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How to Study Mathematics (2017)

264 points by ayoisaiah 2y ago

jayhoon 2y ago
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.
- 609venezia 2y ago
  Possible harmonization of the two ideas: the intuition that we go into math at high school level can help serve us at that level of math. We have some idea of geometry-like objects and 2d-calculus like curves from our everyday life At university level the objects become more abstract, so the intuition we use in normal daily life may no longer apply. New kinds of intuition may develop but it takes work, including lots of time spent with the formal processes and calculations along with reflection on that time, and the active creation of new metaphors to drive the intuition. For example, I still remember a professor using "Ice-9" (from _Cat's Cradle_) as a metaphor for how proving some local property of a holomorphic function on the complex plane made that property true for its global behavior
- tseid 2y ago
  I think he's saying here that intuition is sufficient for high school math, but not sufficient for college. That's not to say that it isn't necessary, only that it isn't sufficient.
- alganet 2y ago
  I read the book. To me, what it says is "intuition can be honed and it is powerful, but hard to pass along to others". Just that. Bessis actually mentions examples of how intuition and technique complement each other nicely.
- chongli 2y ago
  This seems like a contradiction but I don’t think it is. What it’s really saying is that experience is a precondition for intuition. When a high school student looks at a high school math problem they’re drawing on all of their experience in K-12 math to get intuition for how to solve the problem. When they leave high school to study math in undergrad they struggle because their experience is no longer sufficient. They’re faced with a lot more abstract problems and the demands for rigour are much higher. The problems also tend to operate at higher levels on Bloom’s taxonomy [1] than high school math, something with which the average high school student would have little or no experience. It is this unfamiliar territory where intuition is hard to come by. After gaining more experience (later undergrad and into grad school and beyond) the intuition starts to come back. But it’s fundamentally a different kind of intuition. In high school math it was often a visual/geometric intuition that teachers were trying to build. In higher math it’s an intuition for abstractions and for the tools you need to attack problems. This is really no different from a programmer looking at a problem and saying “I need a hash map and then this problem is trivial.” [1] https://en.wikipedia.org/wiki/Bloom's_taxonomy https://en.wikipedia.org/wiki/Bloom's_taxonomy
- Tazerenix 2y ago
  This is related to Terence Tao's notion of the stages of mathematical rigor. As Tao puts it, the value of intuition becomes much higher in the post-rigorous stage once you have sufficiently developed your technical skills. https://terrytao.wordpress.com/career-advice/theres-more-to-mathematics-than-rigour-and-proofs/ https://terrytao.wordpress.com/career-advice/theres-more-to-...
- agumonkey 2y ago
  That topic fascinates me. As a kid, a lot of topics felt intuitive, and college became a pit of darkness. Makes me wonder what in our brains makes something feel natural, obvious, with that feeling of playfulness and certainty .. while some times you're drowning in a blur.
PandaRider 2y ago
I appreciated the article for emphasising memorising definitions and statement of theorems... But not for proofs. For proofs, a general outline would be sufficient.
- xanderlewis 2y ago
  For proofs, I find it a good idea to memorise (or at least implicitly retain) the reason a result is true. So, yes, an outline, but minus any of the implementation details of the proof. I kind of think every book in the definition-theorem-proof style should really be definition-theorem-reason-proof. The reason part being essentially a one or two line natural language summary of ‘why the proof works’ — something that is almost always possible and is enlightening and conducive to efficient memorisation, but that for some reason is very rarely written down explicitly.
  - jll29 2y ago
    > definition-theorem-reason-proof Along the lines of your own argument: even better would be reason0-definition-reason1-theorem-reason2-proof
  - gowld 2y ago
    I think a better word is "motivation" -- why we chose this option at this juncture instead of many other options. Yes, it's a "reason", but "reason" already means something else. The "Reason" as result is true is that it follows from the previously established axioms via logical reasoning.
    - xanderlewis 2y ago
      Motivation is important too, but it’s not what I meant. A very simple example would be Theorem: Every subspace Y of a second-countable topological space X is second-countable. Reason: Intersecting each set in a basis for X with Y yields a basis for Y. Proof: [formal symbolic stuff involving open sets and unions, and mentioning cardinality, etc.] (I’m not claiming ‘reason’ is the best word for this — it probably isn’t. But it’s not the same thing as motivation.) > The "Reason" as result is true is that it follows from the previously established axioms via logical reasoning. One could argue this is not the reason a result is true; it’s the reason we know it’s true. The fact that true statements follow from established truths by logical reasoning is more a property of the formal system (which hopefully is sound and consistent) than it is to do with the notion of truth itself.
