this over and over again, though, you find that absorbing new information takes you less and less time, and you remember it much more easily and for a longer time without additional rehearsal. Sometimes your brain even outruns the learning process and “discovers” a missing part of the structure before you even read about it By reviewing the whole, well-organized structure over and over again, you gradually build a greatly compressed representation of it in your brain and tremendously reduce the amount of work required to flesh out that structure with increasing levels of detail and remember them and be able to work with them for a long, long time. Now let’s understand the second part of doing homework – working problems. As you can probably guess on your own at this point, there are good ways and bad ways to do homework problems. The worst way to do homework (aside from not doing it at all, which is far too common a practice and a bad idea if you have any intention of learning the material) is to do it all in one sitting, right before it is due, and to never again look at it. Doing your homework in a single sitting, working on it just one time fails to repeat and rehearse the material (essential for turning shortterm memoryinto long termin nearlyall cases). It exhausts the neurons in your brain (quite literally – there is metabolic energy consumed in thinking) as one often ends up working on a problem far too long in one sitting just to get done. It fails to incrementally build up in your brain’s long term memory the structures upon which the more complex solutions are based, so you have to constantly go back to the book to get them into short term memory long enough to get through a problem. Even this simple bit of repetition does initiate a learning process. Unfortunately, by not repeating the steps associated with the solution to this kind of problem after this one sitting they soon fade, often without a discernable trace in long term memory. Just as was the case in our experiment with memorizing the number above, the problems almost invariably are not going to be a matter of random noise. They have certain key facts and ideas that are the basis of their solution, and those ideas are used over and over again. There is plenty of pattern and meaning there for your brain to exploit in information compression, and it may well be very cool stuff to know and hence important to you once learned, but it takes time and repetitionPreliminaries 23 and a certain amount of meditation for the “gestalt” of it to spring into your awareness and burn itself into your conceptual memory as “high order understanding”. You have to give it this time, and perform the repetitions, while maintaining an optimistic, philosophical attitude towards the process. You have to do your best to have fun with it. You don’t get strong by lifting light weights a single time. You get strong lifting weights repeatedly, starting with light weights to be sure, but then working up to the heaviest weights you can manage. When you do build up to where you’re lifting hundreds of pounds, the fifty pounds you started with seems light as a feather to you. As with the body, so with the brain. Repeat broad strokes for the big picture with increasingly deep and “heavy” excursions into the material to explore it in detail as the overall picture emerges. Interspersethis with sessionswhereyouwork on problems and tryto usethe materialyou’vefigured out so far. Be sure to discuss it and teach it to others as you go as much as possible, as articulating what you’ve figured out to others both uses a different part of your brain than taking it in (and hence solidifies the memory) and it helps you articulate the ideas to yourself This process will help you learn more, better, faster than you ever have before, and to have fun doing it Your brain is more complicated than you think. You are very likely used to working hard to try to make it figure things out, but you’ve probably observed that this doesn’t work very well. A lot of times you simply cannot “figure things out” because your brain doesn’t yet know the key things required to do this, or doesn’t “see” how those parts you do know fit together. Learning and discovery is not, alas, “intentional” – it is more like trying to get a bird to light on your hand that flits away the moment you try to grasp it. People who do really hard crossword puzzles (one form of great brain exercise) have learned the following. After making a pass through the puzzle and filling in all the words they can “get”, and maybe making a couple of extra passes through thinking hard about ones they can’t get right away, lookingfor patterns, trying partialguesses,they arriveatan impasse. If they continueworkinghard on it, they are unlikely to make further progress, no matter how long they stare at it. Ontheotherhand,iftheyput thepuzzle downanddo something else for a while–especiallyifthe somethingelseisgotobedandsleep–whentheycomebacktothepuzzletheyoftencanimmediately see a dozen or more words that the day before were absolutely invisible to them. Sometimes one of the long theme answers (perhaps 25 characters long) where they have no more than two letters just “gives up” – they can simply “see” what the answer must be. Where do these answers come from? The person has not “figured them out”, they have “recog- nized” them. They come all at once, and they don’t come about as the result of a logical sequential process. 23 Often they come from the person’s right brain . The left brain tries to use logic and simple memory when it works on crosswork puzzles. This is usually good for some words, but for many of the words there are many possible answers and without any insight one can’t even recall one of the possibilities. The clues don’t suffice to connect you up to a word. Even as letters get filled in this continues to be the case, not because you don’t know the word (although in really hard puzzles this can sometimes be the case) but because you don’t know how to recognize the word “all at once” from a cleverly nonlinear clue and a few letters in this context. Therightbrainis(tosomeextent)responsibleforinsightandnon-linearthinking. Itseespatterns, and wholes, not sequential relations between the parts. It isn’t intentional – we can’t “make” our right brains figure something out, it is often the other way around Working hard on a problem, then “sleeping on it” (to get that all important hippocampal involvement going) is actually a great way to develop “insight” that lets you solve it without really working terribly hard after a few tries. 23 Note that this description is at least partly metaphor, for whilethere is some hemispherical specialization of some of these functions, it isn’t always sharp. I’m retaining them here (oh you brain specialists who might be reading this) because they are a valuable metaphor.

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