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This question is about mathematics only. My forethought is that courses are usually based on textbooks, but they have less content than the textbooks. There are less problem sets, and chapters are skipped. Is it true that you can learn more by self-study with textbooks rather than college courses? Given that one could find help in the internet.

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    $\begingroup$ It is indeed true, self-study frees you in many ways. The trouble is, Math is also a social activity, somehow, you have to engage other members of the community. How to accomplish this outside of formal academics... maybe prove something new? $\endgroup$ – James S. Cook Oct 17 '17 at 4:35
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    $\begingroup$ I'd say the answer would depend on you at least as much as on the books/courses. An often-overlooked part of physical classes is that they are harder to procrastinate. $\endgroup$ – Jessica B Oct 17 '17 at 5:47
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    $\begingroup$ If you misunderstand something, who will point you in the right direction if you don't realize you need the help. If you get help on the internet that isn't correct, how will you know? On the other hand, I did independent study in high school and it was a great experience. I remember that I mispronounced math words for months. I took tests at the end to prove my competence. How will you prove that competence if you want to get a job with your knowledge? $\endgroup$ – Amy B Oct 17 '17 at 8:54
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    $\begingroup$ If you are studying on your own, you are learning the content, you are deciding on which exercises to solve and you decide if you did a good job in solving them. Thus, you have no feedback at all (in the way you might get in a course, where homework is graded) and there is a risk of you learning wrong things or, even worse, thinking you understood all while in fact you didn't understand the deeper meaning at all. Personally, I would advice against studying on your own with a textbook, unless you are at least PhD level; then you might have enough experience to avoid the traps. $\endgroup$ – Dirk Oct 17 '17 at 8:55
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    $\begingroup$ I’ve gotten the most from alternating self-study with formal classes. Working in books is great, because of completeness, but no book can do what a good professor can do for you, or a good study group of motivated peers. $\endgroup$ – G Tony Jacobs Oct 17 '17 at 17:30
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Without a constraint of time...

If you mean "because self-study can be done over a longer period than a typical college quarter or semester", then this is as relevant as the other points you make. You don't elaborate on it in the body of your question, but a time-constraint can work both ways -- not having one can be great, because you can work at your own pace and not miss things along the way. [I actually consider the typical term length to be a weapon of math destruction, particularly because students who can't assimilate concepts at the prescribed rate often find themselves in the fail-retake spiral, never completing their degree.] However, working with a time-constraint can be good, particularly if it builds motivation to keep at it. I know I could learn to paint on my own, but having an instructor give me deadlines really keeps me working.

If you mean "let's not consider the constraints of time in this discussion", assuming person A takes the course face-to-face and person B self-studies their way through the material, then I would bet that person A learns more, as I'll explain below.

...courses are usually based on textbooks, but they have less content than the textbooks. There are less problem sets, and chapters are skipped.

Mainstream commercial textbooks try to appeal to a wide audience, so they often include more information than any course requires or has time for. So in this sense, you're right -- a student who self-studies could certainly take on more information by just finishing the book. But wouldn't this also apply to students in a face-to-face course? Every few terms, I have a student who just decides to work through that last chapter because she's interested in the subject, but this outside of the norm by far. My experience, and research I've read, says that students don't read textbooks anyway, except when mimicking a worked example for a homework problem. This may be one downfall of face-to-face classes where instructors explain every last thing in detail -- students are conditioned to not read the text because nothing bad ever happens when they don't, just as people in general are conditioned to not read End User Licensing Agreements.

Given that one could find help in the internet.

If you're an independent learner who knows how to search for content, and you plan to work through a book at a matching level of difficulty, and you set aside time to actually do it, then you could definitely cobble together your own course and learn more than is required of you in a typical college course. You will find enough help on the internet to address nearly every potential issue that could come up with misunderstanding the material. However, a procedure for diagnosing and treating this misunderstanding will be costly time-wise, either for the learner or for the person they found to help them online. One thing a teacher can do better than a machine is to ask leading questions to determine what a student really knows, and then build that toward learning a new thing. If you're on your own and you really misunderstand something, it may be quite difficult to figure out the root of the problem by searching online. Getting on a discussion forum (such as the Mathematics stack exchange) could be the fastest way to get this feedback, but the answers there can vary so widely in their scope that figuring out which one to focus on will be a new challenge, every time you ask a question.

In short, I would say "yes" to your question, in theory. However, I don't see students reading the book on their own, even when they're in my classes. I think having a deadline in an actual course provides motivation for students to actually keep working on the material, and having a human teacher with whom to interact will provide trained, tailored feedback to address misunderstandings and prompting. Finally, a community of other students is a great environment for learning. Having/getting to explain what you're thinking to someone next to you helps solidify your own ideas. This may never happen with self-study.

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  • $\begingroup$ Can sticking with textbook make me misunderstand something? The textbook is there to guide you, so the only possibility is to not understand it. Is this correct? $\endgroup$ – Ooker Oct 19 '17 at 20:33
  • $\begingroup$ @Ooker Can it make you misunderstand something? The obvious answer is "no", but maybe I'm missing something about your question. Could you restate it for me? $\endgroup$ – Nick C Oct 20 '17 at 0:05
  • $\begingroup$ Well, in my opinion getting educated, even by teachers or textbooks only, is to avoid misunderstandings. Surely that in an effort to assimilate the concept, one might misunderstand it a little bit, but that's a part of learning, not be costly time-wise as you said. Teachers can give personalized questions, but a good lecture or textbook should anticipate those questions beforehand. $\endgroup$ – Ooker Oct 20 '17 at 8:13
  • $\begingroup$ hi, what do you think about my question? $\endgroup$ – Ooker Apr 16 '18 at 13:01
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It is a mistake to think that one learning method is best for all students. There may be some students who learn more by self-study. There may be other students who learn more when taking a structured course with the help of an instructor.

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Some students, if studying on their own, will soon quit. But a regular lecture, with homework and deadlines and the threat of a failing grade may be enough to keep them going.

In a similar way, some people hire fitness coaches to force them to do their exercises, since when they are left on their own they don't do it. Or don't do enough of it.

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  • $\begingroup$ I think I agree with you in general, but isn't the book's structure is the structured course, and the text of the author is the instructor (assuming the writing style is clear)? $\endgroup$ – Ooker Oct 17 '17 at 11:29
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    $\begingroup$ @Ooker I think you are assuming a lecture where the lecturer only reads from his book. Such lectures exist and one might argue that in this case, you can just as well skip them - but there are also other lectures, where the presented content goes beyond the one written down, where you get exercises, help and the chance to ask and discuss questions. $\endgroup$ – Dirk Oct 18 '17 at 9:00
  • $\begingroup$ @DirkLiebhold not quite. What I mean is that to learn a concept you really need alone time to assimilate it. Exchanging knowledge can be done with a teacher, a colleague, or a stranger on the internet, but without taking the silent time to assimilate it, you won't know it better $\endgroup$ – Ooker Oct 19 '17 at 10:18
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It can be better in terms of expense. One can pay up to the same price for books but spare potentially several tens of thousands in student debt.

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