litoE 2y ago
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 ..."
- processunknown 2y ago
  “The proof is left as an exercise …”
- InkCanon 2y ago
  I did this but faced enormous problems with transfer/recalling definitions/etc. what helped me improve several fold in approximately exponential value 1) Drill/spaced repetition basic definitions. The Cornell note taking method is convenient to do this while taking notes. 2) Keep a diary of thoughts, things you couldn't solve or did solve. Especially identifying problems, what works or doesn't, why something went wrong. Metacognitive thinking was really useful for transferring problems to solving new ones. 3) A study group involving a lot of us explaining to each other.
- raincole 2y ago
  My issue during my undergraduate days was that the textbook didn't come with solutions, and I really don't know how to unstuck myself when I got stuck (except asking the TA in the office hours).
  - abdullahkhalids 2y ago
    Ask your peers. As an ex-prof, one of the most prominent sources of missed potential I see in undergrads is not working with their peers. Like serious 2-6 hours sessions of collaborative problem solving.
    - deskr 2y ago
dang 2y ago
Related: How to Study Mathematics (2017) - https://news.ycombinator.com/item?id=26524876 https://news.ycombinator.com/item?id=26524876 - March 2021 (73 comments) How to Study Mathematics (2017) - https://news.ycombinator.com/item?id=16392698 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 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 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 https://news.ycombinator.com/item?id=16562173 - March 2018 (211 comments) Others?
- nioj 2y ago
  Another bonus: Ask HN: How to self-learn math? - https://news.ycombinator.com/item?id=16562173 https://news.ycombinator.com/item?id=16562173 - March 2018 (211 comments)
  - dang 2y ago
    Thanks! Added above.
ChaitanyaSai 2y ago
>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?
- wisty 2y ago
  The author also said to internalise it, but the simple fact is that in a proof-y math test you can't just say "well it should converge if everything is well behaved", the edge cases matter and definitions set the boundaries.
- vector_spaces 2y ago
  The reasoning is that, similar to memorization of times tables, being able to recall a definition or a theorem and its context / assumptions "automatically" without needing to use your brain frees you to worry about higher order activities. Being able to apply a theorem over and over again ultimately builds mastery and internalization. Counterintuitively, mathematicians like being "brain-off" as much as possible -- you want to be able to read a phrase like "closed convex subset of a Hilbert space" and effortlessly think to yourself "oh! there's a unique norm minimizer" -- if you have to piece that together from scratch every time, you're going to have a hard time -- reading papers and learning new fields becomes a dreadful slog, similar to how math in general becomes a slog for kids who don't memorize their times tables.
- kmill 2y ago
  The author really does mean memorize. To engage with pure mathematics, you must know the definitions, since the definitions are the bedrock of the subject. If you don't know the axioms of a topology, how can you check for yourself whether something forms a topological space? Or without knowing the exact definition of continuous, how can you know whether a proof of continuity is correct? Without knowing the definitions, you can't really know mathematics. To be clear, this does not mean memorizing all the theorems. Getting to know the theorems (and solving problems) is what helps you internalize the subject. Math is the art of what's certain, and knowing exactly what the objects of the subject are is necessary for that. Theorems are derived from the definitions, but definitions can't be derived. In my experience with a math (undergrad and PhD), I realized I had to know definitions to feel competent at all. In my teaching, it's hard to convince students to actually memorize any definitions — so many times students carry around misconceptions (like that "linearly independent" just means that no vector is a scale multiple of any other vector), but if they just had it memorized, they might realize that the misconception doesn't hold up. Math is weird in that the definitions are actually the exact truth (by definition! tautologically so), so it does take some time to get used to the fact that they're essential.
0xRusty 2y ago
> 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.
- jll29 2y ago
  > "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" This is good advice if the objective is to do well regarding grade results. If you want to get down to the bottom of things, to understand everything and to solve fundamental questions of science, you might well want to invert the advice: If you liked biology, study chemistry, for the processes of life are (electro-)chemical processes. If you liked chemistry, study physics, for the processes of molecules and atoms, their formation and reactions, are physical processes. ...
