by ayoisaiah on 12/1/24, 9:50 PM with 103 comments
by vouaobrasil on 12/2/24, 10:56 AM
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.
by litoE on 12/2/24, 1:55 AM
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 ..."
by lordnacho on 12/2/24, 10:40 AM
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.
by dang on 12/2/24, 2:00 AM
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?
by jayhoon on 12/2/24, 12:43 AM
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.
by sourcepluck on 12/2/24, 1:07 PM
Taken from Gian-Carlo Rota in The Lost Cafe, a quote I found here http://www.romanpress.com/Rota/Rota.php
by youoy on 12/2/24, 5:41 AM
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.
by Syzygies on 12/2/24, 5:50 AM
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.
by yzydserd on 12/2/24, 2:07 PM
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.
by 0xRusty on 12/2/24, 3:16 AM
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.
by adamddev1 on 12/2/24, 6:24 PM
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."
by cod1r on 12/2/24, 4:35 AM
by PandaRider on 12/2/24, 12:46 AM
by ChaitanyaSai on 12/2/24, 2:25 AM
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?
by cubefox on 12/2/24, 10:26 AM
by richrichie on 12/2/24, 7:36 AM
by revskill on 12/2/24, 4:01 AM
by Koshkin on 12/2/24, 4:10 PM
- John Von Neumann