Dear Physics Forum advisers, I am an undergraduate student in US and an inspiring applied mathematician in the fields of theoretical computer science. I recently have been reading a book called "Joy of Learning" by Hironaka Heisuke, a mathematician and Fields Medalist (1970); the book seems to be published only in Japan and South Korea, so I do not think there is no translated version in English. In the book, Professor Heisuke remarked that whenever he asked a question to mathematics students (undergraduates and graduates), most time he got responses like "Oh, I am studying algebraic geometry", "I am studying abstract algebra", "I am studying analytic number theory", etc. He felt little disappointment since he hoped to hear ideas and theories the students formulated. He said that it would be a great way to study the mathematics by formulating one's own conjecture first, investigating about it more by reading books involving them, and developing that conjecture or changing it if it is not validated. I was intrigued by his remark, and I would like to implement his thought to my current method of studying. My method of studying the mathematics is just reading books in topics of my interest, and formulating interesting questions as I read the books and trying to answer them. Do you think it would be a good idea to come up with my own idea before reading the books? For example, before reading a book in the abstract algebra, should I come up with idea like an existence of patterned sets or an interesting question like applications of groups and fields to cryptography, and try to think about ways I can describe those sets (groups, rings, and fields) before diving into the book? Since I am doing a undergraduate research in the theoretical computer science that involves a lot of mathematics (combinatorics, graph theory, analysis, etc.), perhaps I should try to formulate my own theories from the main project, and try to answer them too? I really apologize for the long post, and look forward to hear back from you!