Don’t forget that he also cheated his way through his entire last year in Potions and is willing, despite knowing better, to experiment with unknown spells. Harry’s not really all that sympathetic anymore, even though he has the power of love or whatever. I’m kind of glad Voldemort killed his parents. That’ll learn the little brat. 
OK fuffle, after reading your objections, let me amend part of my version of events.
Aberforth notices that Snape is no longer in the common room, and Snape happened to be sitting near the back staircase leading up to the private rooms. Aberforth, suspicious of any potential Death Eaters, wonders where he went and sneaks upstairs, catching him at the door etc. When Aberforth pushes Snape into the room, the conversation goes like this:
Aberforth: This punk was listening at the door!
Snape: No way!
Albus: Explain yourself, Severus!
Snape: I heard you were at the Hog’s Head looking for teachers so I dropped by the pub to see if you had a spot for me and was waiting downstairs to have a chat with you. I only came up here because I had to obey nature’s call and I thought this was the door to the john. I would never eavesdrop!
Aberforth and Trelawney (rolling eyes): Yeah right.
Albus (fixes Snape with a piercing glance): What did you hear?
Snape (putting on his best occlumency poker face): Well, I heard some voices, but I don’t know what you guys were talking about. Were you discussing salary? Because I would need a raise from my Potions Job at Pendragon’s Cauldrons & More.
Albus: OK then, wait downstairs, I’ll see you in a few.
Aberforth and Trelawney: But… but… can’t you tell from his unwashed hair that he’s a suspicious character?
Albus: Innocent until proven guilty kthxbye.
I have a friend who is a teacher, and after seeing the first Harry Potter movie, that was his reaction: “Harry and co. break all the rules in the school and get rewarded for that? I wish I were headmaster at Hogwarts, things would be run very differently, yes sir!”
(adding something for fuffle, that should have been in my previous post) If you think it’s odd that Dumbledore hears the scuffle but doesn’t get up, then take out that part. Dumbledore doesn’t realize what’s happening on the other side of the door until Snape and Aberforth burst into the room.
Yes, the timing is tricky, but isn’t that the way in the Potterverse? You always overhear just enough so that you can be wondering and guessing as to what the people really meant, and go off half-cocked in the wrong direction most of the time.
How did Harry cheat? He just used what was in his book! There is no rule in the Potions class that says that you can’t use a little initiative and experiment, is there? You could just as well say that Hermione is cheating because she’s only copying exactly what some other dude (Libatius Borage) had written down.
No, of course not. But that would require him to actually be innovating. Copying someone else’s answers is…well, cheating. Hermione isn’t cheating because she’s using the standard teaching text; true innovation takes off from there.
Or did I just get whooshed?
You’re certainly right about that; this exact sort of thing has been a major plot device all the way through. I do like your second version of the story a lot better; it does make a lot more sense. Unfortunately, it puts us right back in the “Dumbledore trusts Snape because he loves Lily or had a vow or something maybe” boat. I’d much rather think I’m onto something big! 
I totally disagree. What if Hermione found a book in the library that had different / better instructions than Borage and started using those in addition to the school textbook? Before the first flying lesson in first year, she checked out “Quidditch Through The Ages” and started getting flying tips from there before class. Was that cheating?
What Harry Potter is doing is using the Second Edition of the book, as revised and commentated upon by Severus Snape. The only difference is that the book has never been officially published and/or isn’t from the library. Saying “there is a better way of doing your graduate work, but you’re not allowed to use this better way because it’s not in The Book I assigned you to read” would be the mark of a bad teacher. The object of the class is not to see how well you follow instructions, the object of the class is to brew good potions.
[edited to add]What I don’t think Harry is cheating, I do think that he is wrong in not telling the teacher “Hey, this book has better instructions than Borage, why don’t you tell the students to use these?”
Those texts are available to everyone, plus she would probably cite her sources if she did so (she’s always talking about the info she found in Hogwarts, a History, for example). If she found a book with better instructions, kept it a secret, and then started claiming credit for the work, like Harry does, then I’d say she’s committing potions plagiarism.
