Elementary Physics lecture notes

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Introductory Physics I Elementary Mechanics by Robert G. Brown Duke University Physics Department Durham, NC 27708-0305 rgbphy.duke.eduI: Getting Ready to Learn Physics 1Preliminaries See, Do, Teach Ifyouarereadingthis, Iassumethatyouareeithertakingacourseinphysicsor wishtolearnphysicsonyourown. Ifthisisthecase,Iwanttobeginbyteaching 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 learningprocess. Ifyouremaindisengaged,detatchedfromthelearningprocess, 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 classroom with some audiovisual aids or some demonstrations. The standard expectation 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 intobecomingalecturetranscriptionmachine, recordingallkindsofthingsthat 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 34 Preliminaries material any more than is strictly required to do the homework, until one day a bigtestdrawsnighandyourealizethatyoureallydon’tunderstandanythingand haveforgottenmostofwhatyoudid,briefly,understand. Doomanddestruction loom. Sound familiar? On the other hand, you may be in a course where the instructor has struc- tured the course with a balanced mix of open lecture (held as a freeform discus- sion where questions aren’t just encouraged but required) and group interactive learning situations such as a carefully structured recitation and lab where dis- cussionanddoingblendtogether, wherestudentsteacheachotherandusewhat they have learned in many ways and contexts. If so, you’re lucky, but luck only goes so far. 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 thequesttolearnanythingyoueverchoosetostudy. Inaword, yousimplymay not give a rodent’s furry behind about learning the material so that studying is always a fight with yourself to “make” yourself do 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 devoutlyhope,becausethat’sallthetrainingyouarelikelytogetuntilyoustart doing the procedure over and over again with real humans and with limited oversight from an attending physician with too many other things 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 signifi- cantly reinforced by research in teaching methodology, especially in physics and mathematics.Preliminaries 5 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 nearlycompletewasteoftime. That’snottosaythat“lecture”intheformofan 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 scheme of learning, but by being passive during lecture you cause it to fail in its 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 instructor presents it, you will have to work much harder to achieve the sense of the material “later”, if later ever comes at all. If you fail toidentifytheimportantconceptsduringthepresentationandseethelectureas 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. Reading is a bit more active – at the very least your brain is more likely to be somewhat engaged if you aren’t “just” transcribing the book onto a piece of paperorlettingthewordsandsymbolshappeninyourmind–butisstillpretty passive. Even watching nifty movies or cool-ee-oh demonstrations 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 yourbrainarelikelytotakechargeandpullyoureyesirresistablytothewindow 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 inthelearningprocess,thingschangedramatically. Insteadofpassivelylistening inlecture,youcanatleasttrytoaskquestionsandinitiatediscussionswhenever an idea is presented that makes no intial sense to you. 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 fun6 Preliminaries 1 In summary, sitting on your ass and writing meaningless (to you, so far) things down as somebody 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 introductory 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. Articulationofideas,whetheritistoyourselfortoothersinadiscussionset- ting, requires you to create tentative patterns that might describe and organize all the details you are being presented with. Using those patterns and apply- ing 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’tmakesense”. Inan“active”lecturestudentsparticipateintheprocess, and can ask questions and kick ideas around until they do make sense. Partici- pation 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. Learning is increased still further by doing, the very essence of activity and engagement. “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)reading the 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 writeF =q(v×B) on the board, it means a great deal to me, but if you are taking this course for the first time 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 tons of barley with a market value ofv and a profit margin of q? Tolearnthisexpression(foryes, thisisaforcelawofnatureandonethatwe very much must learn this semester) we have to learn what the symbols stand 1 I mean, of course, your donkey. What did you think I meant?Preliminaries 7 for – q is the charge of a point-like object in motion at velocityv 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, at least at first). In order to get a gut feeling for what this equation represents, for the directions 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 any- where 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 you can take in college), even if you’re not a super-genius (or have never managed in the past to learn like one, a glance and you’re done): Work in groups 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, and the groups and groupactivitiesmaybehighlystructuredorfreeform. “Studio”or“TeamBased Learning”methodsforteachingphysicshaveevenwrappedthelectureitselfinto agroup-structuredsetting,soeverythingisdoneingroups/teams,and(probably bymakingitnearlyimpossibletobedisengagedandsitpassivelyinclasswaiting for learning to “happen”) this approach yields measureable improvements (all things being equal) on at least some objective instruments for measurement of learning. If you take charge of your own learning, though, you will quickly see that in any course, however taught, you can study in a group This is true even in a course where “the homework” is to be done alone by fiat of the (unfortunately ignorant and misguided) instructor. Just