Open question, respond with your style.
What I found to be extremely detrimental to all my attempts at self study are the lac of repetition and drilling of the material. However, looking back, I'd probably take an assload of notes and go over them, constantly for a few hours a day for a few days, progress, do the same again, review them material, etc. However, I'm a Biology guy, so I really have no useful information in how to study Physics and Math.
This is pretty good advice for physics and math as well (though 'repetition' in these areas really means doing problems).
What I would do is first do a little research to find out what the best textbook/learning material is for the particular area I want to study. Once you buy that treat it like you would any class.
Set aside time every day or pretty regularly to study the text and make sure you take notes while doing it. If you have difficulty understanding a topic use ancillary sources to sure that up. Assign yourself homework from whatever material you have (hopefully from the text you've selected) or can find and make sure solutions are available somewhere (for example for ambitious grad students there are tons of examples of past quals with solutions on the internet at different universities). Do the problem sets and go through the material week by week. The rub is that you don't have a prof. or anyone to turn to when you're having difficulty with a concept but you can turn to other sources when such a problem arises.
For math classes, I usually look over the material once during for homework and once or twice before a test. I had to learn everything about line integrals, surface integrals, curls, vector fields, green and stokes theorems for my final in a day and a half. I used no doz the morning of the test and it went pretty smoothly.
When I actually enjoy the material, however, I sit down with the book, read through each section and do a few easy problems and then a few I find interesting. This is how I learned from Dummit and Foote for my abstract algebra class and it worked great.
Over my winter break I have been reading Felix Hausdorff's book on set theory, which unfortunately, being more an old text for professionals(I suppose, since it does contain original material of Hausdorff's and set theory was very new in 1914) it contains no problems, but it is very clear and very insightful, so; I feel that even if my 'skill' with the material is lacking compared to where it would be reading a modern introductory text, my understanding of the subject will be much greater.
I am also working on Mathematical Logic by Ebbinghaus and Flum. It is very organized and very judicious with notation. All of the fat is cut out of the notations in favor of a more intuitive aesthetic which I think is a very good idea. There are some good exercises in this book which give some insight into the mechanics of the procedures. I find this book to be a bit challenging at times. What I do when I hit a roadblock, I first go back and make sure that it is not due to my glazing over an important point or example. Then I consider what was given to me and see if I can't piece together a more intuitive view(the text is very bare bones). If I don't get it right away, I put the book down and do something else. Sometimes the intuitive view of the material comes to me and I go back and read more. Sometimes it doesn't come to me until I sit back down with the book and read the material over again. So for, I haven't had to labor over anything for too long, and for the most part anything that wasn't clear was only unclear due to the abbreviated proofs or constantly changing notations.
So, I suppose I don't have any specific studying 'technique', I simply immerse myself for hours in material that I find interesting. It gets easier the more you do it, to some extent.
Interesting question and one that I would love to see other people's methods of tackling.
First and foremost, I get multiple sources together. One source might be incomplete in one area, such a physics text that is vague in gravitation. You'd want another text or book of problems to help explain this area to avoid gaps in knowledge. I always make sure that the books I do buy are varied. Like, buy a textbook and a book of problems. For German, let's say, I'd get a textbook, a novel, a grammar book, and a dictionary. Here it's important to avoid the temptation to buy ten books and never ever look at them because it's a waste of money.
Then read through the explanatory sections (take notes or don't, I usually take rough outlines because I'm not a rote memorization person) and then apply that knowledge to the problems and solve them. How many problems? It depends. A person learns almost nothing by doing the same sort of problem a hundred times if only the arithmetic is different but another person who did five completely different problems learns a lot more. Quality, not quantity.
One thing I'd like to say that has helped me immensely is to remember that you must UNDERSTAND what you're doing. Insight and a clear explanation of why a problem has to be solved this way needs to be achieved. The people in my physics course who use brute force and just memorize steps to a problem usually fail when presented with new or even slightly different problems. This is where most of the whining about 'hard tests' comes from; because those people don't actually understand physics.
If I'm stuck or can't figure something out, I'll look at more books (which is why I have them in the first place) or ask someone who would know and try again. Very simple. Self-studying is a lot about discipline but not the sort of monastic, torturous discipline that most people imagine. You need to make a realistic schedule of what to study when and follow it.
The where and when in my schedule is usually in short chunks (say half hour to 50 minutes) in a quiet place like a cafe or a library. You can try it in a dorm, if you're still in school, but I find that the dorm has never helped me get anything done but catching some zs. There's too many distractions!
