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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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Especially since from what we see of the HBP's notes, they are more along the lines of technical impovements of the brewing process. Which apparently isn't something the average student can ever figure out, but only learn from the book or look up elsewhere.
From what we see, the only lesson where an actual understanding of the principles was required was the antidote lesson. And the book couldn't help Harry there - but then, no one but Hermione even understood what they were supposed to do.
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I thought it was all about "trial and error", myself. Potions seems to be a lesson where one learns through making mistakes and learning what not to do. Judging from the fact that Hermione hasn't been able to discover these technical improvements in any other book (had she done so, she'd have been able to keep up with Harry), it would appear that the HBP made all these notes as the result of his own efforts. He did the work, he experimented with strategic changes (probably as self-imposed homework).
Harry sailed to the top of the class on another person's efforts and, to do so, he had to pretend that he was someone who cared enough about Potions to do that extra work in his own time. Whichever way you slice it, it ain't right.
That said, I would be prepared to argue that Harry would not have been cheating had he spent his time memorizing the HBP's work and doing experiments outside of class to confirm the rightness of the notations. How is this not cheating, you ask? Simple: because Harry is spending his own time verifying the information and doing the actual physical experiments. He'd be spending his own time in experimentation and learning. That way, he'd have an arguably unfair advantage, but at least he'd be using those notes to ascertain for himself the basic principles of Potion-making. He'd be WORKING.
As it is, Harry just coasts and learns nothing whilst at the same time soaring to the top of the class by following, monkey-like, another person's detailed hard work. Don't ask me to believe that's not cheating :)
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I won't ;) But I'm still not convinced myself.
I guess what it really comes down to is [as someone else said somewhere down the page], that from what we get to see of potions lessons in canon, it's a very weird subject and everyone has different ideas what it involves.
And depending on these ideas(and of course about a 1000 additional personal factors) he's either cheating or not.
For example I don't see potions as a trial and error lesson. They are supposed to get it right the first time by following their recipe. So a better recipe might make it easier, but shouldn't really make much of a difference.
I still see Harry not telling Slughorn that he had no idea what he was doing as wrong - but as you said before, it's an understandable reaction. And of course Slughorn *wants* it to be true. Otherwise he'd definitely have called Harry on the Bezoar trick.
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I do understand that it was, at the least, dishonest and a little more than slightly low for Harry to fail to correct Slughorn's perception of him as a Potions whizz (although I think it's also been said that it's perfectly understandable).
That said, I would be prepared to argue that Harry would not have been cheating had he spent his time memorizing the HBP's work and doing experiments outside of class to confirm the rightness of the notations. How is this not cheating, you ask? Simple: because Harry is spending his own time verifying the information and doing the actual physical experiments. He'd be spending his own time in experimentation and learning. That way, he'd have an arguably unfair advantage, but at least he'd be using those notes to ascertain for himself the basic principles of Potion-making. He'd be WORKING.
And this is where I have to sit back and think because, correct me if I'm wrong, but didn't Harry indeed take the steps to verify that the instructions worked? When the notations said to "crush [the roots] with flat side of silver dagger, releases juice better than cutting." (p. 189, Scholastic edition), Harry tried it. And it worked. And, if I recall properly, he did test every notation in the book, though more than most of the experimentation was done in class. Not trying to be snarky, but honestly curious - how then does that not meet the qualification of "working" or "not using his own time in experimentation and learning"?
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And this is where I have to sit back and think because, correct me if I'm wrong, but didn't Harry indeed take the steps to verify that the instructions worked? When the notations said to "crush [the roots] with flat side of silver dagger, releases juice better than cutting." (p. 189, Scholastic edition), Harry tried it. And it worked. And, if I recall properly, he did test every notation in the book, though more than most of the experimentation was done in class.
How I wish I had the book with me! If you're right that Harry does work outside of class as well as in class, then you've basically invalidated my argument - I honestly thought he only tested those notes whilst in class. If, however, Harry does only test the HBP's conclusions during class, then he's just coasting along using someone else's work and not doing the extra work needed to justify using his value-added recipe book.
I'm afraid I regard the annotated book as the equivalent of Harry having a personal tutor next to him in a test situation telling him what to do. Had he spent some time in learning the improved instructions by heart, then I would not have a problem with him using the book - and if the book states that Harry did spend his own time testing out the Potions recipes outside of class, then I concede defeat ;)