Ask HN: How to Study Mathematics?
31 comments
For my own part, it was all about finding the right book. I enjoyed my degree in pure maths, and after finishing I really missed studying so I tried out dozens of books. Eventually, I hit upon Calculus by Spivak. This book fundamentally changed me and I solved every problem in it fastidiously - it was just so well made, funny in parts, it had cliffhangers, and the problems were just amazing, linking together over chapters like characters in a novel. For the first time, I was solving problems just for fun and I was improving much faster than I did during my degree. I wrote all my solutions in LaTeX so I had an organised body of work to look back on and watch grow, adding to the satisfaction (I lose paper; pdf is better for me).
So my advice: find a great book, make notes and expand on what the author is saying with diagrams or additional lines of working, focus on solving problems, do it out of love.
So my advice: find a great book, make notes and expand on what the author is saying with diagrams or additional lines of working, focus on solving problems, do it out of love.
Spivak was great, I remember it fondly from my university years.
Is there any kind of crowdsourced "canonical" list of great books? Ideally something that does not bring 5 answers per question to the table. For example, what is the Spivak of discrete mathematics? Of statistics? What is Spivak for multivariate calculus?
Is there any kind of crowdsourced "canonical" list of great books? Ideally something that does not bring 5 answers per question to the table. For example, what is the Spivak of discrete mathematics? Of statistics? What is Spivak for multivariate calculus?
I would love to know the answer to your question. On the one hand, it's too subjective to answer, one student's Spivak is another's Apostol; on the other hand, certain books just seem special in an almost objective way and rise above the scores of other books on the subject.
There is this: https://mathblog.com/mathematics-books/ which looks like a pretty well curated list
There is this: https://mathblog.com/mathematics-books/ which looks like a pretty well curated list
This.
For Algebra, look for Topics in Algebra by Herstein. It reads beautifully, and the problems at the end of each chapters are very good at solidifying the knowledge.
This is very nice to know as I have an ancient copy of this book to hand. Thanks!
Maybe some history of mathematics, to give you some perspective on it all? Something I wish I'd known when I did my degree, Norman Wildberger has a great course:
https://www.youtube.com/watch?v=dW8Cy6WrO94&list=PL55C7C8378...
Visual Group Theory, by Nathan Carter is a book you can get something from no matter your level of maths, imho.
If you know some linear algebra, Gilbert Strang's course on YouTube may be interesting - https://ocw.mit.edu/courses/mathematics/18-06-linear-algebra...
https://www.youtube.com/watch?v=dW8Cy6WrO94&list=PL55C7C8378...
Visual Group Theory, by Nathan Carter is a book you can get something from no matter your level of maths, imho.
If you know some linear algebra, Gilbert Strang's course on YouTube may be interesting - https://ocw.mit.edu/courses/mathematics/18-06-linear-algebra...
Currently finishing my junior year of undergrad in math. I'd give different advice to a casual enthusiast than I would a full-time math student, but here are some tips that have helped me:
- Reread everything. The first time, only read definitions and theorems; proceed steadily and focus on high-level ideas. The second time, try to recall definitions and focus on details of theorems. The third time, try to recall definitions and recall proofs of key theorems; work on memorization and fluency. The fourth time, try to modify the conditions of theorems and see what changes. If you are struggling with reading, get as far as you can and then go around again - it gets easier every time. Take as much time as you can afford before reading the same section again to let the material sink in.
- Do lots of problems - start with trivial examples and slowly scale up the difficulty. Lock down the basics before moving on to harder problems. Ask for recommendations for books that have particularly good problems.
- Practice proving theorems and claims. If you have a question while reading, make a conjecture and try to prove it formally. It took me too long to realize that understanding abstract math is not the same as being able to do proofs and formalize intuition. Following along while reading doesn't mean you can quickly or easily prove facts about the material, and the proof is what counts in the end. Knowing why something is true is not helpful if you cannot formalize it.
- Hone intuition. Proving things is just performing a (directed) graph search on a set of facts with a well-defined start and end goal. Having good internal heuristics will help the search proceed quickly, in the right direction, while avoiding dead-ends; the only way to develop this sense is by practice.