study “around” the actual assignment –assignyourselvesproblems“like”theactualassignment–mosttextbookshave plentyofextraproblemsandthenthereistheInternetandothertextbooks–and do them in a group, then (afterwards) break up and do your actual assignment alone. Note that 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 PDF copy of this book (it’s8 Preliminaries free for instructors, after all, and a click away on the Internet) and point to this page and paragraph containing the following little message from me to them: Yo Teacher Let’s wake up and smell the coffee Don’t prevent your students from doing homework in groups – require it Make the homework correspondingly more difficult Give them quite a lot of course credit for doing it well Construct a recitation or review session where students – in groups – who still cannot get the most difficult problems can get socratic tutorial help after working hard on the problems on their own Integrate discussion and deliberately teach to increase active engagement (instead of passive wandering 2 attention) in lecture . Then watch as student performance and en- gagement spirals into the stratosphere compared to what it was be- fore... Then pray. Some instructors have their egos tied up in things to the point where they cannot learn, and then what can you do? If an instructor lets ego or politics obstruct their search for functional methodology, you’re screwed anyway, and you might as well just tackle the material on your own. Or heck, maybe their expertise and teaching experience vastly exceeds my own so that their naked words are sufficiently golden that any student should be able to learn by just hearing them and doing homework all alone in isolation from any peer-interaction process that might be of use to help them make sense of it all – all data to the contrary. My own words and lecture – in spite of my 31 years of experience in the classroom, in spite of the fact that it has been well over twenty years since I actually used lecture notes to teach the course, in spite of the fact I never, ever prepareforrecitationbecausesolvingthehomeworkproblemswiththestudents “cold” as a peer member of their groups is useful where copying my privately worked out solutions onto a blackboard for them to passively copy on their papers in turn is useless, in spite of the fact that I wrote this book similarly without the use of any outside resource – my words and lecture are not. On the other hand, students who work effectively in groups and learn to use this book (and other resources) and do all of the homework “to perfection” might well learn physics quite well without my involvement at all Let’s understand why working in groups has such a dramatic effect on learn- ing. What happens in a group? Well, a lot of discussion happens, because 2 Perhaps by using Team Based Learning methods to structure and balance student groups and “flipping” classrooms to foist the lecture off onto videos of somebody else lecturing to increase the time spent in the class working in groups, but I’ve found that in mid-sized classes and smaller (less than around fifty students) one can get very good results from traditional lecture without a specially designed classroom by the Chocolate Method – I lecture without notes and offer a piece of chocolate or cheap toy or nifty pencil to any student who catches me making a mistake on the board before I catch it myself, who asks a particularly good question, who looks like they are nodding off to sleep (seriously, chocolate works wonders here, especially when ceremoniously offered). Anything that keeps students focussed during lecture by making it into a game, by allowing/encouraging them to speak out without raising their hands, by praising them and rewarding them for engagement makes a huge difference.Preliminaries 9 humans working on a common problem like to talk. There is plenty of doing go- ingon, presumingthatthegrouphasacommontasklisttoworkthrough, likea smallmountainofreallydifficultproblemsthatnobodycanpossiblysolvework- ing on their own and are barely within their abilities working as a group backed up by the course instructor Finally, in a group 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 This problem you “get”, and teach others (and actually learn it better from teaching it than they do from your presentation – never begrudge the effort required to teach your group peers even if some of them are very slow to under- stand). 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 exploration of false paths, the discovery and articulation of the correct path, the process of discussion, resolution and agree- ment in teaching whereby everybody in the group reaches full understanding. 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. Asanengagedstudentyou don’t have to live with that Putitthereyourself, 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 effectivegroupsoonlearnstoiteratetheteaching–Iteachyou,andtomakesure you got it you immediately use the material I taught you and try to articulate it back to me. Eventually everybody in the group understands, everybody in the group benefits, everybody in the group gets the best possible grade on the material. This process will actually make you (quite literally) more intelligent. You may or may not become smart enough 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 away on 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 group, especially in a relaxed context with food anddrinkpresent. Yes, I’mencouragingyoutohave“physicsstudyparties”(or history study parties, or psychology study parties). Hold contests. Give silly10 Preliminaries prizes. See. Do. Teach. Other Conditions for Learning Learning isn’t only dependent on the engagement pattern implicit in the See, Do,Teachrule. Let’sabsorbafewmoreTrueFactsaboutlearning,inparticular let’s come up with a handful of things 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 notfullyon). Someoftheseswitches,orenvironmentalparameters,acttogether more powerfully than they act alone. We’ll start with the most important pair, a pair that research has shown work together to potentiate or block learning. Insteadofjusttellingyouwhattheyare,arguingthattheyareimportantfor a paragraph or six, and moving on, I’m going to give you an early opportunity topracticeactivelearninginthecontextofreadingachapteronactivelearning. That is, I want you to participate in a tiny mini-experiment. It works a little bit better if it is done verbally in a one-on-one meeting, but it should still work well enough even if it is done in this text that you are reading. I’m 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. At the end of the minute, try to recall the number I gave you and write down what you remember. 