I'm a junior student at a competitive university in my home country and I major in a mathematical field. Last year, I went to the (tenured) professor who matched my interest and she said she normally doesn't give students problems, but we should find our own research problems and write a research proposal. We had some high-level talks about her work but she didn't give me any list of papers to read, etc. After several months I found a problem and wrote a proposal with a method to solve it. I submitted it and she said the problem was not very meaningful. She suggested I talk to some of her Ph.D. students, which I did.
The first semester I attended her group meeting but gradually chose not to attend the second semester. At that time I suffered from some mental health problems, like very low self-esteem and severe procrastination. But by reading papers, taking courses, and talking to my peers, I cultivated my own understanding of the field and gained a lot of technical skills.
I eventually submitted another proposal, which she also rejected (said it was common sense), but she suggested that I should work on one of her projects with her Ph.D. student (~1-year after I first contacted her). It is an ambitious, challenging big problem, but she said it's okay if we don't get the ideal result.
I did learn (in a rather painful way) a lot about the research process. My advisor will reply to my email and schedule a meeting with me whenever I want, and we can talk happily about technical detail and general questions. However most of the time I don't have something meaningful to say to her so I choose not to send an email. She never sends emails/messages to her students to ask how they're doing and said it is the student's duty to take care of their own research and plans etc.
She says what she is doing is cultivating our independence. I think perhaps I was not mature enough and there was some miscommunication between us at least in the beginning. I'm in general enjoying the research and considering applying for a decent Ph.D. program. But I am a little concerned about continuing in research after this experience. Is my experience normal?