- Reread everything. The first time, only read definitions and theorems; proceed steadily and focus on high-level ideas. The second time, try to recall definitions and focus on details of theorems. The third time, try to recall definitions and recall proofs of key theorems; work on memorization and fluency. The fourth time, try to modify the conditions of theorems and see what changes. If you are struggling with reading, get as far as you can and then go around again - it gets easier every time. Take as much time as you can afford before reading the same section again to let the material sink in.
- Do lots of problems - start with trivial examples and slowly scale up the difficulty. Lock down the basics before moving on to harder problems. Ask for recommendations for books that have particularly good problems.
- Practice proving theorems and claims. If you have a question while reading, make a conjecture and try to prove it formally. It took me too long to realize that understanding abstract math is not the same as being able to do proofs and formalize intuition. Following along while reading doesn't mean you can quickly or easily prove facts about the material, and the proof is what counts in the end. Knowing why something is true is not helpful if you cannot formalize it.
- Hone intuition. Proving things is just performing a (directed) graph search on a set of facts with a well-defined start and end goal. Having good internal heuristics will help the search proceed quickly, in the right direction, while avoiding dead-ends; the only way to develop this sense is by practice.
The single most important thing IMHO is to get the basics right
For context, I have a PhD in theoretical computer science and a master in mathematics. I've been teaching mathematics for several years ranging from high school to university.
The biggest problem for students, especially at the university level, is that they don't have the basics down. If you are not fluid in fractions and manipulating equations, you can't focus on the important part of learning induction proofs. This is just one example, but the return on investment on getting the basics down is huge.
For each new subject, you need to take the prerequisites serious. If you can't read a new definition and focus only on the new part, you should go back. The is of course very time consuming if you are missing a lot of prerequisites but if you want to get good, this is the way.
A well planned program will do this for you to some degree but the reason most people struggle is that they don't take prerequisites seriously or simple haven't learned them probably. You should be able to solve exercises in each prerequisite subject to a fairly good degree before tackling the next.
For anybody starting or doing university level mathematics I suggest your start refreshing pre-uni. Make sure you have fractions, linear and quadratic equations, exponentiation and logorithms and basic properties of functions are really down. If you have time, it can be very beneficial for pure mathematics to start out with some formal logic and natural deduction to really understand the way mathematicians reason in proofs and definitions.
Finally, ask all the stupid questions. Nobody, except yourself maybe, care if you misunderstood a basic point and got something wrong because of it. If "you just don't get" something, there is a high chance you missed something. If you don't understand the answer, same thing applies. Mathematics should be, if done properly, straight forward.
For context, I have a PhD in theoretical computer science and a master in mathematics. I've been teaching mathematics for several years ranging from high school to university.
The biggest problem for students, especially at the university level, is that they don't have the basics down. If you are not fluid in fractions and manipulating equations, you can't focus on the important part of learning induction proofs. This is just one example, but the return on investment on getting the basics down is huge.
For each new subject, you need to take the prerequisites serious. If you can't read a new definition and focus only on the new part, you should go back. The is of course very time consuming if you are missing a lot of prerequisites but if you want to get good, this is the way.
A well planned program will do this for you to some degree but the reason most people struggle is that they don't take prerequisites seriously or simple haven't learned them probably. You should be able to solve exercises in each prerequisite subject to a fairly good degree before tackling the next.
For anybody starting or doing university level mathematics I suggest your start refreshing pre-uni. Make sure you have fractions, linear and quadratic equations, exponentiation and logorithms and basic properties of functions are really down. If you have time, it can be very beneficial for pure mathematics to start out with some formal logic and natural deduction to really understand the way mathematicians reason in proofs and definitions.
Finally, ask all the stupid questions. Nobody, except yourself maybe, care if you misunderstood a basic point and got something wrong because of it. If "you just don't get" something, there is a high chance you missed something. If you don't understand the answer, same thing applies. Mathematics should be, if done properly, straight forward.
What online coursework do you recommend for getting the basics / fundamentals down? Is Khan Academy sufficient for this?
Khan Academy is not bad if you like their style of presentation, but as another comment noted, the number of exercises is too few. Unfortunately, I have yet to find a good recourse with focus on exercises, so I tend to create them myself for courses I teach. If you (or any other for that matter) are interested, I can upload a work sheet somewhere.
