PhD in math here with several published papers. And my recommendation is a metaprinciple: enjoy mathematics. Benjamin Finegold said similarly that the secret to chess is to enjoy every move. Personally, I had no trouble in mathematics, ever. And I think the reason for that is that I really enjoy just doing it, writing symbols down, learning about new theories, and even inventing my own.

Not everyone will enjoy mathematics at first sight. But I think at least 50% of that can be explained by the lack of obvious paths to enjoy mathematics. Obviously, most mathematics taught in high-school is not taught as it should be: a cool artistic logical pursuit that has all kinds of fun in it.

So my advice is to really find a mentor who already has found that path and let them show you how to enjoy math.

Believe me, I've tutored a lot of people, many of which initially disliked math and found it difficult. But after a few tutoring sessions, I could see a little sparkle in their eye that said, "hey, this might be cool".

So before you apply logic, studying, and other tedious "productivity" measures to your math learning, make sure you find a way to enjoy it first.

My secret sauce as an undergraduate for all my math courses was solving problems. Solve all the problems at the end of the chapters in the textbook. Find other textbooks in the library and solve all the problems at the end of their chapters.

In graduate school that was expanded: take every chapter of the textbook and rewrite it, filling in all the intermediate steps of every proof, those where the author writes "it follows that ..." or "from which it's obvious that ..."

The first thing you have to get used to when moving from school to uni is being utterly lost and defeated.

At the end of high school, I could do everything. I finished my IB exams with huge amount of time to spare, the only thing holding me back was being able to write fast enough. It had been months since I saw a regular curriculum question that I didn't know how to do. Any marks I lost were just trivial errors.

When I got to university, there would be question sheets where I would look at the questions and wonder what it had to do with the lectures I had just been in. As in "I went to this lecture, and I'm supposed to use the information to answer these questions, but I don't even know what the questions mean".

The learning happens when you are doing this frustrating head-bashing.

You read, you read more, you fill a notebook with useless derivations, and eventually you things start to take shape. This could take the entire week's worth of time, just sitting there fumbling about.

The difference is that in uni, the amount of material is so vast you cannot explain it to someone in the time that you have. The students have to pick up some key ideas, and then fill in all the details themselves by pouring hours into it on their own.

Related:

How to Study Mathematics (2017) - https://news.ycombinator.com/item?id=26524876 - March 2021 (73 comments)

How to Study Mathematics (2017) - https://news.ycombinator.com/item?id=16392698 - Feb 2018 (148 comments)

Bonuses:

Ask HN: How to Study Mathematics? - https://news.ycombinator.com/item?id=23074249 - May 2020 (31 comments)

Ask HN: How to self-study mathematics from the undergrad through graduate level? - https://news.ycombinator.com/item?id=18939913 - Jan 2019 (227 comments)

Ask HN: How to self-learn math? - https://news.ycombinator.com/item?id=16562173 - March 2018 (211 comments)

Others?

> The Germans have aptly called Sitzfleisch the ability to spend endless hours at a desk, doing gruesome work. Sitzfleisch is considered by mathematicians to be a better gauge of success than any of the attractive definitions of talent with which psychologists regale us from time to time. Stan Ulam, however, was able to get by without any Sitzfleisch whatsoever. After his bout with encephalitis, he came to lean on his unimpaired imagination for his ideas, and on the Sitzfleisch of others for technical support. The beauty of his insights and the promise of his proposals kept him amply supplied with young collaborators, willing to lend (and risking to waste) their time.

Taken from Gian-Carlo Rota in The Lost Cafe, a quote I found here http://www.romanpress.com/Rota/Rota.php