>What you shouldn't do is try to self-study by reading a book.
My experience has been that this is exactly how college math works; you pay for self-study. The professor reads directly out of a book (in poor English), or off of pre-made slides provided with the book, for 2-4 hours per week, and then you are left to do the problems from the book on your own time.
Class populations are so large that if everyone had asked clarifying questions, we wouldn't have completed the readings.
If your college experience was different I'm envious of that.
Edit: as a fun side-note, our calc professor was well known for taking up the entire class to draw out a single proof on the chalkboard. As the chalkboard got full, he would incidentally erase his previous writing with his giant belly as he putzed across the room.
There are other benefits beyond being able to ask questions.
Tests and deadlines provide motivation to do the actual work.
Having a curriculum means that the content is laid out in a logical order that the professor believes should be achievable.
There is a stupid amount of information out there. Breaking it down into a progression that students can follow in order to learn and understand it is incredibly important.
If you're motivated enough then sure maybe you can just buy and read the textbook, although sometimes professors deviate from that when it's wrong.
> Class populations are so large that if everyone had asked clarifying questions, we wouldn't have completed the readings.
Good thing not everybody asks, and those that do ask are generally asking questions shared by a good chunk of the class.
> Tests and deadlines provide motivation to do the actual work.
This is so true in my experience. As a young man I could listen to the lecture and say "Yeah, yeah, I got this." and then try to work the homework problems and realize that no, I didn't really understand it. It is hard to force yourself to do the work if you don't have a negative externality for not doing it.
> Having a curriculum means that the content is laid out in a logical order that the professor believes should be achievable.
> There is a stupid amount of information out there. Breaking it down into a progression that students can follow in order to learn and understand it is incredibly important.
You can get all this from a textbook. The curricula of most undergraduate math classes are based textbooks anyway. Many math textbooks even have a roadmap in the preface that tells you which chapters can be skipped for shorter versions of a course, which ones discuss advanced topics, etc.
> Tests and deadlines provide motivation to do the actual work.
The person who asked the original question already seems to have the motivation to learn math for a specific purpose. They're not a random undergrad who has to complete a math course just because it's a requirement for graduation and never expects to use the knowledge.
Also, the downside of studying based on arbitrary deadlines is that if it takes you even slightly longer than average to understand the material, you fall progressively further behind the class. If you learn at your own pace, you can study a topic until you really understand it.
I'm from Germany and so the cost of being enrolled amounts to ~600 euros a year (and I get free public transport), so in cases like mine you'd be absolutely right - I'm mostly working freelance jobs and doing the odd course here when I feel like it, and it's pretty great. I get access to the curriculum, social contacts, a motivation kick and a general sense of immersion.
Though I don't think I'd be still doing that if I have to pay a substantial amount of money, so maybe other people just aren't as lucky? I don't know the exact situation in the US, apart from the ridiculous twenty-grand-a-semester colleges I know of, are there cheaper options?
What you're saying is agreeable: taking a class provides a certain level of motivation that self-study doesn't provide, for most people. That doesn't conflict with the idea that typical college courses have little tangible value added. If you're ok paying a couple of grand for extra motivation, then I hope you truly have a good reason to be doing so, and that it pays off. This type of system doesn't really support people who just want to learn for the sake of it though.
The best class I've ever been to in 5 years of university was a pure maths class (topology and linear analysis) that I took as an elective (I think this is the word). If I'm ever in charge of teaching a class, of mathematics at least, this is how I will do it.
The format was like this: the classes were 2h long, and the professor would begin with an exposition: first a short recap of the previous class, then introducing the new material for the day, new concepts, definitions, the starting point for the day's class. This could take anywhere from 5 minutes to up to 30 or 40.
After that, we were handed a work-sheet. It contained the definitions/summary of the concepts that were just introduced, then a series of exercises. Now this is the core of it: the main part of the class was working through these in order. The exercises were structured so that the rest of the material was learned by doing, by working out through the exercises. They would e.g. ask to prove interesting consequences, or important theorems that followed from the definitions at the start. The professor would point out an exercise, read it aloud, comment on the "meaning" of the problem or what it's meant to demonstrate or similar remarks, then give us some time to figure it out. After a bit, he would ask someone that completed it to present his/her reasoning. Now note that this part required real effort from the part of the professor. I tremendously admire him for this because it required him to listen carefully and think through the proof presented by the student, something which is more difficult than, say, presenting and explaining his own proof on the blackboard. Anyway, he would listen to it, comment ("you could have simplified here", "you didn't consider this case there", etc.), perhaps ask some other students for alternative approaches, or maybe give an alternative approach himself if necessary. By doing this we would learn the rest of the material by working it out from those principles; in a way in a sort of "narrative" that had a thematic "chapter" in each class and an overarching "story" for the whole class.
In summary, only about 10 minutes at the start would be real exposition. This usually amounted to stating definitions or axioms to work with for the rest of the class. The rest of the material -- any theorems, conclusions, etc. -- were worked out.
I have some difficulty concentrating even on 1h or 1h30 lectures. These were 2h and I would be effortlessly engaged for the whole duration. It kind of pains me that maybe the best class I've taken in university wasn't even one in my degree :^)
My experience has been that this is exactly how college math works; you pay for self-study. The professor reads directly out of a book (in poor English), or off of pre-made slides provided with the book, for 2-4 hours per week, and then you are left to do the problems from the book on your own time.
Class populations are so large that if everyone had asked clarifying questions, we wouldn't have completed the readings.
If your college experience was different I'm envious of that.
Edit: as a fun side-note, our calc professor was well known for taking up the entire class to draw out a single proof on the chalkboard. As the chalkboard got full, he would incidentally erase his previous writing with his giant belly as he putzed across the room.