Counterexample: I’m taking a math class, and I find that my book already has all the answers penciled in by another student. I keep it a secret and start getting perfect marks. Am I not cheating? Would you really say that purpose of math is not to learn math, but to produce correct answers? I would flunk a student of mine in a situation like this.
On preview: I see that you added the bit about how Harry should have mentioned the better instructions. I think it’s the claiming credit for the extra notes that shades his actions from unethical into cheating.
I also just realized that maybe I’m starting to hijack a bit. I’m willing to let Harry’s (alleged) misdeeds slide if it’s too much.
Dumbledore loves Harry, feels sorry for him, and feels guilty about putting him in his uncle’s household, which, while it saved his life, also gave him a miserable, abusive childhood. He also feels an obligation to protect Harry from Voldemort, is impressed by his intelligence and skill, and knows that Harry’s destiny is to fight Voldemort. He’s also impressed that Harry has managed to overcome all the things Voldemort has sent after him. He also knows Harry is in pain, and forgives him a lot for that.
Plus, he’s a patient person, and he’s lived long enough, and has been both a teacher and a headmaster long enough that he understands teenagers and that they do sometimes get temperamental or sullen.
And Snape isn’t “mean” or “evil” because he doesn’t put up with Harry Potter. He’s mean because he’s a rotten teacher. He doesn’t like his students, he plays favorites, he’s arbitrary, and he’s constantly mocking or insulting them.
Professor McGonagle doesn’t put up with Harry (or any other student) acting disrespectfully or inappropriately, but people don’t condemn her. That’s because, even though she’s strict, she’s fair about it. She likes teaching, likes her students, and wants them to succeed.
I think the correct counterexample is: My textbook says “use method A to find the result.” But someone wrote in the margin: “use method B”. If I use method B to find the result, and it’s a simpler, more elegant approach, then it’s correct to use method B.
What is relevant is this: Is Harry learning anything by following the better directions? Or is he just copying without understanding? And let’s suppose that he is copying without understanding. If the object is to make an effective potion, it would be nice if you could understand the deep principles and improve upon them, but if you don’t have that skill, you would still deserve a high grade if you knew and could accurately and quickly whip up a specified potion.
I agree that Harry is taking advantage of the Half-Blood Prince’s old book to make himself look more adept at potions than he really is. JK Rowling half-heartedly tries to justify it by having Harry explain that it’s important for him to get on Slughorn’s good side to retrieve the memory Dumbledore wants. Harry is being devious but if you persist in calling it “cheating” I will be forced to use the “Oppugno” spell on you.
Is Harry “cheating” by using Snape’s old notes? To me the larger question is why if Snape was that much better than the book at potions when he was a student, was he not a better teacher. In school I always had teachers who knew things that weren’t in the books, but the difference was that they TOLD these extra facts to us.
In Snape’s case, the fact that his students are never presented with this knowledge, that he seems to hoard it for himself, indicates that he is indeed a dreadful teacher, or at the very least lothes his students.
Yes he is learning. He saved Ron’s life with the bezoar. While this was mentioned in the regular lecture, it’s pretty clear that Harry got the knowledge from the HBP book.
It’s not really textbook 2.0, even an unpublished version. It’s just notes, work done by someone else and not intended for the eyes of others. You can’t really say he’s just using method B, he’s using the work that someone else did and taking credit. If he found Snape’s old History of Magic essays in a book and turned those in for credit, he would still be plagiarizing, which is what he did in Potions, and therefore he’s a cheater cheater pumpkin eater. Or if Malfoy hired a potions master (not Snape) to make notes in his text, and then just followed those instructions to become the best student, we would say that’s cheating. I understand that Harry didn’t set out to cheat, but he did give in to temptation.
