By now in your academic career 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. 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. 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.
Studies show that you are only likely to retain anywhere from 5% to 30% of what you are shown in lecture. Clearly this is not enough to make the information conceptually useful, to learn it. In order to actually learn, 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.
Medical schools have long been aware of this. In the year 1907, medical schools had two years of coursework to prepare a student to be a physician. In the year 2007, they are still two years of coursework - but the amount of science and medicine that is taught in those two years has exploded. They have developed the following mantra to help their students understand the only way the process can still work:
To help facilitate this process, associated with lecture your professor gave you an assignment. Amazingly enough, its purpose is not to torment you or to be the basis of your grade (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.
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 important than attending lecture. You can learn the material with these steps without ever attending 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:
Let's examine these steps.
The first is pretty obvious. You didn't ``get it'' from one lecture. There was too much material. If you were lucky and well prepared and blessed with a good instructor, perhaps you grasped some of it for a moment (and if your instructor was poor or you were particularly poorly prepared you may not have managed even that) but what you did momentarily understand is fading, flitting further and further away with every moment that passes. You need to review the entire topic, as a whole, as well as all its parts. A set of good summary notes might contain all the relative factoids, but there are relations between those factoids - a temporal sequencing, mathematical derviations connecting them to other things you know, a topical association with 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.
Reviewing the material should be done in layers, skimming the textbook and your notes, creating a new set of notes out of the text in combination with your lecture notes, maybe reading in more detail 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.
Your brain is fabulously efficient at storing information in a compressed associative form. It also tends to remember things that are important - whatever that means - and forget things that aren't important to make room for more important stuff. There are lots of experiments that demonstrate this - the simplest being trying to memorize a string of ten or so numbers at a glance (more than the 7 one can typically get into short term memory).
Try memorizing 1357902468 from just the one glance you got reading this sentence. No fair going back and repeating it to yourself, at least while looking at it! Now look at it and try to remember it. One strategy is to just repeat it to yourself until you get it right, but if you stare at it a while and think, you'll see that it has a very simple pattern embedded in it.
In fact, this number ``compresses'' to a single two-step rule - all the odd digits in ascending order followed by all the even digits ditto. You already know what a ``digit'' is, what odd and even numbers are, what ascending versus descending order is. You only need to remember "ascending" and "odd followed by even digits" - everything else is compressed. You will almost certainly be able to remember the digit string tomorrow without further rehearsal because of this rule and the fact that it illustrates an interesting point, where if you didn't notice the pattern and just memorized it as a string of "random" numbers, devoid of any meaning, your brain would have little reason to retain it as it is ``unimportant''. Even if you forget this particular string, you may well remember the point and use a different string like 1212121212 (five repetitions of 12) to illustrate the same point when teaching it to someone else. This is fine! My goal, too, is to teach you this, not some particular patterned set of numbers neither of us really cares about.
This ability to compress goes far beyond what I can explain or you can easily imagine. When I play a game of chess, I've forgotten my first five moves by the time I've made my tenth move. By the time the game finishes, I have no idea how I got into the mess I'm probably in. A chess master, on the other hand, can finish the game and then can recontruct the entire game in order, and can criticize each move as they do so. In fact, they can probably remember the entire game they played yesterday, or the one they played last week. They've built a complex structure of associative memory so that they don't remember moves the same way you or I do.
On the other hand, I can often remember what mistakes a student of mine made a week after grading one of their papers. I many not remember the student's name (no good associative memory there) but I've got great structures for remembering how to solve or not solve physics problems and remember only what the student did wrong - I already know how to do what they did right.
This is the goal of your iterated review process. At first you are memorizing things the hard way, trying to connect what you learn to very simple hierarchical concepts 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.
It is left as a homework exercise for the student to work out why this is a bad idea from the discussion and facts given above. So take a minute and think about it, then turn the page.
Let's see, doing your homework in a single sitting, working on it just one time fails to repeat and rehearse the material (essential for turning short term memory into long term in nearly all 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 them after this one sitting they soon fade, often without a discernable trace in long term memory.
Just as was the case 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 repetition 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. Intersperse this with sessions where you work on problems and try to use the material you've figured 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, looking for patterns, trying partial guesses, they arrive at an impasse. If they continue working hard on it, they are unlikely to make further progress, no matter how long they stare at it.
On the other hand, if they put the puzzle down and do something else for a while - especially if the something else is go to bed and sleep - when they come back to the puzzle they often can immediately 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 ``recognized'' them. They come all at once, and they don't come about as the result of a logical sequential process.
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.
The right brain is (to some extent) responsible for insight and non-linear thinking. It sees patterns, 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'' is actually a great way to develop ``insight'' that lets you solve it without really working terribly hard after a few tries. It also utilizes more of your brain - left and right brain, sequential reasoning and insight, and if you articulate it, or use it, or make something with your hands, then it exercieses these parts of your brain as well, strengthening the memory and your understanding still more. The learning that is associated with this process, and the problem solving power of the method, is much greater than just working on a problem linearly the night before it is due until you hack your way through it using information assembled a part at a time from the book.
The following ``Method of Three Passes'' is a specific strategy that implements many of the tricks discussed above. It is known to be effective for learning by means of doing homework (or in a generalized way, learning anything at all). It is ideal for ``problem oriented homework'', and will pay off big in learning dividends should you adopt it, especially when supported by a group oriented recitation with strong tutorial support and many opportunities for peer discussion and teaching.