How long does it takes to fully understand completely a "page" of math or physics? Now, I have a teacher that tell me that there are in general two style of reading: 1) 15 minutes of reading without writing, and the objective is to get the main points, 2) 45- x hrs of reading with writing, where the person write down each and every step( and perhaps fill in missing steps) to a the chain of reasoning in the page. i asked 3 professors, and 2 said one should speed 45 min to x hrs on step 2, and one said it depends on the book. I pressume the two professor that gave a greatest lower bound estimate made the implicite assumption that the book is of averga quality, and difficulty. The second question i asked is " how do we know that we know after reading a section?". The first two professor said that we should do a lot of exercise and problems. the last professor said that one should imagine giving a lecture to a imaginery audience, or to have someone to lecture to. There are two question i want to ask in this thread. Question 1: how much time does it take you to read an average difficulty "page" of math( subjective), and do you read it by writing down the chain of reasoning as you go along? Do you do some light reading before the pencil-paper intensive reading? Corrollary question: Now, it seems that from the responds of my sample of 3 pro, reading a page of math does consume a bit of time, and that is only a single page. The are two 2 factors here. Factor 1) time complexity: you speed alot of time in reading. 2) sustain concentration to read the page, or the whole section. My corr question is how do you sustain you concentration after reading a 45 min page of math? Question 2: How do you know that you know fully the content of a single "section"( not a page) of math? Do you do alot of problems, and if so, how many problems do you do? Do you try to explain it to an "imaginary", or "real" audience? How do you know that you know?