There properly are tons of sites providing work sheets, but the focus then tends to be on K-12 education and they are often to simple IMO.
For university the importing thing is often manipulating say fractions rather then "calculating" with fractions.
Tangentially, would there be any interest in a service providing this? Pay say 2$ or something and get 50 exercises in the basics with an eye towards university with sample solution? That space is properly swamped but maybe there is a niche? I often think that there is room for textbooks of this short but then again, that market is also swamped with big publishers.
There properly are tons of sites providing work sheets, but the focus then tends to be on K-12 education and they are often to simple IMO.
For university the importing thing is often manipulating say fractions rather then "calculating" with fractions.
Tangentially, would there be any interest in a service providing this? Pay say 2$ or something and get 50 exercises in the basics with an eye towards university with sample solution? That space is properly swamped but maybe there is a niche? I often think that there is room for textbooks of this short but then again, that market is also swamped with big publishers.
I am interested in this. Do you have a website / would you mind posting your contact info?
I think there is absolutely a niche: students (or graduates) who have some experience with undergraduate-level math courses but that need to improve their fundamentals. Most US universities give credit for "Advanced Placement" courses taken in high school, even though these courses are rarely taught to university standards. I skipped the intro calculus sequence at my school, and since the proof-based math courses are usually fairly self-contained, I never got a chance to address the gaps in my fundamentals. I've now found myself in this somewhat awkward position where I can e.g. explain to you what sequential compactness means but can't solve for the roots of a third-degree polynomial.
It doesn't make sense to read a high-school level textbook, and books on competition math are either too hard or the focus is more on exposing you to hard problems rather than teaching fundamentals. There needs to be something which reviews fundamentals in a rigorous way.
I think there is absolutely a niche: students (or graduates) who have some experience with undergraduate-level math courses but that need to improve their fundamentals. Most US universities give credit for "Advanced Placement" courses taken in high school, even though these courses are rarely taught to university standards. I skipped the intro calculus sequence at my school, and since the proof-based math courses are usually fairly self-contained, I never got a chance to address the gaps in my fundamentals. I've now found myself in this somewhat awkward position where I can e.g. explain to you what sequential compactness means but can't solve for the roots of a third-degree polynomial.
It doesn't make sense to read a high-school level textbook, and books on competition math are either too hard or the focus is more on exposing you to hard problems rather than teaching fundamentals. There needs to be something which reviews fundamentals in a rigorous way.
I've added an e-mail to my profile.
It is decent for getting the concepts, but the number of practice questions (about 5 per mini-concept) are far too few to reach the kind of mastery GP is talking about. You probably need to do an order of magnitude more questions to get to the point where you stop having to think when doing these things.
1. Find pens and paper.
2. Find a quiet place.
3. Find a book with lots and lots of exercises suitable for your level. 1/3 easy ones, 1/3 medium ones and 1/3 hard ones could be a good mix. The more motivated you are, the more harder ones you want.
4. Solve exercises.
5. Repeat step 4 forever. Also don't bring any electronic devices with you because they are distracting. If you don't, you'll be much more efficient. And 3-4 hours/day is more than enough to learn whatever you want if you study efficiently.
The standard advice is to get the assigned books (or highly recommended books) for each of the courses in your coming semester and work through them early. You can support them with other resources but you’ll save yourself a lot of time by using them as your main guide - especially for higher level math. The hard choice is how to study from the books. The only general tip I can give you is to not get too caught up in the early chapters. It’s better to do a fast first pass then come back to it later.
if it's something you're self-studying, don't be afraid to put something down and pick it up later. There are topics and materials that I completely failed to understand the first time I encountered them, but months or years later they were easy to understand.
Of course, if your goal is to get good grades in specific courses which have exams on specific dates, then this isn't very useful advice.
Some other advice:
- don't underestimate the value of memorization when it comes to definitions and important theorems.
- do lots of practice problems which are like those on the exams (obviously)
- as much as possible, find peers to study with and talk with. it's easy to fool yourself into thinking you understand something, but when you try to talk about it with a real person, the holes in your knowledge are revealed pretty quickly.