  - jgwil2 2y ago
    The way I heard this phrased back in the day was, "Biology is really chemistry, chemistry is really physics, physics is really math." Never heard psychology or philosophy added to the chain.
- vector_spaces 2y ago
  I agree that the system is garbage. School ruins math -- but I just want to share that nothing stops you from revisiting math later as a fully actualized adult, without the bullshit time constraints and grade pressure. Math isn't well suited to time pressure -- in my view math is about spending lots of time with ideas and playing with them and taking risks and making dumb guesses, it's moonshot through and through. At its core is play, and time pressure doesn't serve anyone well. Anyway, if you were interested in it, you can always revisit it, and even try taking a class again if you ever want to. There are lots of great books out there for self-learners, and lots of communities of folks learning together
revskill 2y ago
I consider college math is all about abstraction.
- jyounker 2y ago
  It's about relationships. Abstraction is the tool for understanding those relationships.
  - Koshkin 2y ago
    That is why we have category theory and homological algebra. https://en.wikipedia.org/wiki/Abstract_nonsense https://en.wikipedia.org/wiki/Abstract_nonsense
TibbityFlanders 2y ago
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cod1r 2y ago
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.
- selimthegrim 2y ago
  UH is famous at least in the high temperature superconductivity world.
youoy 2y ago
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.
- fastasucan 2y ago
  >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. I try to do this as well. If you combine this with understanding the definitions of the various units you more or less will have the textbook in your head, some assembly required.
Syzygies 2y ago
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.
richrichie 2y ago
The biggest obstacle for me was (and is) fear or failure when it comes to solving exercises.
cubefox 2y ago
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.
lordnacho 2y ago
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.
- funcDropShadow 2y ago
  > The first thing you have to get used to when moving from school to uni is being utterly lost and defeated. Very well said, at the university where I studied, there was a pre-semester math repetition course. It was a week long and started with addition of natural numbers. After two days, everything I learned in 13 years of school had been repeated. That was a brutal resetting of expectations. But it made everybody clear, that this is not school anymore. That a different kind of work and focus would be necessary.
vouaobrasil 2y ago
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.
- gcj 2y ago
  I learned math from a lady who would shout all the time, tell us we would amount to nothing and would throw a wooden chalkboard eraser at us. How the hell can I teach myself to enjoy math?
- sourcepluck 2y ago
  Any tips or anecdotes for us (well, me) about the fun things you'd do with students to encourage that joy? I am not a PhD, but have done a fair bit of tutoring young people in maths. I feel similarly to you there, and am always on the lookout for new ways to foster that feeling of mathematics being fun and wonderful. It can be hard. The feeling a lot of young people pick up - of maths being roughly akin to pointless abject suffering - is so strongly rooted in young people sometimes, and can be strongly connected to feelings of inadequacy and shame and so on.
  - sourcepluck 2y ago
    Wrote that reply then clicked on your page to see if there were a blog or anything where you elaborate already on these topics - I've already read a couple of articles from your substack! Oops, hah, I did not realise I was writing to the person behind that. I'd filed it away in my head as something to sink more time into when I've the emotional space to do so, but essentially I find it quite courageous to go against the grain in a world where it's perhaps harder and harder to do so. Everyone has a hot take, of course, but the hot takes are usually very much within the bounds of acceptable discourse. Anyway, more power to you, in your endeavours.
sourcepluck 2y ago
> 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 http://www.romanpress.com/Rota/Rota.php
- deskr 2y ago
  Article on Sitzfleisch: https://www.bbc.com/worklife/article/20180903-to-have-sitzfleisch---its-a-professional-compliment https://www.bbc.com/worklife/article/20180903-to-have-sitzfl...
yzydserd 2y ago
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.
- gmays 2y ago
  I've been using Math Academy daily for over a year now and have been similarly impressed with how much I'm learning with it. Congrats on your pace, 9k XP in 9 weeks is impressive!
Koshkin 2y ago
"Young man, in mathematics you don't understand things. You just get used to them." - John Von Neumann
adamddev1 2y ago
> "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."

