How to Study Mathematics

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?

Interestingly, this guide states that the intuitive understanding of maths is only suitable at the school level but not for the university.

In his recently published book "Mathematica: A Secret World of Intuition and Curiosity", David Bessis argues that the intuition is the "secret" of understanding maths at all levels.

Not sure what conclusion to draw from here, but my (rather dated) experience with university maths tells me that the intuition is a powerful tool in developing the understanding of the subject.

> 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

In my case I realized I achieved a whole new level when I tried to do all of the proofs by myself. I used to read the whole subject a few times and after that, I would go one definition/proof at a time and first to recall the statement and then try to do the proof by myself. By doing that you indirectly achieve all of what the article says.

A nice thing I realised is that once I did that, almost all of the exercises that were complex before for me, turned out to be straightforward. It was like a cheat code where I almost did not need to do any exercises.

I used to teach at the uni at several levels, and every year I would ask if anyone tried to recall the proofs of the theorems at home. and no one did. They were always shocked when I told then they should do it.

This article is missing the "meta" in studying mathematics.

Creative introspection into how one learns begins to really pay off partway through college.

One's relationship to convention becomes as important as one's relationship to technique. Understanding the "whole" of something involves understanding the biases that shape the presentation you're seeing. You'll probably want to shed them.

This applies whether one wants to change math or just learn it. A passive stance, trying to do what others want, is a recipe for frustration.

I’ve been very pleasantly surprised by my recent experience, having signed up for MathAcademy.com after reading about it on HN.

Now in my 50’s, I wanted to relearn high school maths from 35 years ago and I scooted through their Foundations series (now half way through Foundations 3, rapidly accruing like 9000 xp in 9 weeks, 2 hours a day). Planning in 2025 to do 1-3 university level courses with them at a slower pace.

It’s suited my way of learning as an autodidact: enjoyable; measurable; adaptable level of hardness; no hitting of a “wall” or “unmet dependencies”; thorough explanation of problems I didn’t solve.

Perhaps my biggest realisation was that one can learn without needing to document many notes to revise/memorise, because experience and spaced repetition suffices. I’m taking a Xmas hiatus which will be the real test of baked learning.

> Do not let yourself fall behind.

That hit home. I'm afraid I was one of those lazy math undergrads who struggled with a few of the first year topics, didn't get help or put the hours in and never really recovered. I will maintain I think the teaching was very poor in places (lots of "just trust me" handwashing and "this is obvious so I'll leave it to you to complete" which for an 18 year old frankly sucks). A system that lets you get 30% in "analysis 1" and then just marches you straight into "analysis 2" next semester and expects you to just pull your socks up isn't much of a system to me. Honestly I'm afraid my time at university doing maths was miserable. I should have done something more applied like engineering or CS probably.

Someone once told me "If you like biology at school, do psychology at university. If you like chemistry, do biology. If you like physics, do chemistry. If you like maths, do physics and if you like philosophy, do maths". I should have listened.

> "A good way of seeing how a subject works is to examine the proof of a major result and see what previous results were used in it. Then trace these results back to earlier results used to prove them. Eventually you will work your way back to definitions"

I find the parallels between proofs and programs to be fascinating. We could write an analogous thing for programming:

"A good way of seeing how a sort of program works is to examine one of the popular programs/libraries and see what functions were used in it. Then look inside of those functions and see what functions are used inside of those. Eventually you will work your way back to the lower level primitives."

Sometimes it is weird to see a webpage from my university on here. Usually UH doesn't get a lot of attention for its STEM. At least from my anecdotal experience.

I appreciated the article for emphasising memorising definitions and statement of theorems... But not for proofs. For proofs, a general outline would be sufficient.

>Step 3. Memorize the exact wording of the definition.

Huh. Any mathematicians who want share their own opinions and experiences about this?

This pretty much goes completely against my experience with other grad school level neuroscience/ML

You don't want to be so familiar with stuff as to make it second nature but NOT from memorization. That, at least an other areas, leads to surface level recognition

Does the author mean internalize and not memorize?

Does someone have experiences with using o1-preview for mathematics? A while ago I tried to use GPT-4o and Claude Sonnet for certain algebraic questions related to probability theory. The models did help significantly (at least relative to my rather limited ability), though they also often produced wrong results and struggled to make progress on harder questions.

The biggest obstacle for me was (and is) fear or failure when it comes to solving exercises.

I consider college math is all about abstraction.

"Young man, in mathematics you don't understand things. You just get used to them."

- John Von Neumann