- if you can't do that, try to explain the concepts and ideas you're learning to an imaginary friend.
- you might be able to find lecture notes or slides online from the actual versions of classes taught at your institution.
I agree you should ask the professors for materials to look at, or at least try to focus on the materials and courses actually taught at your institution. This is the path of least resistance and will probably make it easier to stay motivated too.
Of course, if your goal is to get good grades in specific courses which have exams on specific dates, then this isn't very useful advice.
Some other advice:
- don't underestimate the value of memorization when it comes to definitions and important theorems.
- do lots of practice problems which are like those on the exams (obviously)
- as much as possible, find peers to study with and talk with. it's easy to fool yourself into thinking you understand something, but when you try to talk about it with a real person, the holes in your knowledge are revealed pretty quickly.
- if you can't do that, try to explain the concepts and ideas you're learning to an imaginary friend.
- you might be able to find lecture notes or slides online from the actual versions of classes taught at your institution.
I agree you should ask the professors for materials to look at, or at least try to focus on the materials and courses actually taught at your institution. This is the path of least resistance and will probably make it easier to stay motivated too.
oh, also -- the advice to skim early chapters is generally good. In general, don't be afraid to skip past things you don't understand, as long as you have some way of testing that you are learning everything you need in the end (via peers and practice problems). Sometimes there are seemingly-extraneous details that can only make sense after you see how they are used later in the text, and sometimes confusing stuff ends up not being very important.
The end goal is of course, a grade; however I will have to take supplementary exams. Although past papers are available, there is a degree of ‘aptitude testing’ involved which is strictly mathematical but still difficult to prepare for directly. Nevertheless practice problems are good for that and I will be doing a lot.
You are right, the advice to skip/quickly pass the first chapters [thank you lurker137] is good. I think this is something I have known subconsciously but I will try to keep aware. Same for those who have suggested seeking further material.
> as much as possible, find peers to study with and talk with
This is good advice. Thank you.
You are right, the advice to skip/quickly pass the first chapters [thank you lurker137] is good. I think this is something I have known subconsciously but I will try to keep aware. Same for those who have suggested seeking further material.
> as much as possible, find peers to study with and talk with
This is good advice. Thank you.
In terms of aptitude testing, going through some books like "How to Prove It" and working on rigorous proofs-based material (perhaps an introductory analysis book with lots of diagrams) will be helpful. As will brushing up on geometry-lots of people memorize a lot of geometry in middle school/high school, do well in their class, and forget it. Geometric concepts continue to appear throughout math and having a deep grasp of them is super helpful for your first "rigorous" math classes, where the secret to finishing a proof often lies in visualizing or drawing it correctly.
Few high-level advices, in arbitrary order.
1. Don't panic. If you don't understand everything immediately, that's normal. Try second or third time - and even fourth and fifth. Not necessarily immediately - take some time.
2. Try different books - for different people different explanations "click" differently.
3. Math is often understood better a few months after you studied the material. In many courses you don't have time to internalize new concepts in real time - but after some pause you feel better adjusted to them.
4. When you see a math expression, read it aloud, for better understanding. The goal is to force you to actually pay attention to every feature of the math notation used. Even reading this once helps. After reading a few times, you may start to see patterns and understandable pieces of the expression.
5. Scientists sometimes need quite a bit of time to understand what their peers - even in close areas - write in their papers. Reading one paper, even not too long, for the whole week to get a good understanding could be normal.
1. Don't panic. If you don't understand everything immediately, that's normal. Try second or third time - and even fourth and fifth. Not necessarily immediately - take some time.
2. Try different books - for different people different explanations "click" differently.
3. Math is often understood better a few months after you studied the material. In many courses you don't have time to internalize new concepts in real time - but after some pause you feel better adjusted to them.
4. When you see a math expression, read it aloud, for better understanding. The goal is to force you to actually pay attention to every feature of the math notation used. Even reading this once helps. After reading a few times, you may start to see patterns and understandable pieces of the expression.
5. Scientists sometimes need quite a bit of time to understand what their peers - even in close areas - write in their papers. Reading one paper, even not too long, for the whole week to get a good understanding could be normal.