264 points by ayoisaiah 2y ago

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.

609venezia 2y ago
Possible harmonization of the two ideas: the intuition that we go into math at high school level can help serve us at that level of math. We have some idea of geometry-like objects and 2d-calculus like curves from our everyday life At university level the objects become more abstract, so the intuition we use in normal daily life may no longer apply. New kinds of intuition may develop but it takes work, including lots of time spent with the formal processes and calculations along with reflection on that time, and the active creation of new metaphors to drive the intuition. For example, I still remember a professor using "Ice-9" (from _Cat's Cradle_) as a metaphor for how proving some local property of a holomorphic function on the complex plane made that property true for its global behavior
tseid 2y ago
I think he's saying here that intuition is sufficient for high school math, but not sufficient for college. That's not to say that it isn't necessary, only that it isn't sufficient.
alganet 2y ago
I read the book. To me, what it says is "intuition can be honed and it is powerful, but hard to pass along to others". Just that. Bessis actually mentions examples of how intuition and technique complement each other nicely.
chongli 2y ago
This seems like a contradiction but I don’t think it is. What it’s really saying is that experience is a precondition for intuition. When a high school student looks at a high school math problem they’re drawing on all of their experience in K-12 math to get intuition for how to solve the problem. When they leave high school to study math in undergrad they struggle because their experience is no longer sufficient. They’re faced with a lot more abstract problems and the demands for rigour are much higher. The problems also tend to operate at higher levels on Bloom’s taxonomy [1] than high school math, something with which the average high school student would have little or no experience. It is this unfamiliar territory where intuition is hard to come by. After gaining more experience (later undergrad and into grad school and beyond) the intuition starts to come back. But it’s fundamentally a different kind of intuition. In high school math it was often a visual/geometric intuition that teachers were trying to build. In higher math it’s an intuition for abstractions and for the tools you need to attack problems. This is really no different from a programmer looking at a problem and saying “I need a hash map and then this problem is trivial.” [1] https://en.wikipedia.org/wiki/Bloom's_taxonomy https://en.wikipedia.org/wiki/Bloom's_taxonomy
Tazerenix 2y ago
This is related to Terence Tao's notion of the stages of mathematical rigor. As Tao puts it, the value of intuition becomes much higher in the post-rigorous stage once you have sufficiently developed your technical skills. https://terrytao.wordpress.com/career-advice/theres-more-to-mathematics-than-rigour-and-proofs/ https://terrytao.wordpress.com/career-advice/theres-more-to-...
agumonkey 2y ago
That topic fascinates me. As a kid, a lot of topics felt intuitive, and college became a pit of darkness. Makes me wonder what in our brains makes something feel natural, obvious, with that feeling of playfulness and certainty .. while some times you're drowning in a blur.

PandaRider 2y ago

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

xanderlewis 2y ago
For proofs, I find it a good idea to memorise (or at least implicitly retain) the reason a result is true. So, yes, an outline, but minus any of the implementation details of the proof. I kind of think every book in the definition-theorem-proof style should really be definition-theorem-reason-proof. The reason part being essentially a one or two line natural language summary of ‘why the proof works’ — something that is almost always possible and is enlightening and conducive to efficient memorisation, but that for some reason is very rarely written down explicitly.
- jll29 2y ago
  > definition-theorem-reason-proof Along the lines of your own argument: even better would be reason0-definition-reason1-theorem-reason2-proof
- gowld 2y ago
  I think a better word is "motivation" -- why we chose this option at this juncture instead of many other options. Yes, it's a "reason", but "reason" already means something else. The "Reason" as result is true is that it follows from the previously established axioms via logical reasoning.
  - xanderlewis 2y ago
    Motivation is important too, but it’s not what I meant. A very simple example would be Theorem: Every subspace Y of a second-countable topological space X is second-countable. Reason: Intersecting each set in a basis for X with Y yields a basis for Y. Proof: [formal symbolic stuff involving open sets and unions, and mentioning cardinality, etc.] (I’m not claiming ‘reason’ is the best word for this — it probably isn’t. But it’s not the same thing as motivation.) > The "Reason" as result is true is that it follows from the previously established axioms via logical reasoning. One could argue this is not the reason a result is true; it’s the reason we know it’s true. The fact that true statements follow from established truths by logical reasoning is more a property of the formal system (which hopefully is sound and consistent) than it is to do with the notion of truth itself.