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 ahead to catch a glimpse of it while reading the instructions 3 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 away that 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 3 1357986420 (one, two, three, quit and do something else for one minute...)Preliminaries 11 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(withoutcheatinganextrasecondorthree, youknowwhoyouare)that it consisted of the string of odd digits 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 themind’scapacityforprocessingandremembering“meaningless”or“random” information. A string of ten measely (apparently) random digits is too much to remember for one lousy minute, especially if you aren’t given time to do rehearsalandalloftheotherthingswehavetomakeourselvesdoto“memorize” meaningless information. Of course things 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 tomorroworeven a year from nowand have an excellent chance of remembering thisparticulardigitstring,becauseitmakes senseofasort,andthereareplenty 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 lecture and class time, and is one reason retention from 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 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’t usually give students a chance to ex- perience this in lecture. No sooner is one seemingly random factoid laid out on the table than along comes a 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.12 Preliminaries The humble and unsurprising conclusion I want you to draw from this silly little mini-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” – theyareatbestmemorized. Informationcanalmostalwaysbecompressedwhen 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. There is one more example I like to use to illustrate how important this information compression 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, blind- folded, 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 derive the Lorentz group from first principles, and I can, and this doesn’t automatically make me smarter than them either. It is because chess makes sense to them – they’ve achieved a deep understanding of the game, 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 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 very much the goal of this course (the one in the textbook you are reading, since I use this intro in many textbooks), and to achieve it you 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,aspatternsandstructureexistinabundanceinallsubjectsofinterest. 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 very likely didn’t do what I asked, you didn’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, justwantedtoreadthedamnchapterandgetitoverwithsoyoucoulddo, well,Preliminaries 13 whatever the hell else it is you were planning to do today that’s more important to you than 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 rememberthelittlemnemonicformulaIgaveaboveforthedigitstring(although frankly, peoplethatarethatdisengagedareprobablynotabouttodothingslike 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 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 what you are studying, and the in- formation you are trying to learn makes sense (if only for a moment, 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 pa- rameters to associate with learning – things that increase your probability of learning monotonically as they vary. Some of them are already apparent from the discussion above. Let’s list a few more of them with explanations just so that you can see how easy it is to sit down to study and try to learn and have14 Preliminaries Figure 1: Relation between sense, care and learning “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 day to day, even more when you are really using it or are somewhat sedentary in your physical habits. Note that your brain runs on pure, energy-rich glucose, so when your blood sugar drops your brain activity drops right along with it. This can happen (paradoxically) because you just ate a carbohydrate rich meal. A balanced diet containing foods 4 with a lower glycemic index tends to be harder to digest and provides a longer period of sustained energy for your brain. A daily multivitamin (and various antioxidant supplements such as alpha lipoic acid) can also help maintain your 4 Wikipedia: http://www.wikipedia.org/wiki/glycemic index.Preliminaries 15 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. On the other hand, a good hearty breakfast, eaten at leastanhourbeforeplungingintoyourstudies,isagreatideaandisafarbetter habittodevelopforalifetimethaneatingnobreakfastandinsteadeatingahuge meal right before bed. Learning requires adequate sleep. Sure this is tough to manage at college – there are no parents 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,theformationoflong term memory of any kindfromaday’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. 