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In the first years it seems like all the students do is follow instructions. I have a scene in my mind where Snape gave the students instructions on the blackboard which makes them hard to read later on in the lesson.

HP6 shows that the potions are at best inaccurate if not intentionally misleading. The linked questions seems to come to the conclusion that some recipes leave room for interpretation and tweaking, but I think the students were previously taught to

... follow [the teachers] instructions to the letter.

It doesn't look like the students are learning much from just following instructions in an error-prone environment and being punished for making mistakes. Nobody can remember the recipes they learned and inventing own potions doesn't sound safe.

In HP2, the trio manages to brew Polyjuicepotion, which is considered highly advanced without having an extensive potions education, so apparently the class is not required to perform the craft.

So what is the point of potions class?

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    The trio doesn't make Polyjuice: Hermione, a genius, does. And as for the use of Potion-making "labs", it's probably about as much use as high-school chemistry labs for the absolute majority of people in life. Hell, since they clearly do learn general uses and properties of ingredients, and potions are things they'd use as adults unlike the average muggle and ability to titrate or identify salts, it's probably way more useful! – Shisa Oct 6 '14 at 12:39
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    Early in chemistry classes, we were to follow instructions specifically to learn about HOW to mix chemicals and how they reacted rather than why. The why came from lectures. I don't see any difference here. – phantom42 Oct 6 '14 at 12:40
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    The same goes for learning anything in school (or anywhere, really). You learn how to do it the prescribed way first, and then learn how, why, and when to deviate from the established norm. – phantom42 Oct 6 '14 at 12:43
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    "You are here to learn the subtle science and exact art of potionmaking," he began. [...] I don't expect you will really understand the beauty of the softly simmering cauldron with its shimmering fumes, the delicate power of liquids that creep through human veins, bewitching the mind, ensnaring the senses ... I can teach you how to bottle fame, brew glory, even stopper death -- if you aren't as big a bunch of dunderheads as I usually have to teach." – Wayne In Yak Oct 6 '14 at 14:38
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Potions class isn't just about learning how to brew specific potions, although that is certainly part of it.

Working through the potions, even if the instructions aren't fantastic, gives the students experience using the tools (cauldron, measuring devices, etc.), familiarizes them with the storage, handling, and use of many potion ingredients, and also provides information on what the various potions do, and how they should be used.

We've seen examples of the Potions professors discussing more than just following instructions to create potions. Snape asks Harry on his first day what a bezoar is for. Slughorn displays several potions and asks the students to identify them.

As was mentioned in comments, saying that taking Potions is not required in order to make advanced potions misses the fact that Hermione is not only an exceptional student, but simply brilliant. It is extremely doubtful that anyone else in the class could have managed the potion, and even with all of her talent, she still managed to make a crucial mistake (misidentifying the hair she used as a sample for her potion).

Also, keep in mind that much of the punishment for making mistakes shown is merely Snape taking out his dislike of Gryffindor in general, and Harry in particular.

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    A good example of all of this is when Slughorn asks the class to make an antidote in book 6. They had to use the principles they had been learning to figure out what was in the poison and then what ingredients would counter-act it. – mikeazo Oct 6 '14 at 12:44
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    Also you can see that even the most inept student by book 5-6ish are able to easily create (maybe except for neville >.<) all of the basic potions that they had been "struggling" with previous years since they obviously have kept progressing to harder and harder potions through the years. so at the end while you may not be able to do advanced potions their are sitll 100s of basic potions that you will be able to easily created (without killing yourself since that seems to be the big danger lol) – Himarm Oct 6 '14 at 12:59
  • @Beofett - I didn't realize hair identification would be part of a Potions class :) – DVK-on-Ahch-To Oct 6 '14 at 17:17
  • @DVK It is on page 126, scribbled on the left-side margin in yellow crayon ;) On a more serious note, I do believe a more experienced potions-maker would have taken extra care to make sure that the hair she took was actually the hair she wanted, instead of just hoping... especially since Millicent's hair was considerably longer than that of most cats, and not black (at least in the movie): "I've already got mine raises vial containing three black hairs, Millicent Bulstrode, Slytherin." – Beofett Oct 6 '14 at 17:22
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Think of potions class as your regular Chemistry class. It's not there to make you a master potion maker (no one comes out of high school as a chemist), but to give you some understanding of what potions are, how they are made and what effects they might have.

For instance, a wizard who goes through his potions lessons might, later, recognize the effects of a sleeping drought. Or recognize a love potion before taking it [1]. As in our chemistry classes, they will learn that two ingredients interact with each other in such a way to, for instance, stop the effects of the first one. This could be very useful if they later needed to quickly stop the effects of a simple poison. Just how we learn that acids will react with bases creating a salt and water, for instance.

And besides that, there are the reasons that have already been mentioned in other answers.

The potion classes (again, much like our chemistry classes) will teach the students how to use some equipment that wouldn't be used in other settings, such as cauldrons. It is also supposed to teach the students the main, simple potions which they will probably need at some point in time.

And about the books being misleading, and potions being a "Follow the Recipe" class, well potions will also take a lot of cooking if you think about it. But with maybe disastrous results when something goes wrong. Which is why it's important to follow the recipes: to make sure nothing goes wrong. However, the better you get at it, and the more intimate you are with the subject, the more room you'd give yourself to experiment. Which is really what make the real masters.

So, in someway, that would also be the aim of the potions class: to make the students intimate enough with potions so that they would be able to later experiment and not HAVE to follow the recipes all the time.

[1]: We see that during the Half Blood Prince, when Slughorn asks the students to recognize different types of potions. That would be useful if they were in a situation, for example, like the one on the first book, where they had to choose from a few potions the one that was safe to drink. No potions class = being clueless, and depending solely on what information is given. With the potions class they'd be able to make some assumptions and rule out some options.

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Potions class corresponds very strongly to chemistry in any normal high school. The students learn the properties of various magical substances, how they interact, and how to get them to react in the desired way. They then do this as a practical matter in the scenes where things blow up. What is boils down to is that Snape is not a very good teacher. He is overly critical, refuses to offer decent instruction, and does not properly observe his students in a lab environment. He may be good at the subject, but he's not good at teaching it, from what we see in the books.

I suppose that the ultimate goal for a student in potions would be to be able to derive his or her own potions, or at least to understand how the potions that he or she used in daily life worked.

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Potion making is probably very comparable to cooking. People will teach how to cook, though memorization of whole recipes is not required, by having them follow provided instructions in class. But in cooking classes, you also learn a lot of techniques, and there's many facts that you should have memorized about cooking in order to be good at it. Such as unit conversion, what is the temperature you must cook various types of raw meat in order for it to be safe, appropriate ingredient substitutions, that sort of thing. It seems to me that there would be analogies in potion making for all of these things. You'll notice also the kids do get tested on memorized knowledge like this regarding potion making in the books. Like in the very first book, Snape asking the students, "What's the difference between monkshood and wolfsbane," and "Where can you find a bezoar?"

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