litoE 2y ago

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 ..."

processunknown 2y ago
“The proof is left as an exercise …”
InkCanon 2y ago
I did this but faced enormous problems with transfer/recalling definitions/etc. what helped me improve several fold in approximately exponential value 1) Drill/spaced repetition basic definitions. The Cornell note taking method is convenient to do this while taking notes. 2) Keep a diary of thoughts, things you couldn't solve or did solve. Especially identifying problems, what works or doesn't, why something went wrong. Metacognitive thinking was really useful for transferring problems to solving new ones. 3) A study group involving a lot of us explaining to each other.
raincole 2y ago
My issue during my undergraduate days was that the textbook didn't come with solutions, and I really don't know how to unstuck myself when I got stuck (except asking the TA in the office hours).
- abdullahkhalids 2y ago
  Ask your peers. As an ex-prof, one of the most prominent sources of missed potential I see in undergrads is not working with their peers. Like serious 2-6 hours sessions of collaborative problem solving.
  - deskr 2y ago

dang 2y ago

Related: How to Study Mathematics (2017) - https://news.ycombinator.com/item?id=26524876 https://news.ycombinator.com/item?id=26524876 - March 2021 (73 comments) How to Study Mathematics (2017) - https://news.ycombinator.com/item?id=16392698 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 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 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 https://news.ycombinator.com/item?id=16562173 - March 2018 (211 comments) Others?

nioj 2y ago
Another bonus: Ask HN: How to self-learn math? - https://news.ycombinator.com/item?id=16562173 https://news.ycombinator.com/item?id=16562173 - March 2018 (211 comments)
- dang 2y ago
  Thanks! Added above.

ChaitanyaSai 2y ago

>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?

wisty 2y ago
The author also said to internalise it, but the simple fact is that in a proof-y math test you can't just say "well it should converge if everything is well behaved", the edge cases matter and definitions set the boundaries.
vector_spaces 2y ago
The reasoning is that, similar to memorization of times tables, being able to recall a definition or a theorem and its context / assumptions "automatically" without needing to use your brain frees you to worry about higher order activities. Being able to apply a theorem over and over again ultimately builds mastery and internalization. Counterintuitively, mathematicians like being "brain-off" as much as possible -- you want to be able to read a phrase like "closed convex subset of a Hilbert space" and effortlessly think to yourself "oh! there's a unique norm minimizer" -- if you have to piece that together from scratch every time, you're going to have a hard time -- reading papers and learning new fields becomes a dreadful slog, similar to how math in general becomes a slog for kids who don't memorize their times tables.
kmill 2y ago
The author really does mean memorize. To engage with pure mathematics, you must know the definitions, since the definitions are the bedrock of the subject. If you don't know the axioms of a topology, how can you check for yourself whether something forms a topological space? Or without knowing the exact definition of continuous, how can you know whether a proof of continuity is correct? Without knowing the definitions, you can't really know mathematics. To be clear, this does not mean memorizing all the theorems. Getting to know the theorems (and solving problems) is what helps you internalize the subject. Math is the art of what's certain, and knowing exactly what the objects of the subject are is necessary for that. Theorems are derived from the definitions, but definitions can't be derived. In my experience with a math (undergrad and PhD), I realized I had to know definitions to feel competent at all. In my teaching, it's hard to convince students to actually memorize any definitions — so many times students carry around misconceptions (like that "linearly independent" just means that no vector is a scale multiple of any other vector), but if they just had it memorized, they might realize that the misconception doesn't hold up. Math is weird in that the definitions are actually the exact truth (by definition! tautologically so), so it does take some time to get used to the fact that they're essential.

