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We can spend a lot of time thinking of material which we have basics for without ever studying the original ideas from a textbook, for example, once one has finished regular derivatives, it is pretty motivated to jump to partial derivatives and other ideas related to it.

Then a person self studying is faced with a choice, either he could think of the material more without opening the book or he could see how it was done before, so, in this dilemma, what is the correct 'choice' to make for maximizing our competence long term?

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    $\begingroup$ What is the objective that you are trying to optimize? Is it the pleasure in learning, or is it the rate of knowledge gained per time, or something else? My thought is that it is entirely dependent on what is in the head of the learner, so that it might be impossible to say a correct choice without a lot of conditions. $\endgroup$
    – Carser
    Feb 17 at 12:48
  • $\begingroup$ The objective is to maximize competence/ become a mathematician who can produce relevant results. As per knowledge vs pleasure of learning.. I guess it depends on if there is a test coming up or not, but even without a test, we need mix of both becuase life is limited $\endgroup$
    – Buraian
    Feb 17 at 17:37
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    $\begingroup$ I think this is an intensely personal question. I know some people who work through texts and papers methodically, turning each page of text into ten or twenty pages of notes. I know others who skim a page of text and then dream about it for a month. There are as many different ways of learning and creating as there are people. I would suggest playing with many styles and seeing what suits you best. $\endgroup$ Feb 17 at 17:53
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My advice (IN GENERAL) is to follow the book VERY explicitly and closely. You still should be doing lots of active things, not just "reading". But use the book as a scaffold. In self instruction, the dangers of flailing around (to include quitting) are way higher than the dangers of missing something book did not cover or not thinking for yourself. So just pick a book and follow it and spend almost all your time there.

  1. Ideally use a "programmed learning" text, such as by Stroud. https://en.wikipedia.org/wiki/Ken_Stroud If not able to find that, try to pick a text that is accessible/pleasant. It should have good problems. Good means including easy drill work not just "prove what we said" or "interesting hard", having answers for checking, significant amount of them. Avoid texts that are overly difficult or that require significant amounts of instruction/tutoring. This includes many of the premier hard books you will see recommended on Q&A sites. The biggest danger in self study is giving up. NOT doing something too easy. Doing a ho-hum program perfectly will get you way more than doing a ballbuster program sporadically. Note, you can always go back and do an extra book or selected sections afterwards. (This is the same principle as in music and sports training. It is a practical pedagogy point, not a math point.)

  2. Work each example, on paper, as you get to it. Try on your own, than look at how book did it, if needed. Then repeat on own, if you had to look.

  3. Work the homework problems after each section. Check your answers. Rework any problems missed. Do the whole missed problem over, writing it out from beginning to end, even if you "only had a dumb mistake".

  4. At the end of your notebook, working backwards, keep a list of questions. This is a parking lot for questions from reading or problems. If the program is appropriately scoped (easy enough), you should definitely be able to keep moving through sections of the book. NOT having to get on the computer and ask a SE question or call a tutor/friend. I.e. keep moving, if at all possible. Finish the section studied in text/problems. BUT have a parking lot for questions (something you didn't understand, something new you wonder, out of scope areas, etc.) Then follow up occasionally to answer the questions. (Leave space in your notebook for the answers. Personally, I find that often as you finish the section or perhaps go to the next one, you will be able to answer your own questions (and then write in your own answer in the space you left). But having this organized list, means you can very efficiently target small amounts of tutor time (or ask on Q&A sites...but tutor more efficient).

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