If the only objective in class is to make a potion that’s up to a certain standard, then it’s only fair for everyone to have the same instructions to work from. It’s not just that he deserves a high grade for ably following instructions; he’s getting a higher grade than even Hermione, the best instructions-follower in town. How is it fair to her or the other students if someone has better instructions and gets the highest grade because of it?
But if the objective of the class is to get people to experiment, then he’s done far more than plagiarize. He’s actually failing class (morally speaking), since he’s not been sufficiently encouraged by his (fake) successes by making any effort to do his own experiments, just passing off someone else’s innovations. In this I agree with you that he’s not actually learning anything from class, so maybe we could agree that he’s at least cheating himself.
This all seems a bit beside the point anyway. Harry has a destiny and isn’t going to defeat the Dark Lord by making potions at him. He has more important lessons to be learning: how to take risks, how to rely on your friends, how to learn from people you hate (the real, authorial point behind the whole Potions text subplot), and how to bear up under unbelievable pressure. I think this is one of the main reasons why, to touch on Captain Amazing’s point, that Dumbledore takes such a free hand with him. He knows Harry doesn’t have to be a great student to do what he’s contractually obligated to do.
And am the only one who is just dying to hear Alan Rickman say, “It was I, the HALF-BLOOD PRINCE?”
A good point. I remember Umbridge said in book 5 that his classes are generally pretty advanced, but he might just be giving them more difficult potions to study, and I agree that there’s a lot that he does that’s just downright indefensible. To put the most charitable possible light on it, maybe he’s trying to teach his students how to work under serious pressure? Given his history, he might consider that the most useful lesson of all…or he might just be an out-and-out bastard.
What I want to know is why Snape was carrying around his mother’s old sixth-year Potions book before he had taken his fifth-year OWLs.
Do we know that he was writing in it in fifth year? We know that he invented Levicorpus then, but it doesn’t necessarily follow that he wrote it in the text that year.
On the other hand, Snape may have been studying ahead.
I agree with you that this is really beside the point, and that the reason for him to be using the Half-Blood Prince’s book is for him to realize that good information can come from unexpected sources. However, you still haven’t convinced me with your reasoning. If the assignment was “improve Libatius Borage’s instructions on producing Euphoria”, and he just copied out of the book, that would be cheating. But if the assignment is “make me a cauldron of Euphoria” then it’s not cheating to use notes even if written by someone who wasn’t the author of the class-approved textbook. What if Malfoy hired Snape to give him extra potions tutoring on the week-ends? Would that be cheating? Would it be fair to the other students?
He’s getting better grades than Hermione, but that’s because Hermione is being stubborn. Harry has given her the opportunity to follow the Half-Blood Prince’s instructions also, but she turns him down. He’s also offered the book to Ron, but Ron has problems reading the hand-writing. So Harry is at least offering his friends the same opportunities that he has.
I think part of it is the Slytherin mentality - you don’t want to share your knowledge with other people, you want to use it for your own advancement. It’s a darn shame that Snape didn’t publish his own potions book. That would have helped him get tenure, and with tenure, he would have been able to overcome the Voldemort curse on the DADA position.
I agree that Snape’s teaching methods are dreadful. But I do think he is trying to get his students to excel. The problem is that, instead of encouraging students to do better, he uses sarcasm and a belittling manner to convince them that they need to work harder. This, along with his obvious favoritism and his dislike of smart-aleck and know-it-all students (such as Harry and Hermione) wil of course make him odious from Harry’s point of view (through which we are seeing the whole story.) When some enterprising revisionist, inspired by the success of Wicked, takes on the task of re-writing the whole epic saga from Snape’s perspective, we will realize that Severus was grossly underappreciated and misunderstood.
He is good at creating his own spells…you’d think Tenurious would have been an obvious choice. 
I wouldn’t doubt that, especially considering comments made by other characters concerning his interests. It makes me wonder if he was keeping notes elsewhere before he started writing in the book.
No, you’re not. Of course, I wouldn’t put it past the filmmakers to cut that out too…they’ve left out quite a few good Snape scenes, IMO.