Sorry if this came across as a 'you' sort of post. It's not directed at anyone in this thread. Lately I've had to talk to a lot of friends and classmates that have failed their classes and are wondering why, so this is the format in which I usually give advice. I hope it helped someone :)
I have only recently (4 months ago) started learning mathematics on my own. My method usually involves reading the next definition or theorem and the proof that accompanies it, then after I understand it copy it down onto a notebook. Copying down information helped me when I was in high school memorizing facts and in lectures, but I am not so sure it helps me now. For those that study pure mathematics or theoretical physics, do you find its useful to copy the theorems after you read them, or is it better to read them, do the questions, then go back and cement the definitions and the theorems.
I find it useful to note down the theorms and definitions in a note book for future refernce.
Going over notes never works for me. Just get out a textbook and hack your way through practice questions. It's the only way I learn. Which is why at school I only got average marks for English and biology, but managed to do much better in maths and physics.
The thing with my notes is, once I write them I never look at them again.
Self study or study-study?
I "self study" much differently than I study for school.
I self study very non-organized and disorderly, but I greatly enjoy it. When I self study, I jump around through text books, spend more time on what I find enjoyable, spend no time on what I find boring, and might even skip to a totally different subject while still mid-sentence on another subject.
Despite how sloppy it is, it seems my self studying is at least somewhat effective. I learned calculus completely through self study. I followed absolutely no pattern or structure, but whatever I did learn held up enough for me to score an A- in a multivariable/vector calculus (IE: Calc III) class as my first math course since pre-algebra in high school 12 years earlier.
When I study for school, I follow the pattern and structure taught by the professor and devote my time on each subject based on importance for the class instead of how "fun" it is.
I don't know that I can say my self-studying is any less effective though. In fact, on an hour per hour basis, I'd have to say it is more effective. I certainly didn't spend 16 weeks differential calculus and 16 weeks on integral calculus. I doubt I spent much more than 16 hours on each.
I'm sure there are many "subtleties" of those subjects that I missed by not learning them in a classroom setting, but it hasn't caused me trouble yet. In fact, I think it may have been a benefit....I spent much more time, relatively, on the "important" parts of those subjects than I would have been able to if I had to learn every subtle nuance with equal vigor in case it might be seen on an exam.
If I were somehow able to "earn" a degree in Physics and Mathematics in a completely self-study manner with a "hired" tutor to answer my questions and confusions as they pop-up.....I would be in nerd heaven.
A lot of the time I feel that I'm just aimlessly taking notes while self studying. When you guys say reading through the material then doing chapters, do you involve note taking? I'm questioning if note taking actually helps. I hear a lot of active learning with generalizing of specializing theorems. While studying from rudin, this was sometimes helpful but most of the time beyond my ability.
For example, even during compact sets I had trouble of thinking of a neat compact set besides the obvious closed set in R^n.
I don't remember much if I don't take notes while I self-study. For classes, I don't need notes because the lecture and homework are enough to brand the concepts onto my brain. My notes usually languish afterward in unread notebooks, but the "active" participation is invaluable for recall.
Read over notes, read over text in text books, do the hardest problems and be persistent about finding the answer without looking for hints until you've tried everything you could possibly think of.
Do this while having the mindset that it's not a chore, but it's more like a HOBBY (eh, like sudoku), and it's good for you in the end. Hating to study is detrimental to the proccess, so you'll just have to lie to yourself that it does not suck.
I don't think I've ever taken a note in a Math lecture that gave me any benefit....
My method is to try to get a grasp of the material before the semester even starts. I take my "notes" before lecture so I have something to follow along with during the lecture. I never take "notes" in lecture. The moment I put pen to paper....my brain shuts off and I'm in copy mode.
The best method I've found is to "self-study" before the material is covered in class. I try to go to the lecture already "knowing" the material. Of course the lecture lets me know how little I actually "knew," but I'm able to follow along with what I already know and pick up the missing pieces during the lecture. Self study at that point is just re-assuring that I know what I thought I did.
I'm horrible at notes. If I go into a lecture to "learn," I'm doomed. My notes don't make sense, I feel lost in the lecture, and I have a major mental problem with learning what I'm supposed to when studying those notes later on.
When I study before the lectures, at my own pace and on my own terms, I never feel like I'm "studying." I just goof around learning the material I love. By the time the lecture comes, I'm actually "studying" what the professor lectures because it's solidifying what I have already tried to teach myself.
I'm sure I spend the same amount of time with either method, but for some reason, my method doesn't feel like work, it feels like fun.
Troponin your method of studying for lectures is very good, I will emulate it during this semester. I too also find you learn so much more from the lectures when you know the material.
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