Is it morally right and pedagogically right to google answers to homework? This is a soft question that I have been struggling with lately.
My professor sets tough questions for homework (around 10 per week).
The difficulty is such that if I attempt the questions entirely on my own, I usually get stuck for over 2 hours per question, with no guarantee of succeeding.
Some of the questions are in fact theorems proved by famous mathematicians like Gauss, or results from research papers.
As much as I dislike to search for answers on the internet, I am often forced to by time constraints if I even expect to complete the homework in time for submission. (I am taking 2 other modules and writing an undergraduate thesis too).
My school does not have explicit rules against googling for homework, so I guess it is not a legal issue.
However, it often goes against my conscience, and I wonder if this practice is counterproductive for my mathematical development.
Any suggestions and experience dealing with this?
 A: Let me explain why I, and almost all faculty members I know, do not want students searching for homework problems online. 


*

*It destroys our ability to calibrate the course difficulty. Twenty hours of homework a week is very high for a math course; higher than I would expect from any course that was not promoted as a "boot camp" style course. Either you are falling behind the rest of the class, or other people are turning in much scantier work than you are, or everyone is googling the problems. The first two situations are obvious, and your professor should be adjusting to it. The last situation is invisible. We had an analysis course at MI last year pedagogically ruined because everyone kept solving the homework problems, so the professor kept increasing his pace, until an in class test revealed that no one was actually doing the homework themselves.

*It forces us to use more obscure, and often not as good, problems. There are some fields where there are computations every student should do -- and, as a result, they are written up in books and online sources everywhere. It hurts my ability to design good problem sets if I can't put this fundamental problems on the problem set. Even in fields where there are not such key problems, there are often only so many ways to set up an example so that it is doable in a reasonable amount of time. If I can't use the examples which are already online, then I need to pick larger and stranger values for my parameters, which makes the problem set harder.

*I do not believe that students will learn as much from reading a solution as finding it themselves; this is probably uncontroversial. Moreover, I think that hearing a solution from a classmate with whom you have been discussing the problem together is better than hearing it from a classmate who solved it separately; hearing it from a classmate is better than hearing it from a faculty member; and hearing it from a faculty member is better than reading it in a textbook or here on math.SE. I think that the more interactive and the less polished the presentation, the more you have to engage your own understanding to process and take in the answer. This is why I almost never leave full answers to questions that look like homework here; I think it is harmful.
Let me quote the policy I will have for the combinatorial representation theory course I will be teaching this Fall:

Homework Policy: You are welcome to consult each other provided (1)
  you list all people and sources who aided you, or whom you aided and
  (2) you write-up the solutions independently, in your own language. If
  you seek help from other mathematicians/math students, you should be
  seeking general advice, not specific solutions, and must disclose this
  help. I am, of course, glad to provide help!
I don't intend for you to need to consult books and papers outside
  your notes. If you do consult such, you should be looking for
  better/other understanding of the definitions and concepts, not
  solutions to the problems.
You MAY NOT post homework problems to internet fora seeking solutions.
  Although I know of cases where such fora are valuable, and I
  participate in some, I feel that they have a major tendency to be too
  explicit in their help. You may post questions asking for clarification and 
  alternate perspectives on concepts and results we have covered.

