Is the skill to learn new math by reading textbook alone (no lectures) required when one becomes a PhD student? I've taken a few math classes in college. I'm wondering as one graduates college and applies to become a PhD student in math, is he/she going to be required to learn math related materials alone without lectures, only by reading books (ie. by following the examples in the book and doing exercises with solutions provided)?
 A: Very much so! Indeed! Although one may be misled by the required courses with homework and exams. Through various mechanisms, a tradition has arisen in which students are "required" (coerced, by fear of rejection) to do things which they should have desired to do. As is usual in human situations, "requiring" a thing generates a certain hostility, which, in this situation is counter-productive, and so on... Despite the (widespread, systematic) inaccuracies in the pedagogical design of graduate programs, people do suffer through these. :)  On the face of it, being able to independently assimilate things from sources is irrelevant... ?!?!? ... since the game appears to be compliance and obedience, going to class, doing homework, reviewing for exams. (For the first year or two of grad school.)
(Yes, obviously intellectually nonsensical, but this certification of nonsense does not say what one should do...)
... however... and somewhat unfortunately... the seemingly required willingness to play along with things that make only fractional sense becomes immediately irrelevant as soon as one has passed the standard hurdles. I note that it is very true that there appears to be a general proclivity to learn too little rather than too much (I would be interested to hear comments on this...), so a responsible mentor might worry that novices misinterpret advice about attending to research rather than "classes".
In that context, in my own life experience, and in observation of others, ... one must (I hesitate to do "MUST", but here it might be warranted) be able to cope with sources. Books. PDFs. Old papers. Essays by people who've thought about things. 
If you can register for a class where the person in the front of the room seems to understand these issues, to address them, and be able to render sense of them, you are fortunate.
A: Getting a Ph.D. in math is (in part) getting an education in how to do research in math. Doing research in math doesn't mean that you necessarily have to do it sitting alone with books. Many researchers work together in groups and support each other.
However, doing research means that you have to move beyond what lectures cover. You will have to "come up with" new math. From what I understand, in many places when you start a Ph.D. program, then you start by taking classes. You will then at some point find an advisor and then gradually move from taking classes to independent studies with your advisor and/or other professors. Doing work with a professor will in the beginning (for most people) just be the professor that teaches you material. But more and more you will have to work independently. A good advisor (IMO) will know how much he/she needs to help. And he/she will be good at asking you questions and giving your problems that are enough beyond what you know and still isn't impossible for you to answer.
So at the end, you will hopefully be at a point where you can work more or less on your own.
Does that mean that you just sit and learn form books all day? No, it doesn't mean just that. You will also be studying research articles and you will try to move beyond what has been written down. 
So, yes, you definitely will have to be able to teach yourself material without much help from anyone. It, however, doesn't mean that research is just a one man show. Researchers collaborate and help teach each other material. 
A: You need to be able to learn independently. You need to be able to develop and asses arguments and solutions without an outside source to correct your work. You will become an outside source to others. This is not an irrelevant skill that you are being forced to do as a mere initiation. This is in fact, the skill you are attempting to master. As an expert, you will without doubt encounter situations for which you have no pre-packaged solution. Often, in order to resolve a problem of this sort, you will need to study new material in order to develop and refine your understanding. This is not a skill that is entirely new. As an undergraduate, and in many ways as a graduate student, your teachers are providing guidance only. They tell you where to look and what is important. They also tell you why things are important but it is only through your own studies and through situations that you encounter that you may finally see the wisdom of the guidance you have received. You dissertation is intended to demonstrate your ability to think independently.
A: Most certainly! Being able to independently learn mathematics (or anything else) is a vital ability any academically oriented (or any other person) needs to acquire. It is certainly required for a PhD and I would also say for a Masters. 
It is my opinion that developing the ability to learn independently is something that needs to be done by the student as early as the first year of series studies (for most people this would be the first year of the Bachelor). A way to do it would be to supplement the dictated textbooks in any given course by books you find on your own in the library or online and reading these books alongside the course. 
A: Yes.  However, it’s not something to be too afraid of; it’s just one of the skills that you should expect/aim to acquire in the course of graduate school.
Different countries and universities have different cultures in this respect (unsurprisingly).  Most (all?) grad programs in the US start with a year or two of taught courses, through which you will gradually transition to more independent study.  For instance, when you are ready to start learning about an area that’s too specialised to warrant a lecture course, you may well be able to take a “private study” course with a faculty member, so they will help guide your reading, recommending what books/papers to read, and discussing them with you as you read them.  Or you may be able to set up a reading group with fellow-students (and/or faculty members), where you read through some book or series of papers together — so then you read mostly on your own, but can discuss and compare notes with colleagues on a regular basis.
Some people are naturally good at learning direct from textbooks; but for the rest of us, things like these are how we learn, bit by bit, how to do it.  I was a comparative latecomer to the skill, and I’m still (as, currently, a post-doc) most at home learning more socially (lectures, discussions with collaborators, etc); but it’s an essential skill to have to at least to some extent, and if it’s something that scares you a little, then it’s worth putting some effort into overcoming that fear.