I wrote about my journey learning mathematics for the last ~6 years quite recently actually! Might be of interest to you as well: https://www.neilwithdata.com/mathematics-self-learner
Obtain Terrence Tao's Analysis I book and start from scratch, there's an ongoing workshop for this https://learnaifromscratch.github.io/math.html
However you should probably ask professors at your school what you should be doing in your free time, since they are the one's who will be determining said significant incentive so probably have a better idea what you should be doing than any of us.
However you should probably ask professors at your school what you should be doing in your free time, since they are the one's who will be determining said significant incentive so probably have a better idea what you should be doing than any of us.
Do what the professors tell you: buy (or download for free) the books, do all the practice exercises and proofs, go to office hours, listen to the professor in class and don't write everything they write, just the key stuff, and get together with other students to go over the harder problems.
That'll get you through the courses, and your level of effort and capabilities will determine how much you learn. Forget about grades, if you just do your best you'll get the best grade you can.
That'll get you through the courses, and your level of effort and capabilities will determine how much you learn. Forget about grades, if you just do your best you'll get the best grade you can.
Some practices and advice that I gained from my time doing a PhD in mathematics:
1. Find a time of day to read and work on problems that works for you and make it interruption-free. For me it was up at 6AM and working until 9 or 10AM with no phone or laptop. If you can do longer or fit in another session, go for it.
2. I found that I process new material in the background, so either going to the gym or taking a long walk gave my mind the break and mental space to process what it had just ingested.
3. In mathematics, it's common to present the most terse, stripped-down version of a result which can rob it of the background and context. I found it was often helpful to find the author's previous work, PhD thesis or talks in order to understand how they got to their result.
4. Don't be afraid to get a little side-tracked or obsessed with something that's not directly related to what you're currently working on. Part of training to be a mathematician is learning the ability to hold seemingly unrelated concepts in your head for long periods of time until the connection becomes apparent. Learn to love difficult things!
1. Find a time of day to read and work on problems that works for you and make it interruption-free. For me it was up at 6AM and working until 9 or 10AM with no phone or laptop. If you can do longer or fit in another session, go for it.
2. I found that I process new material in the background, so either going to the gym or taking a long walk gave my mind the break and mental space to process what it had just ingested.
3. In mathematics, it's common to present the most terse, stripped-down version of a result which can rob it of the background and context. I found it was often helpful to find the author's previous work, PhD thesis or talks in order to understand how they got to their result.
4. Don't be afraid to get a little side-tracked or obsessed with something that's not directly related to what you're currently working on. Part of training to be a mathematician is learning the ability to hold seemingly unrelated concepts in your head for long periods of time until the connection becomes apparent. Learn to love difficult things!
Feedback.
You can try to do math on your own, but you'll have a high chance of failing. You need to get feedback.
Just like you can't learn a foreign language without interacting with the speakers of that language, it's very hard to learn math without speaking with people who know math.
So, by all means, find good books, good youtube videos (you can't go wrong with 3blue1brown), Khan academy, all those will help. But if you don't get feedback from people, it's easy to think you make solid progress when you actually don't. It's easy to not know what direction to pursue among hundreds of options.
How do you get feedback? If you can afford it, just pay for it. Nowadays there should be various online services that offer math tutoring. Get in touch with them, and tell them you'd like to go through Calculus 1, or Ordinary Differential Equations, for example. With a competent tutor, you can learn calculus in, let's say, 3 weeks, on your own, you could spend 3 months and still have serious gaps.
Edit: Because of feedback, it's actually easier to get good at programming than at math. In programming, you write programs, if they run and product the result you expect, that's great, if they don't, you realize right away you made a mistake.
So, if you can find a little coding project that forces you to learn math, you can get valuable math feedback by seeing that the program runs as expected. Ray tracing comes to mind. Maybe building an n-body simulator could be a fun exercise too. The phenomenal book "Turtle Geometry" teaches you math in this way, it asks you to write and run (and play with) various programs to produce quite some pretty pictures. You could go through it, and write all those programs in Python rather than Logo (Python has a turtle package embedded). With that book you can really learn a ton of math, including some Riemannian geometry.
You can try to do math on your own, but you'll have a high chance of failing. You need to get feedback.