You should ask your professor for his or her policy, but I think that this is on the permissive side of what most math professors would write if they thought about a policy.
A: Personal anecdote. In the late 1970's i was taking topology from Munkres, Topology: a first course. The professor was Joel Spencer, a wonderful teacher, who is up for an AMS Trustee position, see the current Notices. In particular, he made up his own assignments that might not be questions in the book, which takes extra care and work. We had gone through compactness and the more intuitive sequential compactness and limit point compactness. We did most of the proof in class, that the product of just two compact spaces was also compact. the homework was to complete the proof for compactness, and throw in proofs that the product of two sequentially compact spaces was also sequentially compact, and the product of two limit point compact spaces was also limit point compact. Two of them were easy enough, but i struggled with the limit point one for at least a couple of days. Eventually I handed in a paper saying just that "I couldn't do this one." It came back from the grader with "Excellent" written on top, because the supposed fact is false. I was mystified, I asked Prof. Spencer what was so great about it. It took years for me to understand that not being able to prove something false was exactly right. 
I still have the book. I see on page 182, problem 5(e) that Munkres was well aware of this, referring to Counterexamples in Topology by Steen and Seebach. 
Putting it together, two hours on a mathematics problem does not seem very much to me. Oh, meanwhile, I am not in favor of cheating, or asking (anonymously) for others to support cheating.  
A: First of all, be relax and take things easier. If some problems are hard 
and you cannot solve them then I see no problem to ask for help as long as you 
want to understand and learn the tackling ways of the problems, not only to hand in a solution. The real fact is that some teachers do really poor at their classes and expect a lot from their students. I don't know how things in your
case are, but if you like mathematics you may learn a lot on your own. Moreover, if you study the problems and the solutions posted on this site you'll learn a lot!    
A: The aim of doing homework is practicing what you theoretically learned at school. When you do your homework, you gain experience in the subject. If you are having difficulties on doing your homework, that is the sign of your lack of comprehension in the subject.
If every homework question is taking your two hours, then that means that you seriously are lacking theoretical background on the subject. I suggest you repeat it before rushing into homework. Or first try solving easier questions, and gain experience in the progress.
Don't leave your homework on your difficult courses to the last night before the dead-line. If you do so, when you realize that it won't finish in time, you will most probably give up doing it, or try to make it solved on the internet. Start doing your homework a few days earlier than the dead-line. If you start earlier, you will have enough time to overcome any hindrances on your path.
If you think won't be able to solve a problem, don't ask the problem itself to the other people. Ask the part of the problem in which you are stuck. If you ask the whole question to someone else, the solution won't be your original work and you won't get much benefit from that homework.
If you had no choice and made your question solved to someone else, at least try to solve a similar question (e.g.; change the numbers in the original question) yourself.
Remember that homework is for your own benefit. If you want to succeed, don't cheat.
A: Firstly, you should always appropriately reference any information you find out in this way.
Secondly, I think this process can actually be helpful to your learning, provided you spend a reasonable amount of time thinking about the problem first, as you are likely to collaterally learn other things while looking for the information you want. I would also recommend talking to other people on your course (and/or the professor) about the problem before you search the net.
Thirdly, if you don't understand what you read online, then don't hand it in as a solution. It's usually better to give whoever is reading your homework assignments an accurate idea of what you do and don't understand.
As an aside, there are a number of classical theorems proved by mathematicians like Gauss that are not unreasonable to set as homework exercises. You will likely have been presented with a completely different theoretical framework to the one that existed historically, which can make these results much easier to prove than they would have been at the time.
A: I remember a class from graduate school. Microbial Physiology. There was one exam that we all had two weeks to work on and we were told it was fine to collaborate with others. The answers to the questions were not multiple choice or fill in the blank rather they were answers that required detailed explanations. The exam was also very difficult. I remember working with other grad students throughout those two weeks trying to help each other research the answers. We would gather periodically in someones lab or perhaps get together for lunch to go over what we had figured out so far. We divided up tasks so different people would look at different aspects of the problem. The end result was that we all learned a lot about the physiologic structures we were studying. I will always remember that exam as being very difficult but one of the most enjoyable and challenging to complete.
If the intent of the homework is to research and discover an answer, then by all means do so by seeking out advice from others, but if the intent was to give a student practice in learning a particular subject then it should be done by that student on their own.
A: The following excerpt from an answer JDH gave on a thread on meta might serve as a useful standard of comparison. It is much more permissive than the approach of David Speyer:

My opinion is that there is nothing wrong at all with posting homework
  questions here, particularly interesting ones, and I find much of the
  negative reaction to homework-question posters to be somewhat strange,
  alien to my way of learning mathematics in a give-and-take exchange of
  mathematical ideas. Surely posting questions here and studying the
  answers is not much different than studying hard in the library,
  talking mathematics with one's colleagues at math tea or talking to
  one's professor, which are all excellent ways to learn mathematics. In
  particular, I expect that students who post questions here might learn
  just as much if not more from the resulting answers as from their
  professors---we have a number of talented mathematicians, who are very
  good at explaining things---and that math.SE provides a valuable
  service to students having unapproachable professors, having
  professors who do not explain well, or who have few colleagues able to
  help them. Furthermore, the math.SE community strongly benefits from
  the questions and the insightful answers that might be posted.
(...)In particular, I hereby give all of my own students complete
  permission to post any and all their homework problems here, and
  indeed I encourage them to post their questions here and to study the
  answers well and thereby to learn some mathematics. I will be testing
  them on their understanding at the exam.
I would also encourage all mathematics professors to adopt a policy of
  encouraging collaboration on homework among their students, as talking
  about mathematics with one's colleagues is assuredly one of the best
  ways to learn mathematics. Indeed, I recommend that all professors
  should actively encourage their students to form study groups in order
  to work on their homework problems together. Learning as a group, they
  will go very far.