0xRusty 2y ago

> 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.

jll29 2y ago
> "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" This is good advice if the objective is to do well regarding grade results. If you want to get down to the bottom of things, to understand everything and to solve fundamental questions of science, you might well want to invert the advice: If you liked biology, study chemistry, for the processes of life are (electro-)chemical processes. If you liked chemistry, study physics, for the processes of molecules and atoms, their formation and reactions, are physical processes. ...
- jgwil2 2y ago
  The way I heard this phrased back in the day was, "Biology is really chemistry, chemistry is really physics, physics is really math." Never heard psychology or philosophy added to the chain.
vector_spaces 2y ago
I agree that the system is garbage. School ruins math -- but I just want to share that nothing stops you from revisiting math later as a fully actualized adult, without the bullshit time constraints and grade pressure. Math isn't well suited to time pressure -- in my view math is about spending lots of time with ideas and playing with them and taking risks and making dumb guesses, it's moonshot through and through. At its core is play, and time pressure doesn't serve anyone well. Anyway, if you were interested in it, you can always revisit it, and even try taking a class again if you ever want to. There are lots of great books out there for self-learners, and lots of communities of folks learning together

revskill 2y ago

I consider college math is all about abstraction.

jyounker 2y ago
It's about relationships. Abstraction is the tool for understanding those relationships.
- Koshkin 2y ago
  That is why we have category theory and homological algebra. https://en.wikipedia.org/wiki/Abstract_nonsense https://en.wikipedia.org/wiki/Abstract_nonsense

TibbityFlanders 2y ago

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cod1r 2y ago

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.

selimthegrim 2y ago
UH is famous at least in the high temperature superconductivity world.

youoy 2y ago

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.

fastasucan 2y ago
>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. I try to do this as well. If you combine this with understanding the definitions of the various units you more or less will have the textbook in your head, some assembly required.

Syzygies 2y ago

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.

richrichie 2y ago

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

cubefox 2y ago

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.

lordnacho 2y ago

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.

funcDropShadow 2y ago
> The first thing you have to get used to when moving from school to uni is being utterly lost and defeated. Very well said, at the university where I studied, there was a pre-semester math repetition course. It was a week long and started with addition of natural numbers. After two days, everything I learned in 13 years of school had been repeated. That was a brutal resetting of expectations. But it made everybody clear, that this is not school anymore. That a different kind of work and focus would be necessary.

vouaobrasil 2y ago

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.

gcj 2y ago
I learned math from a lady who would shout all the time, tell us we would amount to nothing and would throw a wooden chalkboard eraser at us. How the hell can I teach myself to enjoy math?
sourcepluck 2y ago
Any tips or anecdotes for us (well, me) about the fun things you'd do with students to encourage that joy? I am not a PhD, but have done a fair bit of tutoring young people in maths. I feel similarly to you there, and am always on the lookout for new ways to foster that feeling of mathematics being fun and wonderful. It can be hard. The feeling a lot of young people pick up - of maths being roughly akin to pointless abject suffering - is so strongly rooted in young people sometimes, and can be strongly connected to feelings of inadequacy and shame and so on.
- sourcepluck 2y ago
  Wrote that reply then clicked on your page to see if there were a blog or anything where you elaborate already on these topics - I've already read a couple of articles from your substack! Oops, hah, I did not realise I was writing to the person behind that. I'd filed it away in my head as something to sink more time into when I've the emotional space to do so, but essentially I find it quite courageous to go against the grain in a world where it's perhaps harder and harder to do so. Everyone has a hot take, of course, but the hot takes are usually very much within the bounds of acceptable discourse. Anyway, more power to you, in your endeavours.

sourcepluck 2y ago

> 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 http://www.romanpress.com/Rota/Rota.php

deskr 2y ago
Article on Sitzfleisch: https://www.bbc.com/worklife/article/20180903-to-have-sitzfleisch---its-a-professional-compliment https://www.bbc.com/worklife/article/20180903-to-have-sitzfl...

yzydserd 2y ago

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.

gmays 2y ago
I've been using Math Academy daily for over a year now and have been similarly impressed with how much I'm learning with it. Congrats on your pace, 9k XP in 9 weeks is impressive!

Koshkin 2y ago

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

adamddev1 2y ago

> "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."