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Apropos of nothing - a random HP thought.
Where does half the fandom get the impression that Harry is cheating by using a book with notes in it?
They are allowed to use their book while brewing and are supposed to have read up on the potion they are preparing in class.
And in written tests, I doubt they are allowed to use their books anyway, so the notes won't be any help to him there.
The only difference I see between Harry and a Ravenclaw (in this instance) is that the Ravenclaw would have made the notes himself - and probably not in the book, but on a spare bit of paper.

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IIRC, the notes in the book all deal with procedural aspects of potion-making: how to stir, the best way to skin a shrivel fig, etc. If Harry needs to write essays about the theoretical or alchemical aspects of potion-making, he's still going to have to find that information for himself.
In fact, I would argue that part of why Harry's doing so much better in Potions is that because he finds the Half-Blood Prince so interesting, he's actually reading his textbook outside of class for a change. Snape makes studying fun!
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First of all, it's not just that Harry finishes first, it's that his potions are better, by any objective criterion, than anyone else's in the class. It's not a question of two different sets of algebra instructions both leading to the right answer, it's more a question of (to use a parallel from literature lessons) Harry getting his answers from an advanced-level, well-annotated York Notes crib of JANE EYRE when everyone else in the class is still reading the teacher's intermediate-level notes.
I know what you're going to say - "well, couldn't the rest of the class find their own advanced-level crib in the library?" Since Hermione can't compete with Harry despite her hard work and diligent research, it's obvious that the answer to that one is "no".
OK then, you might ask, shouldn't Harry be allowed to use his unfair advantage? If he just leaves the book on the shelf, how does this help anyone? To which I would have to reply, sure he can use it, but the honorable thing to do would be to memorize the book's contents and practise the procedural aspects outside of class. As I said earlier on this thread, what matters is that the student puts in the effort in his/her own time to memorize facts and figure out how things work. Harry isn't doing that. Without that book to copy he's back to being Mr Average in Potions class, which is why he feels so awkward when Hermione accuses him of cheating. He knows he isn't being honorable.
IIRC, the notes in the book all deal with procedural aspects of potion-making: how to stir, the best way to skin a shrivel fig, etc. If Harry needs to write essays about the theoretical or alchemical aspects of potion-making, he's still going to have to find that information for himself.
I haven't the book with me so I take your word! I also agree with what you say, but from Slughorn's reaction it seems to be that practical knowledge which is the most important element of Potions lessons.
In fact, I would argue that part of why Harry's doing so much better in Potions is that because he finds the Half-Blood Prince so interesting, he's actually reading his textbook outside of class for a change. Snape makes studying fun!
If Snape makes studying so much fun, why don't the kids he's been teaching for five years already know the best way to chop ingredients, skin a shrivelfig, etc? ;) It seems to me entirely typical of Snape that he wants to keep his superior knowledge all to himself, hidden away in his own textbook, leaving his students to study from the most basic recipes so that he can shout at them when they screw up out of ignorance. (No, I don't believe he's a vile murderer :) but I do believe he's a petty piece of work...)
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If Snape makes studying so much fun, why don't the kids he's been teaching for five years already know the best way to chop ingredients, skin a shrivelfig, etc? ;) It seems to me entirely typical of Snape that he wants to keep his superior knowledge all to himself, hidden away in his own textbook, leaving his students to study from the most basic recipes so that he can shout at them when they screw up out of ignorance. (No, I don't believe he's a vile murderer :) but I do believe he's a petty piece of work...)
Oh heavens, yes. Snape is a dreadful teacher who shouldn't be allowed to talk to children, let alone try to impart knowledge to them. I was talking about the SnapeChild who wrote in the textbook making studying fun, mostly because I enjoy the irony of some old notes of Snape's teaching a student more than Snape himself probably ever has as a teacher. I also wonder how, with teachers like Snape and Hagrid and Trelawney and at least half of the DADA teachers, Hogwarts manages to turn out any students who are equipped for the real world.