Just like you can't learn a foreign language without interacting with the speakers of that language, it's very hard to learn math without speaking with people who know math.
So, by all means, find good books, good youtube videos (you can't go wrong with 3blue1brown), Khan academy, all those will help. But if you don't get feedback from people, it's easy to think you make solid progress when you actually don't. It's easy to not know what direction to pursue among hundreds of options.
How do you get feedback? If you can afford it, just pay for it. Nowadays there should be various online services that offer math tutoring. Get in touch with them, and tell them you'd like to go through Calculus 1, or Ordinary Differential Equations, for example. With a competent tutor, you can learn calculus in, let's say, 3 weeks, on your own, you could spend 3 months and still have serious gaps.
Edit: Because of feedback, it's actually easier to get good at programming than at math. In programming, you write programs, if they run and product the result you expect, that's great, if they don't, you realize right away you made a mistake.
So, if you can find a little coding project that forces you to learn math, you can get valuable math feedback by seeing that the program runs as expected. Ray tracing comes to mind. Maybe building an n-body simulator could be a fun exercise too. The phenomenal book "Turtle Geometry" teaches you math in this way, it asks you to write and run (and play with) various programs to produce quite some pretty pictures. You could go through it, and write all those programs in Python rather than Logo (Python has a turtle package embedded). With that book you can really learn a ton of math, including some Riemannian geometry.
A few things that helped me during my studies as a math major:
- get a small-ish whiteboard; one that you can fit in a larger bag. Do problems on that whiteboard first, then work them out on paper for submission once you have either solved the problem or know you are on the right track. It’s way easier to wipe the whiteboard clean to start over than it is to erase your pencil work.
- invest in a quality mechanical pencil and a separate eraser.
-work at least 3 extra problems beyond what was assigned for each category of problem
- don’t memorize anything that you can derive quickly - memorize the derivation. Use your scratch paper to derive whatever you need, when you need it, instead of fretting over memorizing the right formulas, etc.
- read up on the topic of lecture, before lecture, so that you can listen to the meta-information and ask questions during lecture instead of scrambling to follow the topic.
- make a notebook of techniques and tricks of the trade
- definitions are everything.
- get a small-ish whiteboard; one that you can fit in a larger bag. Do problems on that whiteboard first, then work them out on paper for submission once you have either solved the problem or know you are on the right track. It’s way easier to wipe the whiteboard clean to start over than it is to erase your pencil work.
- invest in a quality mechanical pencil and a separate eraser.
-work at least 3 extra problems beyond what was assigned for each category of problem
- don’t memorize anything that you can derive quickly - memorize the derivation. Use your scratch paper to derive whatever you need, when you need it, instead of fretting over memorizing the right formulas, etc.
- read up on the topic of lecture, before lecture, so that you can listen to the meta-information and ask questions during lecture instead of scrambling to follow the topic.
- make a notebook of techniques and tricks of the trade
- definitions are everything.
I recently bought Alex in numberlansld. Quite an interesting book. Also try What is mathematics? The book helped Einstein to grasp math concepts.
Could you tell me details about 'What is mathematics' book ? Which book specifically & Who's the author ?
The authors of "What is Mathematics" are Richard Courant and Herbert Robbins. This one is good as well: "Mathematics: Its Content, Methods and Meaning" by Aleksandrov, Kolmogorov and Lavrent'ev (Dover publications). It's a big book covering a wide amount of topics.
Learn from the countless of people who went before you.
Read the book, A Mind For Numbers by Dr. Barbara Oakley.
This depends super heavily on the math courses you’ve already taken.
My institution gives a very significant incentive to those who excel academically in their early years. I would beat myself up if I did not use the unique opportunity of vast free time this summer to advantage myself in this regard.
My question is essentially for those who have experience studying the subject at university level, or self-studying to an approximately equivalent degree, with emphasis on the pure and problem-solving end over the applied. What strategies, resources, and tips would you advise for learners?
Also, I understand that this type of question is asked fairly regularly. Due to the nature of this forum, much has been said about approaching applications of mathematics for those in other disciplines or for the purpose of self-study. But I think that most would agree that from the outset, pure mathematics and abstract problem skills appear arcane from university-level on.