A: In my opinion, restricting study materials is counterproductive (particularly if no computer-searchable version of the course textbooks exist.) I realize that blindly copying answers is bad, but cheating on coursework has always been a problem and it is an issue that is independent of the Internet.
One common complaint is that students will learn less by Googling than they will by reading the textbook. This may or may not be true, but being able to search gives the learner access to much more targeted information. The difference between needing to skim through fifty pages you already understand in hopes of finding a paragraph you didn't, and being able to immediately enrich yourself on the topic desired, is phenomenal.
The thing is, the anti-Google teachers are right about one thing - you aren't going to remember how to use it practically if you don't actually use it. One answer here said that the degree of the problem became apparent when an in-class test revealed that the students, who up until then had been passing relentlessly difficult questions with ease, knew next to nothing.
This is actually a really useful thing to know, because armed with that knowledge the real problem becomes apparent - the students aren't using their research, which is why it isn't 'sticking.' A great option would be to hold a brief, three- or five-question test before each class - placing numerous, smaller checkpoints along the way will teach the students how to learn the material and retain it for use far better than either cramming or Googling together a paper.
I'm going to go one step further, though, and say that this also illustrates a deeper need for education to evolve. We don't live in the dark anymore - we live in an age of effulgence, where learning of any sort is a phrase away. To educate successfully, it will become necessary to embrace this by teaching more applied mathematics and asking more questions. To wit, if the course itself demands knowledge, the students will learn.
That said, I do not at all approve of students asking for (or receiving) verbatim solutions, either online or from classmates. This is cheating no matter where it takes place.
A: In the end, the real danger in googling the answer is that in the end your understanding won't be good enough.  You are in school to learn and you want to make darned sure, given tuition rates today, that you are getting your money's worth.  From this perspective, the goal isn't to get the right answer so much as to understand the problem. If you can't understand the work well enough to do it now, then what happens the next time?
So I don't know about the moral issues.  After all is it morally wrong to cheat yourself?  I think the pedagogical issues are very real and you want to be careful about what you are doing.  The thing your question says to me though is that you already are somewhat lost in the class in which case meeting with your professor is a very good idea and seeing what sort of help is recommended.  A second thing is I would recommend getting a study buddy you can bounce thoughts and ideas off.  That's a good way to develop understanding too.
Personally I don't have any problem with doing methodology research for homework (I am not a professor though).  But googling for how to solve a specific kind of problem is not the same thing as googling the answer.  With the first at least you can hold out hope that you will understand the process better at the end.  The latter, not so much.
A: Be your own book, create your own. You have a free mind to explore. Discover your potentials, if you resort to googling, you will not be complete and you will be missing the fun of failures as it breeds success..
A: Write up your homework solutions in tex and cite your sources.  Soon the professor will want you to help with research.
A: You go to a University to learn something. At first it's your decision. Noone forced you.
There are many ways to obtain a degree. What matters is if you gain what you require for your planned future ahead.
For example, if your plan is to become a maths professor and you chose a maths degree as a step of the journey towards your goal, then it is useless to go around and Google for the answer because you will have a hard-time when you actually become a professor which will be the time your ability to fix those will matter.
But if your goal is to become an analytical engineer then googling may not harm you as much. Because you can use the time you save to learn something else that matters. It doesn't have to be reading a textbook.
For example, the skill of finding the answer for a tough maths question from the Internet can become handy in your future carrier as it may improve your ability to deliver in short time spans and think out of the box.
After all what matters is not how you learn. What matters is how you use what you learnt and use it to make money.
Weather we like to accept or not, we do everything like learning to earn money. So ideally everything and practically at least most of the things we do must increase our value in the targetted future job market. If you keep that end in mind then you will not get lost. ever.