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. Common sense (and yes, experience)willrapidlyconvinceyouthatlearninggenerallyworksbetterif you’re not stoned or drunk when you study. Learning works much better when youhavetimetolearnandhaven’tputeverythingofftothelastminute. Infact, 5 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 compression and 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 research that attempts to 5 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.16 Preliminaries find benefit in one teaching/learning methodology over another. Some students do improve, even dramatically improve – when this or that teaching/learning methodology 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. The point is that finding an optimal teaching and learning strategy is tech- nically an optimization problem on a high dimensional space. We’ve discussed some of the important dimensions above, isolating a few that appear to have a monotonic effect on the desired outcome in at least some range (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 for experience to fix into long termmemorywithallofthatsleep). We’veomittedonethatiscrucial,however. That is your brain Your Brain and Learning Your brain is more than just a unique instrument. In some sense it is you. You could imagine having your brain removed from your body and being hooked up to machinary that provided it with sight, sound, and touch in such a way that 6 “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. Itdoes“work”. Itis, infact, abiologicalmachineofsublimecomplexity and subtlety, one of the true wonders of 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. For all of that, few indeed are the people who bother to learn to actually use their brain effectively 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. Sohere’satleastoneusermanualforyourbrain. Itisbynomeanscomplete orauthoritative,butitshouldbeenoughtogetyoustarted,tohelpyoudiscover 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, 7 let’ssimplyitemizeinnoparticularorderthevariousfeaturesofthebrain that 6 Imagine very easily if you’ve ever seen The Matrix movie trilogy... 7 Wikipedia: http://www.wikipedia.org/wiki/brain.Preliminaries 17 bearontheprocessoflearning. Bearinmindthatsuchaminimalpresentationis more of a metaphor than anything else because simple (and extremely common) generalizationssuchas“creativityisaright-brainfunction”arenotstrictlytrue as the brain is far more complex than that. 8 • 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. • The human brain consists of layers with a structure that recapitulates evolutionary phylogeny; 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. 9 • The cerebral cortex – especially the outermost layer of it called the neo- cortex – is where “higher thought” activities associated with learning and problem solving take place, although the brain is a very complex instru- ment with functions spread out over many regions. 10 • An important brain model is a neural network . Computer simulated neural networks provide us with insight into how the brain can remember past events and process new information. • The fundamental operational units of the brain’s information processing 11 functionalityarecalledneurons . Neuronsreceiveelectrochemicalsignals 12 fromotherneuronsthataretransmittedthroughlongfiberscalledaxons 13 Neurotransmitters aretheactualchemicalsresponsibleforthetriggered functioning of neurons and hence the neural network in 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 world as seen by the associated sense mechanism. For example, there exists a topographic map 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 8 Wikipedia: http://www.wikipedia.org/wiki/cerebral hemisphere. 9 Wikipedia: http://www.wikipedia.org/wiki/Cerebral cortex. 10 Wikipedia: http://www.wikipedia.org/wiki/Neural network. 11 Wikipedia: http://www.wikipedia.org/wiki/Neurons. 12 Wikipedia: http://www.wikipedia.org/wiki/axon. . 13 Wikipedia: http://www.wikipedia.org/wiki/neurotransmitters.18 Preliminaries • Similar mapsexistfortheothersenses, although sensationsfromtheright side of your body are generally processed in a laterally inverted way by the oppositehemisphere of the brain. What your right eye sees, what your righthandtouches, isultimately transmitted toasensoryareainyourleft 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 neurotransmitters by chemical means, put neurotrans- mitter analogues into your system, and alter the chemical trigger poten- tials of your neurons by taking various drugs, poisons, or hormones. The biochemistry of your brainisextremelyimportanttoitsfunction, and(un- fortunately) 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. 14 • 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. • Intelligenceisnotevenfixedwithinapopulationovertime. Aphenomenon 15 known as “the 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 the lower half of the distribution of intelligence. This is an active area of research (as one might 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 directly with 14 Wikipedia: http://www.wikipedia.org/wiki/intelligence. 15 Wikipedia: http://www.wikipedia.org/wiki/flynn effect.Preliminaries 19 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. 16 • Sensory memory corresponds to the roughly 0.5 second (for most peo- ple) that a sensory impression remains in the brain’s “active sensory reg- ister”, thesensorycortex. Itcantypicallyholdlessthan12“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 sen- sorymemoryistransferred. Justwhichinformationistransferreddepends on where one’s “attention” is, and the mechanics of the attention process are not well understood and are an area of active research. Attention acts like a filtering process, as there is a wealth of parallel information in our sensory memory at any given instant in time but the thread of our aware- ness and experience of time is serial. We tend to “pay attention” to one thingatatime. Shorttermmemorylastsfromafewsecondstoaslongasa 17 minutewithoutrehearsal, andfornearlyallpeopleitholds4−5objects . However,itscapacitycanbeincreasedbyaprocesscalled“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 usual three second look you’ll almost certainly find 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 government agencies in the general category of “intelligence and law enforcement” and the last three are all market symbols for information technology mega-corporations, you can once again recall the information a day later with only the most cur- sory of rehearsals. You’ve taken eighteen ”random” objects that were meaningless and could hence be recalled only through the most arduous ofrehearsalprocesses,convertedthemtosix“chunks”ofthreethatcanbe easily tagged by the brain’s existing long term memory (note that you are 16 Wikipedia: http://www.wikipedia.org/wiki/memory. Several items in a row are con- nected to this page. 17 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.

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