[jorgenveisdal]

How to study mathematics, according to Niels Henrik Abel's high school teacher: 1. Never study more than one book at a time and never abandon a book you have chosen without working all the way through it 2. If you face difficulties, do not give up but instead go back twenty times if that should prove necessary and only then allow yourself to investigate another mathematician's solution 3. Skip over those parts that are of no challenge in order to get at what is new to you 4. Reflect over the reading, in particularly how the writer came to the solution and moreover, what the solution leads to 5. Investigate whether or not another transformation or substitution would have solved it in a better manner 6. Always read with pen in hand so that you can work out all the calculations and practice all the questions you encounter 7. Write up lists of subjects that afford you an opportunity to develop your own theories 8. Geometric reflections can be a suitable way of strengthening and securing one's judgement

[BeetleB]

> never abandon a book you have chosen without working all the way through it

Extremely bad idea, and I learned it the hard way. A lot (most?) textbooks have more details than you need to know. It's better to find out the most important material in the book, focus on that first, and then go to the next tier of important material in the book, and so on. At some level (perhaps after the first), feel free to try other books without completing the current one.

Unfortunately, it's not easier to know which material in the textbook is important and which are merely detailed examples on your own. You need someone with mastery to tell you that.

> If you face difficulties, do not give up but instead go back twenty times if that should prove necessary and only then allow yourself to investigate another mathematician's solution

Partially agree. Pick an N that is large enough, but not too large. Another mistake I would make is refusing to move on until I've solved the problem. A more practical approach is to give it N times over M hours/days, and then move on and/or look up the solution. You'll learn more this way.

[cgriswald]

Most high schools and high school teachers don’t teach in a way that particularly enables students to take this advice. “These are the rules. This is how the math works. Do the math. Also these are the classes you have to take to be an $X.” A relatively good teacher will engage with a student who is enthusiastic about the material in this way but many won’t: “You’ll learn about that in $future_course” without even an attempt to talk about it in the abstract.

Without some guidance on which book to choose, (1) could condemn a reader to mathematical purgatory. In any case, I’d also say that taking multiple math courses in high school helped me to understand that math is most often about relationships and that there were multiple ways to describe the same relationships. This was in spite of no one bothering to really teach that idea. Previously I was mostly focused on axioms and rules (probably because of how I was taught), which, while useful, missed some important stuff. I also prefer to read through multiple texts because it gives me a break, let’s me see new connections, and if I come back to the text after working on something else and still know what’s up I know I really grok the material. (It’s insanely easy for me to memorize and apply rules; that’s not indicative of understanding.)

(3) is good reading practice in general, but also, it’s a skill most people haven’t developed. It’s very easy to say, “Pfft, I know all this,” and just skip something without actually really grokking the information; it’s too easy to over-estimate understanding. For most people I’d say at least work through any proofs and the most difficult questions in each section. This is actually something I think we should be learning in middle school at the latest.

[joe_the_user]

The advice of one teacher of one moderately known mathematician hardly seems definitive.

The advice implies a completely linear process of learning. I think that works for some portion of learners and isn't useful at all for another portion.

Edit: never abandon a book you have chosen without working all the way through it

I've read a number of math books on my own. There are plenty of bad books out there. Some will have you wasting an awful long time if you never abandon them (since they are filled with extremely difficult problems or simply don't explain themselves well). This is advice for someone who, at minimum, has a teacher feeding them good books.

[jorgenveisdal]

How does Abel’s level of fame 200 years later matter?

[joe_the_user]

The post I responded to above cited the high school math teacher of Abel. As far as I can tell, the authority of the claim rested on Abel indeed having significant mathematical achievements ("A Norwegian high school teacher from from 200 years once said" doesn't sound convincing as claim by itself to me). My response was just pacing the situation - Niels Abel is just one example of someone achieving a lot, not insignificant but not the last word.

[watwut]

Not at all. There is reason no contemporary teachers give this particular advice.

Abel is famous because of his algebra results. He had awful good abstract thinking and was indeed very good at math. That does not only everything he ever said is correct. In particular, it does not imply general teaching skills.

Being good at proving theorems does not imply being good at teaching students.

[watwut]

I think that 1. is very bad idea and so is 2.

Both means that if the book you picked assumes you already knew something you don't really, you are stucked. If your book happen too explain something later or is not perfectly written, you are set up to fail.

Very often, later concept let's you figure out the first one. Very often, another book or source adds details to have been missing. Very often, you need to do something else for a while and when you come back to chapter, it suddenly clicks.